VaKeR CYBER ARMY
Apache/2.4.41 (Ubuntu) Linux absol.cf 5.4.0-198-generic #218-Ubuntu SMP Fri Sep 27 20:18:53 UTC 2024 x86_64 www-data ( 33 )7.4.33 pcntl_alarm,pcntl_fork,pcntl_waitpid,pcntl_wait,pcntl_wifexited,pcntl_wifstopped,pcntl_wifsignaled,pcntl_wifcontinued,pcntl_wexitstatus,pcntl_wtermsig,pcntl_wstopsig,pcntl_signal,pcntl_signal_get_handler,pcntl_signal_dispatch,pcntl_get_last_error,pcntl_strerror,pcntl_sigprocmask,pcntl_sigwaitinfo,pcntl_sigtimedwait,pcntl_exec,pcntl_getpriority,pcntl_setpriority,pcntl_async_signals,pcntl_unshare, / usr /local /lib /python3.6 /dist-packages /sympy /integrals /rubi /rules /
Current File : //usr/local/lib/python3.6/dist-packages/sympy/integrals/rubi/rules/binomial_products.py """
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
"""
from sympy.external import import_module
matchpy = import_module("matchpy")
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii, Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def binomial_products():
from sympy.integrals.rubi.constraints import cons461, cons3, cons4, cons5, cons462, cons2, cons463, cons56, cons464, cons89, cons465, cons40, cons466, cons150, cons13, cons165, cons467, cons468, cons45, cons450, cons69, cons139, cons469, cons470, cons471, cons472, cons473, cons474, cons475, cons476, cons477, cons478, cons479, cons480, cons481, cons482, cons483, cons484, cons485, cons486, cons107, cons487, cons488, cons489, cons490, cons198, cons491, cons130, cons359, cons492, cons493, cons494, cons495, cons70, cons71, cons57, cons496, cons59, cons60, cons61, cons62, cons497, cons498, cons499, cons500, cons149, cons8, cons19, cons501, cons502, cons503, cons21, cons504, cons505, cons68, cons506, cons507, cons508, cons509, cons20, cons246, cons96, cons510, cons511, cons512, cons513, cons514, cons515, cons516, cons517, cons518, cons519, cons520, cons521, cons522, cons523, cons64, cons524, cons525, cons526, cons527, cons528, cons529, cons530, cons531, cons33, cons532, cons533, cons534, cons535, cons536, cons537, cons538, cons369, cons539, cons540, cons541, cons542, cons358, cons543, cons25, cons544, cons545, cons546, cons547, cons548, cons549, cons550, cons551, cons552, cons553, cons554, cons555, cons556, cons73, cons557, cons29, cons222, cons52, cons558, cons87, cons559, cons397, cons405, cons65, cons560, cons561, cons562, cons563, cons564, cons565, cons566, cons567, cons568, cons569, cons570, cons571, cons72, cons572, cons573, cons574, cons575, cons404, cons576, cons577, cons578, cons407, cons579, cons580, cons581, cons582, cons583, cons179, cons584, cons585, cons119, cons586, cons587, cons588, cons589, cons388, cons590, cons591, cons592, cons593, cons50, cons55, cons594, cons595, cons596, cons597, cons598, cons95, cons599, cons600, cons601, cons602, cons603, cons604, cons605, cons606, cons90, cons607, cons608, cons609, cons610, cons611, cons612, cons613, cons614, cons615, cons616, cons617, cons618, cons619, cons620, cons621, cons622, cons623, cons624, cons625, cons626, cons627, cons628, cons629, cons48, cons630, cons127, cons631, cons632, cons633, cons155, cons634, cons635, cons178, cons636, cons637, cons638, cons639, cons640, cons180, cons641, cons642, cons398, cons643, cons54, cons644, cons645, cons646, cons647, cons648, cons649, cons650, cons651, cons652, cons653, cons654, cons655, cons656, cons657, cons658, cons210, cons659, cons660, cons661, cons662, cons663, cons382, cons664, cons665
pattern692 = Pattern(Integral((x_**n_*WC('b', S(1)))**p_, x_), cons3, cons4, cons5, cons461)
rule692 = ReplacementRule(pattern692, replacement692)
pattern693 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons4, cons5, cons462)
rule693 = ReplacementRule(pattern693, replacement693)
pattern694 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons4, cons5, cons463, cons56)
rule694 = ReplacementRule(pattern694, replacement694)
pattern695 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons4, cons464)
rule695 = ReplacementRule(pattern695, replacement695)
pattern696 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons89, cons465, cons40)
rule696 = ReplacementRule(pattern696, replacement696)
pattern697 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons466)
rule697 = ReplacementRule(pattern697, replacement697)
pattern698 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons150, cons13, cons165, cons467)
rule698 = ReplacementRule(pattern698, replacement698)
pattern699 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-5)/4), x_), cons2, cons3, cons468, cons45)
rule699 = ReplacementRule(pattern699, replacement699)
pattern700 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-5)/4), x_), cons2, cons3, cons468, cons450)
rule700 = ReplacementRule(pattern700, replacement700)
pattern701 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-7)/6), x_), cons2, cons3, cons69)
rule701 = ReplacementRule(pattern701, replacement701)
pattern702 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons150, cons13, cons139, cons467)
rule702 = ReplacementRule(pattern702, replacement702)
pattern703 = Pattern(Integral(S(1)/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons69)
rule703 = ReplacementRule(pattern703, replacement703)
pattern704 = Pattern(Integral(S(1)/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons469, cons470)
rule704 = ReplacementRule(pattern704, With704)
pattern705 = Pattern(Integral(S(1)/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons469, cons471)
rule705 = ReplacementRule(pattern705, With705)
pattern706 = Pattern(Integral(S(1)/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons470, cons472)
rule706 = ReplacementRule(pattern706, replacement706)
pattern707 = Pattern(Integral(S(1)/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons470, cons473)
rule707 = ReplacementRule(pattern707, replacement707)
pattern708 = Pattern(Integral(S(1)/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons470)
rule708 = ReplacementRule(pattern708, replacement708)
pattern709 = Pattern(Integral(S(1)/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons471, cons474)
rule709 = ReplacementRule(pattern709, replacement709)
pattern710 = Pattern(Integral(S(1)/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons471, cons475)
rule710 = ReplacementRule(pattern710, replacement710)
pattern711 = Pattern(Integral(S(1)/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons471)
rule711 = ReplacementRule(pattern711, replacement711)
pattern712 = Pattern(Integral(S(1)/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons476, cons470)
rule712 = ReplacementRule(pattern712, With712)
pattern713 = Pattern(Integral(S(1)/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons476, cons471)
rule713 = ReplacementRule(pattern713, With713)
pattern714 = Pattern(Integral(S(1)/(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons477)
rule714 = ReplacementRule(pattern714, With714)
pattern715 = Pattern(Integral(S(1)/(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons478)
rule715 = ReplacementRule(pattern715, With715)
pattern716 = Pattern(Integral(S(1)/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons479, cons480)
rule716 = ReplacementRule(pattern716, With716)
pattern717 = Pattern(Integral(S(1)/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons479, cons478)
rule717 = ReplacementRule(pattern717, With717)
pattern718 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons45, cons481)
rule718 = ReplacementRule(pattern718, replacement718)
pattern719 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons45, cons482)
rule719 = ReplacementRule(pattern719, replacement719)
pattern720 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons450)
rule720 = ReplacementRule(pattern720, replacement720)
pattern721 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons483)
rule721 = ReplacementRule(pattern721, With721)
pattern722 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons484)
rule722 = ReplacementRule(pattern722, With722)
pattern723 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons468)
rule723 = ReplacementRule(pattern723, With723)
pattern724 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons485, cons45)
rule724 = ReplacementRule(pattern724, replacement724)
pattern725 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons486, cons107, CustomConstraint(With725))
rule725 = ReplacementRule(pattern725, replacement725)
pattern726 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons486, cons107)
rule726 = ReplacementRule(pattern726, With726)
pattern727 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons485, cons450)
rule727 = ReplacementRule(pattern727, replacement727)
pattern728 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(6)*WC('b', S(1))), x_), cons2, cons3, cons69)
rule728 = ReplacementRule(pattern728, With728)
pattern729 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(8)*WC('b', S(1))), x_), cons2, cons3, cons69)
rule729 = ReplacementRule(pattern729, replacement729)
pattern730 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-1)/4), x_), cons2, cons3, cons468)
rule730 = ReplacementRule(pattern730, replacement730)
pattern731 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-1)/4), x_), cons2, cons3, cons485, cons45)
rule731 = ReplacementRule(pattern731, replacement731)
pattern732 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-1)/4), x_), cons2, cons3, cons485, cons450)
rule732 = ReplacementRule(pattern732, replacement732)
pattern733 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-3)/4), x_), cons2, cons3, cons45, cons468)
rule733 = ReplacementRule(pattern733, replacement733)
pattern734 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-3)/4), x_), cons2, cons3, cons45, cons485)
rule734 = ReplacementRule(pattern734, replacement734)
pattern735 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-3)/4), x_), cons2, cons3, cons450)
rule735 = ReplacementRule(pattern735, replacement735)
pattern736 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-1)/3), x_), cons2, cons3, cons69)
rule736 = ReplacementRule(pattern736, replacement736)
pattern737 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-2)/3), x_), cons2, cons3, cons69)
rule737 = ReplacementRule(pattern737, replacement737)
pattern738 = Pattern(Integral((a_ + x_**S(4)*WC('b', S(1)))**(S(-3)/4), x_), cons2, cons3, cons69)
rule738 = ReplacementRule(pattern738, replacement738)
pattern739 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-1)/6), x_), cons2, cons3, cons69)
rule739 = ReplacementRule(pattern739, replacement739)
pattern740 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons150, cons13, cons487, cons488, cons489)
rule740 = ReplacementRule(pattern740, replacement740)
pattern741 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons150, cons13, cons487, cons488, cons490)
rule741 = ReplacementRule(pattern741, replacement741)
pattern742 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons198)
rule742 = ReplacementRule(pattern742, replacement742)
pattern743 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons491)
rule743 = ReplacementRule(pattern743, With743)
pattern744 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons4, cons130)
rule744 = ReplacementRule(pattern744, replacement744)
pattern745 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons4, cons5, cons359, cons492, cons493, cons494)
rule745 = ReplacementRule(pattern745, replacement745)
pattern746 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons4, cons5, cons359, cons492, cons493, cons495)
rule746 = ReplacementRule(pattern746, replacement746)
pattern747 = Pattern(Integral((u_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons4, cons5, cons70, cons71)
rule747 = ReplacementRule(pattern747, replacement747)
pattern748 = Pattern(Integral((x_**n_*WC('b1', S(1)) + WC('a1', S(0)))**WC('p', S(1))*(x_**n_*WC('b2', S(1)) + WC('a2', S(0)))**WC('p', S(1)), x_), cons59, cons60, cons61, cons62, cons4, cons5, cons57, cons496)
rule748 = ReplacementRule(pattern748, replacement748)
pattern749 = Pattern(Integral((a1_ + x_**WC('n', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('n', S(1))*WC('b2', S(1)))**WC('p', S(1)), x_), cons59, cons60, cons61, cons62, cons57, cons497, cons13, cons165, cons498)
rule749 = ReplacementRule(pattern749, replacement749)
pattern750 = Pattern(Integral((a1_ + x_**WC('n', S(1))*WC('b1', S(1)))**p_*(a2_ + x_**WC('n', S(1))*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons57, cons497, cons13, cons139, cons498)
rule750 = ReplacementRule(pattern750, replacement750)
pattern751 = Pattern(Integral((a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons5, cons57, cons499)
rule751 = ReplacementRule(pattern751, replacement751)
pattern752 = Pattern(Integral((a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons5, cons57, cons500)
rule752 = ReplacementRule(pattern752, With752)
pattern753 = Pattern(Integral((x_**n_*WC('b1', S(1)) + WC('a1', S(0)))**p_*(x_**n_*WC('b2', S(1)) + WC('a2', S(0)))**p_, x_), cons59, cons60, cons61, cons62, cons4, cons5, cons57, cons149)
rule753 = ReplacementRule(pattern753, replacement753)
pattern754 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons4, cons5, cons57, cons496)
rule754 = ReplacementRule(pattern754, replacement754)
pattern755 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**n_*WC('b', S(1)))**p_, x_), cons3, cons8, cons19, cons4, cons5, cons501, cons502)
rule755 = ReplacementRule(pattern755, replacement755)
pattern756 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons3, cons8, cons19, cons4, cons5, cons501, cons503)
rule756 = ReplacementRule(pattern756, replacement756)
pattern757 = Pattern(Integral((c_*x_)**m_*(x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons3, cons8, cons19, cons4, cons5, cons21)
rule757 = ReplacementRule(pattern757, replacement757)
pattern758 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons19, cons4, cons40, cons504)
rule758 = ReplacementRule(pattern758, replacement758)
pattern759 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons4, cons5, cons505, cons68)
rule759 = ReplacementRule(pattern759, replacement759)
pattern760 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons4, cons5, cons57, cons506, cons68)
rule760 = ReplacementRule(pattern760, replacement760)
pattern761 = Pattern(Integral(x_**WC('m', S(1))*(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons19, cons4, cons5, cons502)
rule761 = ReplacementRule(pattern761, replacement761)
pattern762 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons19, cons4, cons5, cons57, cons507)
rule762 = ReplacementRule(pattern762, replacement762)
pattern763 = Pattern(Integral((c_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons4, cons5, cons502)
rule763 = ReplacementRule(pattern763, replacement763)
pattern764 = Pattern(Integral((c_*x_)**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons4, cons5, cons57, cons507)
rule764 = ReplacementRule(pattern764, replacement764)
pattern765 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons19, cons4, cons130)
rule765 = ReplacementRule(pattern765, replacement765)
pattern766 = Pattern(Integral(x_**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons19, cons4, cons5, cons508, cons68)
rule766 = ReplacementRule(pattern766, replacement766)
pattern767 = Pattern(Integral(x_**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons19, cons4, cons5, cons57, cons509, cons68)
rule767 = ReplacementRule(pattern767, replacement767)
pattern768 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons4, cons5, cons508, cons56)
rule768 = ReplacementRule(pattern768, replacement768)
pattern769 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons4, cons5, cons57, cons509, cons56)
rule769 = ReplacementRule(pattern769, replacement769)
pattern770 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons150, cons20, CustomConstraint(With770))
rule770 = ReplacementRule(pattern770, replacement770)
pattern771 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons5, cons57, cons497, cons20, CustomConstraint(With771))
rule771 = ReplacementRule(pattern771, replacement771)
pattern772 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons150, cons246, cons165, cons96, cons510, cons511)
rule772 = ReplacementRule(pattern772, replacement772)
pattern773 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons57, cons497, cons246, cons165, cons512, cons513)
rule773 = ReplacementRule(pattern773, replacement773)
pattern774 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons150, cons246, cons165, cons514, cons511)
rule774 = ReplacementRule(pattern774, replacement774)
pattern775 = Pattern(Integral(x_**S(2)/(a_ + x_**S(4)*WC('b', S(1)))**(S(5)/4), x_), cons2, cons3, cons468)
rule775 = ReplacementRule(pattern775, replacement775)
pattern776 = Pattern(Integral(x_**m_/(a_ + x_**S(4)*WC('b', S(1)))**(S(5)/4), x_), cons2, cons3, cons468, cons515)
rule776 = ReplacementRule(pattern776, replacement776)
pattern777 = Pattern(Integral(x_**m_/(a_ + x_**S(4)*WC('b', S(1)))**(S(5)/4), x_), cons2, cons3, cons468, cons516)
rule777 = ReplacementRule(pattern777, replacement777)
pattern778 = Pattern(Integral(sqrt(x_*WC('c', S(1)))/(a_ + x_**S(2)*WC('b', S(1)))**(S(5)/4), x_), cons2, cons3, cons8, cons468)
rule778 = ReplacementRule(pattern778, replacement778)
pattern779 = Pattern(Integral((x_*WC('c', S(1)))**m_/(a_ + x_**S(2)*WC('b', S(1)))**(S(5)/4), x_), cons2, cons3, cons8, cons468, cons517, cons518)
rule779 = ReplacementRule(pattern779, replacement779)
pattern780 = Pattern(Integral((x_*WC('c', S(1)))**m_/(a_ + x_**S(2)*WC('b', S(1)))**(S(5)/4), x_), cons2, cons3, cons8, cons468, cons517, cons96)
rule780 = ReplacementRule(pattern780, replacement780)
pattern781 = Pattern(Integral(x_**S(2)/(a_ + x_**S(4)*WC('b', S(1)))**(S(5)/4), x_), cons2, cons3, cons485)
rule781 = ReplacementRule(pattern781, replacement781)
pattern782 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons150, cons246, cons139, cons519, cons520, cons511)
rule782 = ReplacementRule(pattern782, replacement782)
pattern783 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons57, cons497, cons246, cons139, cons521, cons522, cons513)
rule783 = ReplacementRule(pattern783, replacement783)
pattern784 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons150, cons246, cons139, cons511)
rule784 = ReplacementRule(pattern784, replacement784)
pattern785 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons57, cons497, cons246, cons139, cons513)
rule785 = ReplacementRule(pattern785, replacement785)
pattern786 = Pattern(Integral(x_/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons69)
rule786 = ReplacementRule(pattern786, replacement786)
pattern787 = Pattern(Integral(x_**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons523, cons64, cons524, cons470)
rule787 = ReplacementRule(pattern787, With787)
pattern788 = Pattern(Integral(x_**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons525, cons64, cons524, cons471)
rule788 = ReplacementRule(pattern788, With788)
pattern789 = Pattern(Integral(x_**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons526, cons64, cons524, cons470)
rule789 = ReplacementRule(pattern789, With789)
pattern790 = Pattern(Integral(x_**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons526, cons64, cons524, cons471)
rule790 = ReplacementRule(pattern790, With790)
pattern791 = Pattern(Integral(x_**S(2)/(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons477)
rule791 = ReplacementRule(pattern791, With791)
pattern792 = Pattern(Integral(x_**S(2)/(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons478)
rule792 = ReplacementRule(pattern792, With792)
pattern793 = Pattern(Integral(x_**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons527, cons64, cons524, cons480)
rule793 = ReplacementRule(pattern793, With793)
pattern794 = Pattern(Integral(x_**m_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons527, cons64, cons528, cons478)
rule794 = ReplacementRule(pattern794, With794)
pattern795 = Pattern(Integral(x_**m_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons527, cons64, cons529, cons478)
rule795 = ReplacementRule(pattern795, With795)
pattern796 = Pattern(Integral(x_**m_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons530, cons531)
rule796 = ReplacementRule(pattern796, replacement796)
pattern797 = Pattern(Integral(x_/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons483)
rule797 = ReplacementRule(pattern797, With797)
pattern798 = Pattern(Integral(x_/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons484)
rule798 = ReplacementRule(pattern798, With798)
pattern799 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons468)
rule799 = ReplacementRule(pattern799, With799)
pattern800 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons486, cons107)
rule800 = ReplacementRule(pattern800, With800)
pattern801 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons485)
rule801 = ReplacementRule(pattern801, With801)
pattern802 = Pattern(Integral(x_**S(4)/sqrt(a_ + x_**S(6)*WC('b', S(1))), x_), cons2, cons3, cons69)
rule802 = ReplacementRule(pattern802, With802)
pattern803 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(8)*WC('b', S(1))), x_), cons2, cons3, cons69)
rule803 = ReplacementRule(pattern803, replacement803)
pattern804 = Pattern(Integral(x_**S(2)/(a_ + x_**S(4)*WC('b', S(1)))**(S(1)/4), x_), cons2, cons3, cons468)
rule804 = ReplacementRule(pattern804, replacement804)
pattern805 = Pattern(Integral(x_**S(2)/(a_ + x_**S(4)*WC('b', S(1)))**(S(1)/4), x_), cons2, cons3, cons485)
rule805 = ReplacementRule(pattern805, replacement805)
pattern806 = Pattern(Integral(S(1)/(x_**S(2)*(a_ + x_**S(4)*WC('b', S(1)))**(S(1)/4)), x_), cons2, cons3, cons468)
rule806 = ReplacementRule(pattern806, replacement806)
pattern807 = Pattern(Integral(S(1)/(x_**S(2)*(a_ + x_**S(4)*WC('b', S(1)))**(S(1)/4)), x_), cons2, cons3, cons485)
rule807 = ReplacementRule(pattern807, replacement807)
pattern808 = Pattern(Integral(sqrt(c_*x_)/(a_ + x_**S(2)*WC('b', S(1)))**(S(1)/4), x_), cons2, cons3, cons8, cons468)
rule808 = ReplacementRule(pattern808, replacement808)
pattern809 = Pattern(Integral(sqrt(c_*x_)/(a_ + x_**S(2)*WC('b', S(1)))**(S(1)/4), x_), cons2, cons3, cons8, cons485)
rule809 = ReplacementRule(pattern809, replacement809)
pattern810 = Pattern(Integral(S(1)/((x_*WC('c', S(1)))**(S(3)/2)*(a_ + x_**S(2)*WC('b', S(1)))**(S(1)/4)), x_), cons2, cons3, cons8, cons468)
rule810 = ReplacementRule(pattern810, replacement810)
pattern811 = Pattern(Integral(S(1)/((x_*WC('c', S(1)))**(S(3)/2)*(a_ + x_**S(2)*WC('b', S(1)))**(S(1)/4)), x_), cons2, cons3, cons8, cons485)
rule811 = ReplacementRule(pattern811, replacement811)
pattern812 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons5, cons150, cons33, cons532, cons514, cons511)
rule812 = ReplacementRule(pattern812, replacement812)
pattern813 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons5, cons150, cons533, cons514, cons534)
rule813 = ReplacementRule(pattern813, replacement813)
pattern814 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons5, cons57, cons497, cons33, cons531, cons512, cons513)
rule814 = ReplacementRule(pattern814, replacement814)
pattern815 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons5, cons57, cons497, cons535, cons512, cons536)
rule815 = ReplacementRule(pattern815, replacement815)
pattern816 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons5, cons150, cons33, cons96, cons511)
rule816 = ReplacementRule(pattern816, replacement816)
pattern817 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons5, cons150, cons537, cons534)
rule817 = ReplacementRule(pattern817, replacement817)
pattern818 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons5, cons57, cons497, cons33, cons96, cons513)
rule818 = ReplacementRule(pattern818, replacement818)
pattern819 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons5, cons57, cons497, cons538, cons536)
rule819 = ReplacementRule(pattern819, replacement819)
pattern820 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons5, cons150, cons369, cons511)
rule820 = ReplacementRule(pattern820, With820)
pattern821 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons5, cons57, cons497, cons369, cons513)
rule821 = ReplacementRule(pattern821, With821)
pattern822 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons150, cons13, cons487, cons488, cons539)
rule822 = ReplacementRule(pattern822, replacement822)
pattern823 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons57, cons497, cons13, cons487, cons488, cons540)
rule823 = ReplacementRule(pattern823, replacement823)
pattern824 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons150, cons13, cons487, cons488, cons20, cons541)
rule824 = ReplacementRule(pattern824, replacement824)
pattern825 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons57, cons497, cons13, cons487, cons488, cons20, cons542)
rule825 = ReplacementRule(pattern825, replacement825)
pattern826 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons198, cons20)
rule826 = ReplacementRule(pattern826, replacement826)
pattern827 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons5, cons57, cons499, cons20)
rule827 = ReplacementRule(pattern827, replacement827)
pattern828 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons5, cons198, cons369)
rule828 = ReplacementRule(pattern828, With828)
pattern829 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons5, cons57, cons499, cons369)
rule829 = ReplacementRule(pattern829, With829)
pattern830 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons5, cons198, cons358)
rule830 = ReplacementRule(pattern830, replacement830)
pattern831 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons5, cons57, cons499, cons358)
rule831 = ReplacementRule(pattern831, replacement831)
pattern832 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons19, cons5, cons491)
rule832 = ReplacementRule(pattern832, With832)
pattern833 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons19, cons5, cons57, cons500)
rule833 = ReplacementRule(pattern833, With833)
pattern834 = Pattern(Integral((c_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons5, cons491)
rule834 = ReplacementRule(pattern834, replacement834)
pattern835 = Pattern(Integral((c_*x_)**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons5, cons57, cons500)
rule835 = ReplacementRule(pattern835, replacement835)
pattern836 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons19, cons4, cons5, cons543, cons25)
rule836 = ReplacementRule(pattern836, replacement836)
pattern837 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons19, cons4, cons5, cons57, cons544, cons545)
rule837 = ReplacementRule(pattern837, replacement837)
pattern838 = Pattern(Integral((c_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons4, cons5, cons543, cons25)
rule838 = ReplacementRule(pattern838, replacement838)
pattern839 = Pattern(Integral((c_*x_)**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons4, cons5, cons57, cons544, cons545)
rule839 = ReplacementRule(pattern839, replacement839)
pattern840 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons19, cons4, cons546, cons13, cons165)
rule840 = ReplacementRule(pattern840, replacement840)
pattern841 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons19, cons4, cons57, cons547, cons13, cons165)
rule841 = ReplacementRule(pattern841, replacement841)
pattern842 = Pattern(Integral((c_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons4, cons546, cons13, cons165)
rule842 = ReplacementRule(pattern842, replacement842)
pattern843 = Pattern(Integral((c_*x_)**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons4, cons57, cons547, cons13, cons165)
rule843 = ReplacementRule(pattern843, replacement843)
pattern844 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons4, cons548, cons13, cons165, cons514)
rule844 = ReplacementRule(pattern844, replacement844)
pattern845 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons4, cons57, cons549, cons13, cons165, cons512)
rule845 = ReplacementRule(pattern845, replacement845)
pattern846 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons19, cons4, cons548, cons13, cons487)
rule846 = ReplacementRule(pattern846, With846)
pattern847 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons19, cons4, cons57, cons549, cons13, cons487)
rule847 = ReplacementRule(pattern847, With847)
pattern848 = Pattern(Integral((c_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons4, cons548, cons13, cons487)
rule848 = ReplacementRule(pattern848, replacement848)
pattern849 = Pattern(Integral((c_*x_)**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons4, cons57, cons549, cons13, cons487)
rule849 = ReplacementRule(pattern849, replacement849)
pattern850 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons4, cons548, cons13, cons139)
rule850 = ReplacementRule(pattern850, replacement850)
pattern851 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons4, cons57, cons548, cons13, cons139)
rule851 = ReplacementRule(pattern851, replacement851)
pattern852 = Pattern(Integral(x_**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons19, cons4, cons550, cons533)
rule852 = ReplacementRule(pattern852, With852)
pattern853 = Pattern(Integral(x_**m_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons19, cons4, cons550, cons537)
rule853 = ReplacementRule(pattern853, replacement853)
pattern854 = Pattern(Integral((c_*x_)**m_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons8, cons19, cons4, cons550, cons551)
rule854 = ReplacementRule(pattern854, replacement854)
pattern855 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons4, cons5, cons359, cons552)
rule855 = ReplacementRule(pattern855, replacement855)
pattern856 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons4, cons5, cons359, cons553)
rule856 = ReplacementRule(pattern856, replacement856)
pattern857 = Pattern(Integral(x_**WC('m', S(1))*(a_ + v_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons4, cons5, cons554, cons20, cons555)
rule857 = ReplacementRule(pattern857, replacement857)
pattern858 = Pattern(Integral(u_**WC('m', S(1))*(a_ + v_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons19, cons4, cons5, cons556)
rule858 = ReplacementRule(pattern858, replacement858)
pattern859 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons59, cons60, cons61, cons62, cons8, cons19, cons4, cons5, cons57, cons149)
rule859 = ReplacementRule(pattern859, replacement859)
pattern860 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons73, cons557)
rule860 = ReplacementRule(pattern860, replacement860)
pattern861 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons73, cons222, cons504)
rule861 = ReplacementRule(pattern861, replacement861)
pattern862 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons5, cons52, cons73, cons198)
rule862 = ReplacementRule(pattern862, replacement862)
pattern863 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons5, cons52, cons73, cons491)
rule863 = ReplacementRule(pattern863, With863)
pattern864 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_/(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons73, cons558, cons87)
rule864 = ReplacementRule(pattern864, replacement864)
pattern865 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons73, cons559, cons397, cons405, cons56)
rule865 = ReplacementRule(pattern865, replacement865)
pattern866 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons4, cons52, cons73, cons559, cons65)
rule866 = ReplacementRule(pattern866, replacement866)
pattern867 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons4, cons5, cons52, cons73, cons559)
rule867 = ReplacementRule(pattern867, replacement867)
pattern868 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons4, cons5, cons52, cons73, cons560, cons561)
rule868 = ReplacementRule(pattern868, replacement868)
pattern869 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons4, cons52, cons73, cons560, cons562, cons56)
rule869 = ReplacementRule(pattern869, replacement869)
pattern870 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons73, cons563)
rule870 = ReplacementRule(pattern870, replacement870)
pattern871 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons73, cons564)
rule871 = ReplacementRule(pattern871, replacement871)
pattern872 = Pattern(Integral((c_ + x_**n_*WC('d', S(1)))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons4, cons73, cons89, cons465)
rule872 = ReplacementRule(pattern872, replacement872)
pattern873 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons4, cons73, cons565)
rule873 = ReplacementRule(pattern873, replacement873)
pattern874 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons73, cons466, cons566, cons567)
rule874 = ReplacementRule(pattern874, replacement874)
pattern875 = Pattern(Integral(S(1)/((a_ + x_**n_*WC('b', S(1)))*(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons4, cons73)
rule875 = ReplacementRule(pattern875, replacement875)
pattern876 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))**(S(1)/3)*(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons73, cons568, cons468)
rule876 = ReplacementRule(pattern876, replacement876)
pattern877 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))**(S(1)/3)*(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons73, cons568, cons485)
rule877 = ReplacementRule(pattern877, replacement877)
pattern878 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(2)/3)/(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons73, cons568)
rule878 = ReplacementRule(pattern878, replacement878)
pattern879 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))**(S(1)/4)*(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons73)
rule879 = ReplacementRule(pattern879, replacement879)
pattern880 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))**(S(3)/4)*(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons73)
rule880 = ReplacementRule(pattern880, replacement880)
pattern881 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**WC('p', S(1))/(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons73, cons13, cons165, cons569)
rule881 = ReplacementRule(pattern881, replacement881)
pattern882 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**p_/(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons73, cons13, cons139, cons570, cons571)
rule882 = ReplacementRule(pattern882, replacement882)
pattern883 = Pattern(Integral(sqrt(a_ + x_**S(4)*WC('b', S(1)))/(c_ + x_**S(4)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons72, cons572)
rule883 = ReplacementRule(pattern883, replacement883)
pattern884 = Pattern(Integral(sqrt(a_ + x_**S(4)*WC('b', S(1)))/(c_ + x_**S(4)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons72, cons573)
rule884 = ReplacementRule(pattern884, With884)
pattern885 = Pattern(Integral(sqrt(a_ + x_**S(4)*WC('b', S(1)))/(c_ + x_**S(4)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons73)
rule885 = ReplacementRule(pattern885, replacement885)
pattern886 = Pattern(Integral((a_ + x_**S(4)*WC('b', S(1)))**(S(1)/4)/(c_ + x_**S(4)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons73)
rule886 = ReplacementRule(pattern886, replacement886)
pattern887 = Pattern(Integral((a_ + x_**S(4)*WC('b', S(1)))**p_/(c_ + x_**S(4)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons73, cons13, cons574)
rule887 = ReplacementRule(pattern887, replacement887)
pattern888 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(4)*WC('b', S(1)))*(c_ + x_**S(4)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons73)
rule888 = ReplacementRule(pattern888, replacement888)
pattern889 = Pattern(Integral(S(1)/((a_ + x_**S(4)*WC('b', S(1)))**(S(3)/4)*(c_ + x_**S(4)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons73)
rule889 = ReplacementRule(pattern889, replacement889)
pattern890 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons468, cons575)
rule890 = ReplacementRule(pattern890, replacement890)
pattern891 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons4, cons73, cons404, cons139, cons576, cons577)
rule891 = ReplacementRule(pattern891, replacement891)
pattern892 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons4, cons73, cons404, cons139, cons578, cons577)
rule892 = ReplacementRule(pattern892, replacement892)
pattern893 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons4, cons52, cons73, cons13, cons139, cons407, cons577)
rule893 = ReplacementRule(pattern893, replacement893)
pattern894 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons73, cons150, cons222, cons579)
rule894 = ReplacementRule(pattern894, replacement894)
pattern895 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons4, cons5, cons73, cons397, cons578, cons580, cons581, cons577)
rule895 = ReplacementRule(pattern895, replacement895)
pattern896 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons4, cons73, cons404, cons405, cons165, cons577)
rule896 = ReplacementRule(pattern896, replacement896)
pattern897 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons575, cons468, cons582)
rule897 = ReplacementRule(pattern897, replacement897)
pattern898 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons583, cons179, cons45, cons584)
rule898 = ReplacementRule(pattern898, replacement898)
pattern899 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons583, cons179, cons585)
rule899 = ReplacementRule(pattern899, replacement899)
pattern900 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons119)
rule900 = ReplacementRule(pattern900, replacement900)
pattern901 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/sqrt(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons575, cons468)
rule901 = ReplacementRule(pattern901, replacement901)
pattern902 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/sqrt(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons575, cons485)
rule902 = ReplacementRule(pattern902, replacement902)
pattern903 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/sqrt(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons583, cons179, cons45)
rule903 = ReplacementRule(pattern903, replacement903)
pattern904 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/sqrt(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons583, cons179, cons585)
rule904 = ReplacementRule(pattern904, replacement904)
pattern905 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/sqrt(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons583, cons179, cons450)
rule905 = ReplacementRule(pattern905, replacement905)
pattern906 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/sqrt(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons583, cons119)
rule906 = ReplacementRule(pattern906, replacement906)
pattern907 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons4, cons52, cons73, cons130)
rule907 = ReplacementRule(pattern907, replacement907)
pattern908 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons4, cons5, cons52, cons73, cons586, cons45, cons179)
rule908 = ReplacementRule(pattern908, replacement908)
pattern909 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons4, cons5, cons52, cons73, cons586, cons450)
rule909 = ReplacementRule(pattern909, replacement909)
pattern910 = Pattern(Integral((a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons587, cons588, cons589)
rule910 = ReplacementRule(pattern910, replacement910)
pattern911 = Pattern(Integral((a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons4, cons5, cons52, cons587, cons388, cons149)
rule911 = ReplacementRule(pattern911, replacement911)
pattern912 = Pattern(Integral((u_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(u_**n_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons52, cons70, cons71)
rule912 = ReplacementRule(pattern912, replacement912)
pattern913 = Pattern(Integral(u_**WC('p', S(1))*v_**WC('q', S(1)), x_), cons5, cons52, cons590)
rule913 = ReplacementRule(pattern913, replacement913)
pattern914 = Pattern(Integral(u_**WC('p', S(1))*v_**WC('q', S(1))*x_**WC('m', S(1)), x_), cons5, cons52, cons591, cons592)
rule914 = ReplacementRule(pattern914, replacement914)
pattern915 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons52, cons593, cons502)
rule915 = ReplacementRule(pattern915, replacement915)
pattern916 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons52, cons593, cons503)
rule916 = ReplacementRule(pattern916, replacement916)
pattern917 = Pattern(Integral((e_*x_)**m_*(x_**WC('n', S(1))*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons52, cons21)
rule917 = ReplacementRule(pattern917, replacement917)
pattern918 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons52, cons73, cons55)
rule918 = ReplacementRule(pattern918, replacement918)
pattern919 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons73, cons222, cons504)
rule919 = ReplacementRule(pattern919, replacement919)
pattern920 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons52, cons73, cons502)
rule920 = ReplacementRule(pattern920, replacement920)
pattern921 = Pattern(Integral((e_*x_)**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons52, cons73, cons502)
rule921 = ReplacementRule(pattern921, replacement921)
pattern922 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons73, cons557)
rule922 = ReplacementRule(pattern922, replacement922)
pattern923 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons73, cons594, cons68)
rule923 = ReplacementRule(pattern923, replacement923)
pattern924 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons59, cons60, cons61, cons62, cons8, cons29, cons50, cons19, cons4, cons5, cons595, cons57, cons596, cons68)
rule924 = ReplacementRule(pattern924, replacement924)
pattern925 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons5, cons73, cons597, cons598, cons95, cons599)
rule925 = ReplacementRule(pattern925, replacement925)
pattern926 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons73, cons597, cons68)
rule926 = ReplacementRule(pattern926, replacement926)
pattern927 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons5, cons73, cons598, cons95, cons599, cons600)
rule927 = ReplacementRule(pattern927, replacement927)
pattern928 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons59, cons60, cons61, cons62, cons8, cons29, cons50, cons5, cons595, cons57, cons598, cons95, cons599, cons600)
rule928 = ReplacementRule(pattern928, replacement928)
pattern929 = Pattern(Integral(x_**m_*(a_ + x_**S(2)*WC('b', S(1)))**p_*(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons73, cons13, cons139, cons601, cons602)
rule929 = ReplacementRule(pattern929, replacement929)
pattern930 = Pattern(Integral(x_**m_*(a_ + x_**S(2)*WC('b', S(1)))**p_*(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons73, cons13, cons139, cons603, cons602)
rule930 = ReplacementRule(pattern930, replacement930)
pattern931 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons73, cons13, cons139, cons604)
rule931 = ReplacementRule(pattern931, replacement931)
pattern932 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons59, cons60, cons61, cons62, cons8, cons29, cons50, cons19, cons4, cons595, cons57, cons13, cons139, cons604)
rule932 = ReplacementRule(pattern932, replacement932)
pattern933 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons73, cons605)
rule933 = ReplacementRule(pattern933, replacement933)
pattern934 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons59, cons60, cons61, cons62, cons8, cons29, cons50, cons19, cons4, cons5, cons595, cons57, cons605)
rule934 = ReplacementRule(pattern934, replacement934)
pattern935 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_/(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons73, cons150, cons130, cons606)
rule935 = ReplacementRule(pattern935, replacement935)
pattern936 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons5, cons73, cons150, cons95, cons96, cons90)
rule936 = ReplacementRule(pattern936, replacement936)
pattern937 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons73, cons150, cons13, cons139)
rule937 = ReplacementRule(pattern937, replacement937)
pattern938 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons73, cons150, cons607)
rule938 = ReplacementRule(pattern938, replacement938)
pattern939 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons5, cons52, cons73, cons150, cons20, CustomConstraint(With939))
rule939 = ReplacementRule(pattern939, replacement939)
pattern940 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons5, cons52, cons73, cons150, cons369, cons40)
rule940 = ReplacementRule(pattern940, With940)
pattern941 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons73, cons150, cons608, cons139, cons405, cons609, cons610)
rule941 = ReplacementRule(pattern941, replacement941)
pattern942 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons73, cons150, cons404, cons139, cons578, cons610)
rule942 = ReplacementRule(pattern942, replacement942)
pattern943 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons73, cons150, cons404, cons139, cons576, cons610)
rule943 = ReplacementRule(pattern943, replacement943)
pattern944 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons52, cons73, cons150, cons246, cons139, cons611, cons610)
rule944 = ReplacementRule(pattern944, replacement944)
pattern945 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons52, cons73, cons150, cons246, cons139, cons612, cons610)
rule945 = ReplacementRule(pattern945, replacement945)
pattern946 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons52, cons73, cons150, cons13, cons139, cons610)
rule946 = ReplacementRule(pattern946, replacement946)
pattern947 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons73, cons150, cons608, cons405, cons96, cons165, cons610)
rule947 = ReplacementRule(pattern947, replacement947)
pattern948 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons5, cons73, cons150, cons613, cons578, cons96, cons610)
rule948 = ReplacementRule(pattern948, replacement948)
pattern949 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons5, cons73, cons150, cons613, cons576, cons96, cons610)
rule949 = ReplacementRule(pattern949, replacement949)
pattern950 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons73, cons150, cons404, cons405, cons165, cons610)
rule950 = ReplacementRule(pattern950, replacement950)
pattern951 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons5, cons73, cons150, cons397, cons578, cons610)
rule951 = ReplacementRule(pattern951, replacement951)
pattern952 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons5, cons73, cons150, cons613, cons405, cons609, cons610)
rule952 = ReplacementRule(pattern952, replacement952)
pattern953 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons5, cons52, cons73, cons150, cons33, cons611, cons610)
rule953 = ReplacementRule(pattern953, replacement953)
pattern954 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons5, cons52, cons73, cons150, cons33, cons96, cons610)
rule954 = ReplacementRule(pattern954, replacement954)
pattern955 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))/((a_ + x_**n_*WC('b', S(1)))*(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons73, cons150, cons33, cons614)
rule955 = ReplacementRule(pattern955, replacement955)
pattern956 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))/((a_ + x_**n_*WC('b', S(1)))*(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons73, cons150)
rule956 = ReplacementRule(pattern956, replacement956)
pattern957 = Pattern(Integral(x_**m_/((a_ + x_**n_*WC('b', S(1)))*sqrt(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons73, cons615, cons616, cons617)
rule957 = ReplacementRule(pattern957, replacement957)
pattern958 = Pattern(Integral(x_**S(2)/((a_ + x_**S(4)*WC('b', S(1)))*sqrt(c_ + x_**S(4)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons73)
rule958 = ReplacementRule(pattern958, With958)
pattern959 = Pattern(Integral(x_/((a_ + x_**S(3)*WC('b', S(1)))*sqrt(c_ + x_**S(3)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons73, cons618)
rule959 = ReplacementRule(pattern959, With959)
pattern960 = Pattern(Integral(x_**m_/((a_ + x_**S(3)*WC('b', S(1)))*sqrt(c_ + x_**S(3)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons73, cons618, cons619)
rule960 = ReplacementRule(pattern960, replacement960)
pattern961 = Pattern(Integral(x_**m_/((a_ + x_**S(3)*WC('b', S(1)))*sqrt(c_ + x_**S(3)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons73, cons618, cons620)
rule961 = ReplacementRule(pattern961, replacement961)
pattern962 = Pattern(Integral(x_**S(2)*sqrt(c_ + x_**S(4)*WC('d', S(1)))/(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons73)
rule962 = ReplacementRule(pattern962, replacement962)
pattern963 = Pattern(Integral(x_**WC('m', S(1))*sqrt(c_ + x_**S(3)*WC('d', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons73, cons618, cons621)
rule963 = ReplacementRule(pattern963, replacement963)
pattern964 = Pattern(Integral(x_**S(2)/(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons73, cons468, cons575, cons582)
rule964 = ReplacementRule(pattern964, replacement964)
pattern965 = Pattern(Integral(x_**n_/(sqrt(a_ + x_**n_*WC('b', S(1)))*sqrt(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons73, cons622, cons623)
rule965 = ReplacementRule(pattern965, replacement965)
pattern966 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons150, cons246, cons624, cons487)
rule966 = ReplacementRule(pattern966, With966)
pattern967 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons5, cons52, cons73, cons198, cons20)
rule967 = ReplacementRule(pattern967, replacement967)
pattern968 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons5, cons52, cons198, cons369)
rule968 = ReplacementRule(pattern968, With968)
pattern969 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons5, cons52, cons73, cons198, cons358)
rule969 = ReplacementRule(pattern969, replacement969)
pattern970 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons19, cons5, cons52, cons73, cons491)
rule970 = ReplacementRule(pattern970, With970)
pattern971 = Pattern(Integral((e_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons5, cons52, cons73, cons491)
rule971 = ReplacementRule(pattern971, replacement971)
pattern972 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons52, cons73, cons543, cons25)
rule972 = ReplacementRule(pattern972, replacement972)
pattern973 = Pattern(Integral((e_*x_)**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons52, cons73, cons543, cons25)
rule973 = ReplacementRule(pattern973, replacement973)
pattern974 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons73, cons404, cons139, cons578, cons610)
rule974 = ReplacementRule(pattern974, replacement974)
pattern975 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons73, cons404, cons139, cons576, cons610)
rule975 = ReplacementRule(pattern975, replacement975)
pattern976 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons52, cons73, cons13, cons139, cons610)
rule976 = ReplacementRule(pattern976, replacement976)
pattern977 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons73, cons404, cons405, cons165, cons610)
rule977 = ReplacementRule(pattern977, replacement977)
pattern978 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons73, cons397, cons578, cons610)
rule978 = ReplacementRule(pattern978, replacement978)
pattern979 = Pattern(Integral(x_**m_/((a_ + x_**n_*WC('b', S(1)))*(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons73, cons625)
rule979 = ReplacementRule(pattern979, replacement979)
pattern980 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))/((a_ + x_**n_*WC('b', S(1)))*(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons19, cons73)
rule980 = ReplacementRule(pattern980, replacement980)
pattern981 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons73, cons626, cons627, cons628)
rule981 = ReplacementRule(pattern981, replacement981)
pattern982 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons587, cons588, cons589)
rule982 = ReplacementRule(pattern982, replacement982)
pattern983 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons52, cons587, cons388, cons149)
rule983 = ReplacementRule(pattern983, replacement983)
pattern984 = Pattern(Integral((e_*x_)**m_*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons52, cons587)
rule984 = ReplacementRule(pattern984, replacement984)
pattern985 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons52, cons73, cons68, cons629, cons45, cons179)
rule985 = ReplacementRule(pattern985, replacement985)
pattern986 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons52, cons73, cons68, cons629, cons450)
rule986 = ReplacementRule(pattern986, replacement986)
pattern987 = Pattern(Integral(x_**WC('m', S(1))*(v_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(v_**n_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons52, cons554, cons20, cons555)
rule987 = ReplacementRule(pattern987, replacement987)
pattern988 = Pattern(Integral(u_**WC('m', S(1))*(v_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(v_**n_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons52, cons556)
rule988 = ReplacementRule(pattern988, replacement988)
pattern989 = Pattern(Integral((a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**WC('p', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('u', S(1)), x_), cons59, cons60, cons61, cons62, cons8, cons29, cons4, cons5, cons52, cons595, cons57, cons496)
rule989 = ReplacementRule(pattern989, replacement989)
pattern990 = Pattern(Integral((a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**WC('p', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)) + x_**WC('n2', S(1))*WC('e', S(1)))**WC('q', S(1))*WC('u', S(1)), x_), cons59, cons60, cons61, cons62, cons8, cons29, cons50, cons4, cons5, cons52, cons595, cons48, cons57, cons496)
rule990 = ReplacementRule(pattern990, replacement990)
pattern991 = Pattern(Integral((a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**p_*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**p_*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('u', S(1)), x_), cons59, cons60, cons61, cons62, cons8, cons29, cons4, cons5, cons52, cons595, cons57)
rule991 = ReplacementRule(pattern991, replacement991)
pattern992 = Pattern(Integral((a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**WC('p', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)) + x_**WC('n2', S(1))*WC('e', S(1)))**WC('q', S(1))*WC('u', S(1)), x_), cons59, cons60, cons61, cons62, cons8, cons29, cons50, cons4, cons5, cons52, cons595, cons48, cons57)
rule992 = ReplacementRule(pattern992, replacement992)
pattern993 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons630)
rule993 = ReplacementRule(pattern993, replacement993)
pattern994 = Pattern(Integral((e_ + x_**n_*WC('f', S(1)))/((a_ + x_**n_*WC('b', S(1)))*(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons631)
rule994 = ReplacementRule(pattern994, replacement994)
pattern995 = Pattern(Integral((e_ + x_**n_*WC('f', S(1)))/((a_ + x_**n_*WC('b', S(1)))*sqrt(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons631)
rule995 = ReplacementRule(pattern995, replacement995)
pattern996 = Pattern(Integral((e_ + x_**n_*WC('f', S(1)))/(sqrt(a_ + x_**n_*WC('b', S(1)))*sqrt(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons632)
rule996 = ReplacementRule(pattern996, replacement996)
pattern997 = Pattern(Integral((e_ + x_**S(2)*WC('f', S(1)))/(sqrt(a_ + x_**S(2)*WC('b', S(1)))*(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons468, cons575)
rule997 = ReplacementRule(pattern997, replacement997)
pattern998 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons404, cons139, cons405)
rule998 = ReplacementRule(pattern998, replacement998)
pattern999 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons52, cons13, cons139)
rule999 = ReplacementRule(pattern999, replacement999)
pattern1000 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons397, cons405, cons633)
rule1000 = ReplacementRule(pattern1000, replacement1000)
pattern1001 = Pattern(Integral((e_ + x_**S(4)*WC('f', S(1)))/((a_ + x_**S(4)*WC('b', S(1)))**(S(3)/4)*(c_ + x_**S(4)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons155)
rule1001 = ReplacementRule(pattern1001, replacement1001)
pattern1002 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(e_ + x_**n_*WC('f', S(1)))/(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons5, cons4, cons634)
rule1002 = ReplacementRule(pattern1002, replacement1002)
pattern1003 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons52, cons635)
rule1003 = ReplacementRule(pattern1003, replacement1003)
pattern1004 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))*(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons155)
rule1004 = ReplacementRule(pattern1004, replacement1004)
pattern1005 = Pattern(Integral(S(1)/(x_**S(2)*(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons8, cons29, cons50, cons127, cons178)
rule1005 = ReplacementRule(pattern1005, replacement1005)
pattern1006 = Pattern(Integral(sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons636, cons637, cons638)
rule1006 = ReplacementRule(pattern1006, replacement1006)
pattern1007 = Pattern(Integral(sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons639)
rule1007 = ReplacementRule(pattern1007, replacement1007)
pattern1008 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons575, cons640, cons638)
rule1008 = ReplacementRule(pattern1008, replacement1008)
pattern1009 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons583, cons179, cons180, cons641)
rule1009 = ReplacementRule(pattern1009, replacement1009)
pattern1010 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons119)
rule1010 = ReplacementRule(pattern1010, replacement1010)
pattern1011 = Pattern(Integral(sqrt(c_ + x_**S(2)*WC('d', S(1)))/((a_ + x_**S(2)*WC('b', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons575)
rule1011 = ReplacementRule(pattern1011, replacement1011)
pattern1012 = Pattern(Integral(sqrt(c_ + x_**S(2)*WC('d', S(1)))/((a_ + x_**S(2)*WC('b', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons583)
rule1012 = ReplacementRule(pattern1012, replacement1012)
pattern1013 = Pattern(Integral(sqrt(e_ + x_**S(2)*WC('f', S(1)))/((a_ + x_**S(2)*WC('b', S(1)))*(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons575, cons640)
rule1013 = ReplacementRule(pattern1013, replacement1013)
pattern1014 = Pattern(Integral((e_ + x_**S(2)*WC('f', S(1)))**(S(3)/2)/((a_ + x_**S(2)*WC('b', S(1)))*(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons575, cons640)
rule1014 = ReplacementRule(pattern1014, replacement1014)
pattern1015 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2)*sqrt(e_ + x_**S(2)*WC('f', S(1)))/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons575, cons640)
rule1015 = ReplacementRule(pattern1015, replacement1015)
pattern1016 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**q_*(e_ + x_**S(2)*WC('f', S(1)))**r_/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons642, cons398, cons643)
rule1016 = ReplacementRule(pattern1016, replacement1016)
pattern1017 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**q_*(e_ + x_**S(2)*WC('f', S(1)))**r_/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons54, cons397, cons578)
rule1017 = ReplacementRule(pattern1017, replacement1017)
pattern1018 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**q_*(e_ + x_**S(2)*WC('f', S(1)))**r_/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons54, cons397, cons398)
rule1018 = ReplacementRule(pattern1018, replacement1018)
pattern1019 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**q_*(e_ + x_**S(2)*WC('f', S(1)))**r_/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons54, cons397, cons644)
rule1019 = ReplacementRule(pattern1019, replacement1019)
pattern1020 = Pattern(Integral(sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))/(a_ + x_**S(2)*WC('b', S(1)))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons155)
rule1020 = ReplacementRule(pattern1020, replacement1020)
pattern1021 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))**S(2)*sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons155)
rule1021 = ReplacementRule(pattern1021, replacement1021)
pattern1022 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1)))**r_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons54, cons65, cons397, cons405)
rule1022 = ReplacementRule(pattern1022, replacement1022)
pattern1023 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1)))**r_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons52, cons65, cons397, cons644)
rule1023 = ReplacementRule(pattern1023, replacement1023)
pattern1024 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons155)
rule1024 = ReplacementRule(pattern1024, replacement1024)
pattern1025 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/(sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons155)
rule1025 = ReplacementRule(pattern1025, replacement1025)
pattern1026 = Pattern(Integral(sqrt(c_ + x_**S(2)*WC('d', S(1)))/((a_ + x_**S(2)*WC('b', S(1)))**(S(3)/2)*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons155)
rule1026 = ReplacementRule(pattern1026, replacement1026)
pattern1027 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))/sqrt(e_ + x_**S(2)*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons645)
rule1027 = ReplacementRule(pattern1027, replacement1027)
pattern1028 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))/sqrt(e_ + x_**S(2)*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons646)
rule1028 = ReplacementRule(pattern1028, replacement1028)
pattern1029 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))/(e_ + x_**S(2)*WC('f', S(1)))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons155)
rule1029 = ReplacementRule(pattern1029, replacement1029)
pattern1030 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1)))**r_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons5, cons52, cons54, cons150, CustomConstraint(With1030))
rule1030 = ReplacementRule(pattern1030, replacement1030)
pattern1031 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1)))**r_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons5, cons52, cons54, cons198)
rule1031 = ReplacementRule(pattern1031, replacement1031)
pattern1032 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons52, cons54, cons647)
rule1032 = ReplacementRule(pattern1032, replacement1032)
pattern1033 = Pattern(Integral((u_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(v_**n_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1))*(w_**n_*WC('f', S(1)) + WC('e', S(0)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons5, cons4, cons52, cons54, cons648, cons649, cons70, cons71)
rule1033 = ReplacementRule(pattern1033, replacement1033)
pattern1034 = Pattern(Integral((c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**WC('n', S(1))*WC('f', S(1)))**WC('r', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons54, cons587, cons588)
rule1034 = ReplacementRule(pattern1034, replacement1034)
pattern1035 = Pattern(Integral((c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**WC('n', S(1))*WC('f', S(1)))**WC('r', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons52, cons587, cons40, cons650)
rule1035 = ReplacementRule(pattern1035, replacement1035)
pattern1036 = Pattern(Integral((c_ + x_**WC('mn', S(1))*WC('d', S(1)))**q_*(e_ + x_**WC('n', S(1))*WC('f', S(1)))**WC('r', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons52, cons54, cons587, cons388)
rule1036 = ReplacementRule(pattern1036, replacement1036)
pattern1037 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e1_ + x_**WC('n2', S(1))*WC('f1', S(1)))**WC('r', S(1))*(e2_ + x_**WC('n2', S(1))*WC('f2', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons654, cons655, cons656, cons657, cons4, cons5, cons52, cons54, cons651, cons652, cons653)
rule1037 = ReplacementRule(pattern1037, replacement1037)
pattern1038 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e1_ + x_**WC('n2', S(1))*WC('f1', S(1)))**WC('r', S(1))*(e2_ + x_**WC('n2', S(1))*WC('f2', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons654, cons655, cons656, cons657, cons4, cons5, cons52, cons54, cons651, cons652)
rule1038 = ReplacementRule(pattern1038, replacement1038)
pattern1039 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons52, cons54, cons658, cons502)
rule1039 = ReplacementRule(pattern1039, replacement1039)
pattern1040 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons52, cons54, cons658, cons503)
rule1040 = ReplacementRule(pattern1040, replacement1040)
pattern1041 = Pattern(Integral((g_*x_)**m_*(x_**WC('n', S(1))*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons52, cons54, cons21)
rule1041 = ReplacementRule(pattern1041, replacement1041)
pattern1042 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons659)
rule1042 = ReplacementRule(pattern1042, replacement1042)
pattern1043 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons52, cons54, cons55)
rule1043 = ReplacementRule(pattern1043, replacement1043)
pattern1044 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons660, cons504)
rule1044 = ReplacementRule(pattern1044, replacement1044)
pattern1045 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons52, cons54, cons502)
rule1045 = ReplacementRule(pattern1045, replacement1045)
pattern1046 = Pattern(Integral((g_*x_)**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons52, cons54, cons502)
rule1046 = ReplacementRule(pattern1046, replacement1046)
pattern1047 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons5, cons52, cons54, cons150, cons20, CustomConstraint(With1047))
rule1047 = ReplacementRule(pattern1047, replacement1047)
pattern1048 = Pattern(Integral((x_*WC('g', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1)))**r_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons52, cons54, cons150, cons369)
rule1048 = ReplacementRule(pattern1048, With1048)
pattern1049 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons150, cons404, cons139, cons405, cons661)
rule1049 = ReplacementRule(pattern1049, replacement1049)
pattern1050 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons52, cons150, cons246, cons139, cons609)
rule1050 = ReplacementRule(pattern1050, replacement1050)
pattern1051 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons52, cons150, cons13, cons139)
rule1051 = ReplacementRule(pattern1051, replacement1051)
pattern1052 = Pattern(Integral((x_*WC('g', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons150, cons613, cons405, cons96, cons662)
rule1052 = ReplacementRule(pattern1052, replacement1052)
pattern1053 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons150, cons397, cons405, cons662)
rule1053 = ReplacementRule(pattern1053, replacement1053)
pattern1054 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons52, cons150, cons33, cons532)
rule1054 = ReplacementRule(pattern1054, replacement1054)
pattern1055 = Pattern(Integral((x_*WC('g', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons52, cons150, cons33, cons96)
rule1055 = ReplacementRule(pattern1055, replacement1055)
pattern1056 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(e_ + x_**n_*WC('f', S(1)))/(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons150)
rule1056 = ReplacementRule(pattern1056, replacement1056)
pattern1057 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons52, cons150)
rule1057 = ReplacementRule(pattern1057, replacement1057)
pattern1058 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons52, cons150, cons663)
rule1058 = ReplacementRule(pattern1058, replacement1058)
pattern1059 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons5, cons52, cons54, cons198, cons20)
rule1059 = ReplacementRule(pattern1059, replacement1059)
pattern1060 = Pattern(Integral((x_*WC('g', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons52, cons54, cons198, cons369)
rule1060 = ReplacementRule(pattern1060, With1060)
pattern1061 = Pattern(Integral((x_*WC('g', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons52, cons54, cons198, cons358)
rule1061 = ReplacementRule(pattern1061, replacement1061)
pattern1062 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons5, cons52, cons54, cons491)
rule1062 = ReplacementRule(pattern1062, With1062)
pattern1063 = Pattern(Integral((g_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons52, cons54, cons491)
rule1063 = ReplacementRule(pattern1063, replacement1063)
pattern1064 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons52, cons54, cons543)
rule1064 = ReplacementRule(pattern1064, replacement1064)
pattern1065 = Pattern(Integral((g_*x_)**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons52, cons54, cons543)
rule1065 = ReplacementRule(pattern1065, replacement1065)
pattern1066 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons404, cons139, cons405, cons661)
rule1066 = ReplacementRule(pattern1066, replacement1066)
pattern1067 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons52, cons13, cons139)
rule1067 = ReplacementRule(pattern1067, replacement1067)
pattern1068 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons397, cons405, cons662)
rule1068 = ReplacementRule(pattern1068, replacement1068)
pattern1069 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(e_ + x_**n_*WC('f', S(1)))/(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons382)
rule1069 = ReplacementRule(pattern1069, replacement1069)
pattern1070 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons52, cons664)
rule1070 = ReplacementRule(pattern1070, replacement1070)
pattern1071 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**WC('n', S(1))*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons54, cons587, cons588)
rule1071 = ReplacementRule(pattern1071, replacement1071)
pattern1072 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**WC('n', S(1))*WC('f', S(1)))**WC('r', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons52, cons587, cons40, cons650)
rule1072 = ReplacementRule(pattern1072, replacement1072)
pattern1073 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**q_*(e_ + x_**WC('n', S(1))*WC('f', S(1)))**WC('r', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons52, cons54, cons587, cons388)
rule1073 = ReplacementRule(pattern1073, replacement1073)
pattern1074 = Pattern(Integral((g_*x_)**m_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**WC('n', S(1))*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons52, cons54, cons587)
rule1074 = ReplacementRule(pattern1074, replacement1074)
pattern1075 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons52, cons54, cons665)
rule1075 = ReplacementRule(pattern1075, replacement1075)
pattern1076 = Pattern(Integral(u_**WC('m', S(1))*(e_ + v_**n_*WC('f', S(1)))**WC('r', S(1))*(v_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(v_**n_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons52, cons54, cons556)
rule1076 = ReplacementRule(pattern1076, replacement1076)
pattern1077 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e1_ + x_**WC('n2', S(1))*WC('f1', S(1)))**WC('r', S(1))*(e2_ + x_**WC('n2', S(1))*WC('f2', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons654, cons655, cons656, cons657, cons210, cons19, cons4, cons5, cons52, cons54, cons651, cons652, cons653)
rule1077 = ReplacementRule(pattern1077, replacement1077)
pattern1078 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e1_ + x_**WC('n2', S(1))*WC('f1', S(1)))**WC('r', S(1))*(e2_ + x_**WC('n2', S(1))*WC('f2', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons8, cons29, cons654, cons655, cons656, cons657, cons210, cons19, cons4, cons5, cons52, cons54, cons651, cons652)
rule1078 = ReplacementRule(pattern1078, replacement1078)
return [rule692, rule693, rule694, rule695, rule696, rule697, rule698, rule699, rule700, rule701, rule702, rule703, rule704, rule705, rule706, rule707, rule708, rule709, rule710, rule711, rule712, rule713, rule714, rule715, rule716, rule717, rule718, rule719, rule720, rule721, rule722, rule723, rule724, rule725, rule726, rule727, rule728, rule729, rule730, rule731, rule732, rule733, rule734, rule735, rule736, rule737, rule738, rule739, rule740, rule741, rule742, rule743, rule744, rule745, rule746, rule747, rule748, rule749, rule750, rule751, rule752, rule753, rule754, rule755, rule756, rule757, rule758, rule759, rule760, rule761, rule762, rule763, rule764, rule765, rule766, rule767, rule768, rule769, rule770, rule771, rule772, rule773, rule774, rule775, rule776, rule777, rule778, rule779, rule780, rule781, rule782, rule783, rule784, rule785, rule786, rule787, rule788, rule789, rule790, rule791, rule792, rule793, rule794, rule795, rule796, rule797, rule798, rule799, rule800, rule801, rule802, rule803, rule804, rule805, rule806, rule807, rule808, rule809, rule810, rule811, rule812, rule813, rule814, rule815, rule816, rule817, rule818, rule819, rule820, rule821, rule822, rule823, rule824, rule825, rule826, rule827, rule828, rule829, rule830, rule831, rule832, rule833, rule834, rule835, rule836, rule837, rule838, rule839, rule840, rule841, rule842, rule843, rule844, rule845, rule846, rule847, rule848, rule849, rule850, rule851, rule852, rule853, rule854, rule855, rule856, rule857, rule858, rule859, rule860, rule861, rule862, rule863, rule864, rule865, rule866, rule867, rule868, rule869, rule870, rule871, rule872, rule873, rule874, rule875, rule876, rule877, rule878, rule879, rule880, rule881, rule882, rule883, rule884, rule885, rule886, rule887, rule888, rule889, rule890, rule891, rule892, rule893, rule894, rule895, rule896, rule897, rule898, rule899, rule900, rule901, rule902, rule903, rule904, rule905, rule906, rule907, rule908, rule909, rule910, rule911, rule912, rule913, rule914, rule915, rule916, rule917, rule918, rule919, rule920, rule921, rule922, rule923, rule924, rule925, rule926, rule927, rule928, rule929, rule930, rule931, rule932, rule933, rule934, rule935, rule936, rule937, rule938, rule939, rule940, rule941, rule942, rule943, rule944, rule945, rule946, rule947, rule948, rule949, rule950, rule951, rule952, rule953, rule954, rule955, rule956, rule957, rule958, rule959, rule960, rule961, rule962, rule963, rule964, rule965, rule966, rule967, rule968, rule969, rule970, rule971, rule972, rule973, rule974, rule975, rule976, rule977, rule978, rule979, rule980, rule981, rule982, rule983, rule984, rule985, rule986, rule987, rule988, rule989, rule990, rule991, rule992, rule993, rule994, rule995, rule996, rule997, rule998, rule999, rule1000, rule1001, rule1002, rule1003, rule1004, rule1005, rule1006, rule1007, rule1008, rule1009, rule1010, rule1011, rule1012, rule1013, rule1014, rule1015, rule1016, rule1017, rule1018, rule1019, rule1020, rule1021, rule1022, rule1023, rule1024, rule1025, rule1026, rule1027, rule1028, rule1029, rule1030, rule1031, rule1032, rule1033, rule1034, rule1035, rule1036, rule1037, rule1038, rule1039, rule1040, rule1041, rule1042, rule1043, rule1044, rule1045, rule1046, rule1047, rule1048, rule1049, rule1050, rule1051, rule1052, rule1053, rule1054, rule1055, rule1056, rule1057, rule1058, rule1059, rule1060, rule1061, rule1062, rule1063, rule1064, rule1065, rule1066, rule1067, rule1068, rule1069, rule1070, rule1071, rule1072, rule1073, rule1074, rule1075, rule1076, rule1077, rule1078, ]
def replacement692(b, n, p, x):
return Dist(b**IntPart(p)*x**(-n*FracPart(p))*(b*x**n)**FracPart(p), Int(x**(n*p), x), x)
def replacement693(a, b, n, p, x):
return Simp(x*(a + b*x**n)**(p + S(1))/a, x)
def replacement694(a, b, n, p, x):
return Dist((n*(p + S(1)) + S(1))/(a*n*(p + S(1))), Int((a + b*x**n)**(p + S(1)), x), x) - Simp(x*(a + b*x**n)**(p + S(1))/(a*n*(p + S(1))), x)
def replacement695(a, b, n, x):
return Int(a**S(2) + S(2)*a*b*x**n + b**S(2)*x**(S(2)*n), x)
def replacement696(a, b, n, p, x):
return Int(x**(n*p)*(a*x**(-n) + b)**p, x)
def replacement697(a, b, n, p, x):
return Int(ExpandIntegrand((a + b*x**n)**p, x), x)
def replacement698(a, b, n, p, x):
return Dist(a*n*p/(n*p + S(1)), Int((a + b*x**n)**(p + S(-1)), x), x) + Simp(x*(a + b*x**n)**p/(n*p + S(1)), x)
def replacement699(a, b, x):
return Simp(S(2)*EllipticE(ArcTan(x*Rt(b/a, S(2)))/S(2), S(2))/(a**(S(5)/4)*Rt(b/a, S(2))), x)
def replacement700(a, b, x):
return Dist((S(1) + b*x**S(2)/a)**(S(1)/4)/(a*(a + b*x**S(2))**(S(1)/4)), Int((S(1) + b*x**S(2)/a)**(S(-5)/4), x), x)
def replacement701(a, b, x):
return Dist(S(1)/((a/(a + b*x**S(2)))**(S(2)/3)*(a + b*x**S(2))**(S(2)/3)), Subst(Int((-b*x**S(2) + S(1))**(S(-1)/3), x), x, x/sqrt(a + b*x**S(2))), x)
def replacement702(a, b, n, p, x):
return Dist((n*(p + S(1)) + S(1))/(a*n*(p + S(1))), Int((a + b*x**n)**(p + S(1)), x), x) - Simp(x*(a + b*x**n)**(p + S(1))/(a*n*(p + S(1))), x)
def replacement703(a, b, x):
return Dist(S(1)/(S(3)*Rt(a, S(3))**S(2)), Int((-x*Rt(b, S(3)) + S(2)*Rt(a, S(3)))/(x**S(2)*Rt(b, S(3))**S(2) - x*Rt(a, S(3))*Rt(b, S(3)) + Rt(a, S(3))**S(2)), x), x) + Dist(S(1)/(S(3)*Rt(a, S(3))**S(2)), Int(S(1)/(x*Rt(b, S(3)) + Rt(a, S(3))), x), x)
def With704(a, b, n, x):
r = Numerator(Rt(a/b, n))
s = Denominator(Rt(a/b, n))
k = Symbol('k')
u = Symbol('u')
u = Int((r - s*x*cos(Pi*(S(2)*k + S(-1))/n))/(r**S(2) - S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x)
u = Int((r - s*x*cos(Pi*(2*k - 1)/n))/(r**2 - 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x)
return Simp(Dist(2*r/(a*n), Sum_doit(u, List(k, 1, n/2 - 1/2)), x) + r*Int(1/(r + s*x), x)/(a*n), x)
def With705(a, b, n, x):
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
k = Symbol('k')
u = Symbol('u')
u = Int((r + s*x*cos(Pi*(S(2)*k + S(-1))/n))/(r**S(2) + S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x)
u = Int((r + s*x*cos(Pi*(2*k - 1)/n))/(r**2 + 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x)
return Simp(Dist(2*r/(a*n), Sum_doit(u, List(k, 1, n/2 - 1/2)), x) + r*Int(1/(r - s*x), x)/(a*n), x)
def replacement706(a, b, x):
return Simp(ArcTan(x*Rt(b, S(2))/Rt(a, S(2)))/(Rt(a, S(2))*Rt(b, S(2))), x)
def replacement707(a, b, x):
return -Simp(ArcTan(x*Rt(-b, S(2))/Rt(-a, S(2)))/(Rt(-a, S(2))*Rt(-b, S(2))), x)
def replacement708(a, b, x):
return Simp(ArcTan(x/Rt(a/b, S(2)))*Rt(a/b, S(2))/a, x)
def replacement709(a, b, x):
return Simp(atanh(x*Rt(-b, S(2))/Rt(a, S(2)))/(Rt(a, S(2))*Rt(-b, S(2))), x)
def replacement710(a, b, x):
return -Simp(atanh(x*Rt(b, S(2))/Rt(-a, S(2)))/(Rt(-a, S(2))*Rt(b, S(2))), x)
def replacement711(a, b, x):
return Simp(Rt(-a/b, S(2))*atanh(x/Rt(-a/b, S(2)))/a, x)
def With712(a, b, n, x):
r = Numerator(Rt(a/b, n))
s = Denominator(Rt(a/b, n))
k = Symbol('k')
u = Symbol('u')
v = Symbol('v')
u = Int((r - s*x*cos(Pi*(S(2)*k + S(-1))/n))/(r**S(2) - S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x) + Int((r + s*x*cos(Pi*(S(2)*k + S(-1))/n))/(r**S(2) + S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x)
u = Int((r - s*x*cos(Pi*(2*k - 1)/n))/(r**2 - 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x) + Int((r + s*x*cos(Pi*(2*k - 1)/n))/(r**2 + 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x)
return Simp(Dist(2*r/(a*n), Sum_doit(u, List(k, 1, n/4 - 1/2)), x) + 2*r**2*Int(1/(r**2 + s**2*x**2), x)/(a*n), x)
def With713(a, b, n, x):
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
k = Symbol('k')
u = Symbol('u')
u = Int((r - s*x*cos(S(2)*Pi*k/n))/(r**S(2) - S(2)*r*s*x*cos(S(2)*Pi*k/n) + s**S(2)*x**S(2)), x) + Int((r + s*x*cos(S(2)*Pi*k/n))/(r**S(2) + S(2)*r*s*x*cos(S(2)*Pi*k/n) + s**S(2)*x**S(2)), x)
u = Int((r - s*x*cos(2*Pi*k/n))/(r**2 - 2*r*s*x*cos(2*Pi*k/n) + s**2*x**2), x) + Int((r + s*x*cos(2*Pi*k/n))/(r**2 + 2*r*s*x*cos(2*Pi*k/n) + s**2*x**2), x)
return Simp(Dist(2*r/(a*n), Sum_doit(u, List(k, 1, n/4 - 1/2)), x) + 2*r**2*Int(1/(r**2 - s**2*x**2), x)/(a*n), x)
def With714(a, b, x):
r = Numerator(Rt(a/b, S(2)))
s = Denominator(Rt(a/b, S(2)))
return Dist(S(1)/(S(2)*r), Int((r - s*x**S(2))/(a + b*x**S(4)), x), x) + Dist(S(1)/(S(2)*r), Int((r + s*x**S(2))/(a + b*x**S(4)), x), x)
def With715(a, b, x):
r = Numerator(Rt(-a/b, S(2)))
s = Denominator(Rt(-a/b, S(2)))
return Dist(r/(S(2)*a), Int(S(1)/(r - s*x**S(2)), x), x) + Dist(r/(S(2)*a), Int(S(1)/(r + s*x**S(2)), x), x)
def With716(a, b, n, x):
r = Numerator(Rt(a/b, S(4)))
s = Denominator(Rt(a/b, S(4)))
return Dist(sqrt(S(2))*r/(S(4)*a), Int((sqrt(S(2))*r - s*x**(n/S(4)))/(r**S(2) - sqrt(S(2))*r*s*x**(n/S(4)) + s**S(2)*x**(n/S(2))), x), x) + Dist(sqrt(S(2))*r/(S(4)*a), Int((sqrt(S(2))*r + s*x**(n/S(4)))/(r**S(2) + sqrt(S(2))*r*s*x**(n/S(4)) + s**S(2)*x**(n/S(2))), x), x)
def With717(a, b, n, x):
r = Numerator(Rt(-a/b, S(2)))
s = Denominator(Rt(-a/b, S(2)))
return Dist(r/(S(2)*a), Int(S(1)/(r - s*x**(n/S(2))), x), x) + Dist(r/(S(2)*a), Int(S(1)/(r + s*x**(n/S(2))), x), x)
def replacement718(a, b, x):
return Simp(asinh(x*Rt(b, S(2))/sqrt(a))/Rt(b, S(2)), x)
def replacement719(a, b, x):
return Simp(asin(x*Rt(-b, S(2))/sqrt(a))/Rt(-b, S(2)), x)
def replacement720(a, b, x):
return Subst(Int(S(1)/(-b*x**S(2) + S(1)), x), x, x/sqrt(a + b*x**S(2)))
def With721(a, b, x):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
return Simp(S(2)*S(3)**(S(3)/4)*sqrt((r**S(2)*x**S(2) - r*s*x + s**S(2))/(r*x + s*(S(1) + sqrt(S(3))))**S(2))*sqrt(sqrt(S(3)) + S(2))*(r*x + s)*EllipticF(asin((r*x + s*(S(1) - sqrt(S(3))))/(r*x + s*(S(1) + sqrt(S(3))))), S(-7) - S(4)*sqrt(S(3)))/(S(3)*r*sqrt(s*(r*x + s)/(r*x + s*(S(1) + sqrt(S(3))))**S(2))*sqrt(a + b*x**S(3))), x)
def With722(a, b, x):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
return Simp(S(2)*S(3)**(S(3)/4)*sqrt((r**S(2)*x**S(2) - r*s*x + s**S(2))/(r*x + s*(S(1) - sqrt(S(3))))**S(2))*sqrt(S(2) - sqrt(S(3)))*(r*x + s)*EllipticF(asin((r*x + s*(S(1) + sqrt(S(3))))/(r*x + s*(S(1) - sqrt(S(3))))), S(-7) + S(4)*sqrt(S(3)))/(S(3)*r*sqrt(-s*(r*x + s)/(r*x + s*(S(1) - sqrt(S(3))))**S(2))*sqrt(a + b*x**S(3))), x)
def With723(a, b, x):
q = Rt(b/a, S(4))
return Simp(sqrt((a + b*x**S(4))/(a*(q**S(2)*x**S(2) + S(1))**S(2)))*(q**S(2)*x**S(2) + S(1))*EllipticF(S(2)*ArcTan(q*x), S(1)/2)/(S(2)*q*sqrt(a + b*x**S(4))), x)
def replacement724(a, b, x):
return Simp(EllipticF(asin(x*Rt(-b, S(4))/Rt(a, S(4))), S(-1))/(Rt(a, S(4))*Rt(-b, S(4))), x)
def With725(a, b, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-a*b, S(2))
if IntegerQ(q):
return True
return False
def replacement725(a, b, x):
q = Rt(-a*b, S(2))
return Simp(sqrt(S(2))*sqrt((a + q*x**S(2))/q)*sqrt(-a + q*x**S(2))*EllipticF(asin(sqrt(S(2))*x/sqrt((a + q*x**S(2))/q)), S(1)/2)/(S(2)*sqrt(-a)*sqrt(a + b*x**S(4))), x)
def With726(a, b, x):
q = Rt(-a*b, S(2))
return Simp(sqrt(S(2))*sqrt((a + q*x**S(2))/q)*sqrt((a - q*x**S(2))/(a + q*x**S(2)))*EllipticF(asin(sqrt(S(2))*x/sqrt((a + q*x**S(2))/q)), S(1)/2)/(S(2)*sqrt(a/(a + q*x**S(2)))*sqrt(a + b*x**S(4))), x)
def replacement727(a, b, x):
return Dist(sqrt(S(1) + b*x**S(4)/a)/sqrt(a + b*x**S(4)), Int(S(1)/sqrt(S(1) + b*x**S(4)/a), x), x)
def With728(a, b, x):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
return Simp(S(3)**(S(3)/4)*x*sqrt((r**S(2)*x**S(4) - r*s*x**S(2) + s**S(2))/(r*x**S(2)*(S(1) + sqrt(S(3))) + s)**S(2))*(r*x**S(2) + s)*EllipticF(acos((r*x**S(2)*(S(1) - sqrt(S(3))) + s)/(r*x**S(2)*(S(1) + sqrt(S(3))) + s)), sqrt(S(3))/S(4) + S(1)/2)/(S(6)*s*sqrt(r*x**S(2)*(r*x**S(2) + s)/(r*x**S(2)*(S(1) + sqrt(S(3))) + s)**S(2))*sqrt(a + b*x**S(6))), x)
def replacement729(a, b, x):
return Dist(S(1)/2, Int((-x**S(2)*Rt(b/a, S(4)) + S(1))/sqrt(a + b*x**S(8)), x), x) + Dist(S(1)/2, Int((x**S(2)*Rt(b/a, S(4)) + S(1))/sqrt(a + b*x**S(8)), x), x)
def replacement730(a, b, x):
return -Dist(a, Int((a + b*x**S(2))**(S(-5)/4), x), x) + Simp(S(2)*x/(a + b*x**S(2))**(S(1)/4), x)
def replacement731(a, b, x):
return Simp(S(2)*EllipticE(asin(x*Rt(-b/a, S(2)))/S(2), S(2))/(a**(S(1)/4)*Rt(-b/a, S(2))), x)
def replacement732(a, b, x):
return Dist((S(1) + b*x**S(2)/a)**(S(1)/4)/(a + b*x**S(2))**(S(1)/4), Int((S(1) + b*x**S(2)/a)**(S(-1)/4), x), x)
def replacement733(a, b, x):
return Simp(S(2)*EllipticF(ArcTan(x*Rt(b/a, S(2)))/S(2), S(2))/(a**(S(3)/4)*Rt(b/a, S(2))), x)
def replacement734(a, b, x):
return Simp(S(2)*EllipticF(asin(x*Rt(-b/a, S(2)))/S(2), S(2))/(a**(S(3)/4)*Rt(-b/a, S(2))), x)
def replacement735(a, b, x):
return Dist((S(1) + b*x**S(2)/a)**(S(3)/4)/(a + b*x**S(2))**(S(3)/4), Int((S(1) + b*x**S(2)/a)**(S(-3)/4), x), x)
def replacement736(a, b, x):
return Dist(S(3)*sqrt(b*x**S(2))/(S(2)*b*x), Subst(Int(x/sqrt(-a + x**S(3)), x), x, (a + b*x**S(2))**(S(1)/3)), x)
def replacement737(a, b, x):
return Dist(S(3)*sqrt(b*x**S(2))/(S(2)*b*x), Subst(Int(S(1)/sqrt(-a + x**S(3)), x), x, (a + b*x**S(2))**(S(1)/3)), x)
def replacement738(a, b, x):
return Dist(x**S(3)*(a/(b*x**S(4)) + S(1))**(S(3)/4)/(a + b*x**S(4))**(S(3)/4), Int(S(1)/(x**S(3)*(a/(b*x**S(4)) + S(1))**(S(3)/4)), x), x)
def replacement739(a, b, x):
return -Dist(a/S(2), Int((a + b*x**S(2))**(S(-7)/6), x), x) + Simp(S(3)*x/(S(2)*(a + b*x**S(2))**(S(1)/6)), x)
def replacement740(a, b, n, p, x):
return Dist(a**(p + S(1)/n), Subst(Int((-b*x**n + S(1))**(-p + S(-1) - S(1)/n), x), x, x*(a + b*x**n)**(-S(1)/n)), x)
def replacement741(a, b, n, p, x):
return Dist((a/(a + b*x**n))**(p + S(1)/n)*(a + b*x**n)**(p + S(1)/n), Subst(Int((-b*x**n + S(1))**(-p + S(-1) - S(1)/n), x), x, x*(a + b*x**n)**(-S(1)/n)), x)
def replacement742(a, b, n, p, x):
return -Subst(Int((a + b*x**(-n))**p/x**S(2), x), x, S(1)/x)
def With743(a, b, n, p, x):
k = Denominator(n)
return Dist(k, Subst(Int(x**(k + S(-1))*(a + b*x**(k*n))**p, x), x, x**(S(1)/k)), x)
def replacement744(a, b, n, p, x):
return Int(ExpandIntegrand((a + b*x**n)**p, x), x)
def replacement745(a, b, n, p, x):
return Simp(a**p*x*Hypergeometric2F1(-p, S(1)/n, S(1) + S(1)/n, -b*x**n/a), x)
def replacement746(a, b, n, p, x):
return Dist(a**IntPart(p)*(S(1) + b*x**n/a)**(-FracPart(p))*(a + b*x**n)**FracPart(p), Int((S(1) + b*x**n/a)**p, x), x)
def replacement747(a, b, n, p, u, x):
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*x**n)**p, x), x, u), x)
def replacement748(a1, a2, b1, b2, n, p, x):
return Int((a1*a2 + b1*b2*x**(S(2)*n))**p, x)
def replacement749(a1, a2, b1, b2, n, p, x):
return Dist(S(2)*a1*a2*n*p/(S(2)*n*p + S(1)), Int((a1 + b1*x**n)**(p + S(-1))*(a2 + b2*x**n)**(p + S(-1)), x), x) + Simp(x*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p/(S(2)*n*p + S(1)), x)
def replacement750(a1, a2, b1, b2, n, p, x):
return Dist((S(2)*n*(p + S(1)) + S(1))/(S(2)*a1*a2*n*(p + S(1))), Int((a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1)), x), x) - Simp(x*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(S(2)*a1*a2*n*(p + S(1))), x)
def replacement751(a1, a2, b1, b2, n, p, x):
return -Subst(Int((a1 + b1*x**(-n))**p*(a2 + b2*x**(-n))**p/x**S(2), x), x, S(1)/x)
def With752(a1, a2, b1, b2, n, p, x):
k = Denominator(S(2)*n)
return Dist(k, Subst(Int(x**(k + S(-1))*(a1 + b1*x**(k*n))**p*(a2 + b2*x**(k*n))**p, x), x, x**(S(1)/k)), x)
def replacement753(a1, a2, b1, b2, n, p, x):
return Dist((a1 + b1*x**n)**FracPart(p)*(a2 + b2*x**n)**FracPart(p)*(a1*a2 + b1*b2*x**(S(2)*n))**(-FracPart(p)), Int((a1*a2 + b1*b2*x**(S(2)*n))**p, x), x)
def replacement754(a1, a2, b1, b2, c, m, n, p, x):
return Int((c*x)**m*(a1*a2 + b1*b2*x**(S(2)*n))**p, x)
def replacement755(b, c, m, n, p, x):
return Dist(b**(S(1) - (m + S(1))/n)*c**m/n, Subst(Int((b*x)**(p + S(-1) + (m + S(1))/n), x), x, x**n), x)
def replacement756(b, c, m, n, p, x):
return Dist(b**IntPart(p)*c**m*x**(-n*FracPart(p))*(b*x**n)**FracPart(p), Int(x**(m + n*p), x), x)
def replacement757(b, c, m, n, p, x):
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(b*x**n)**p, x), x)
def replacement758(a, b, m, n, p, x):
return Int(x**(m + n*p)*(a*x**(-n) + b)**p, x)
def replacement759(a, b, c, m, n, p, x):
return Simp((c*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*c*(m + S(1))), x)
def replacement760(a1, a2, b1, b2, c, m, n, p, x):
return Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(a1*a2*c*(m + S(1))), x)
def replacement761(a, b, m, n, p, x):
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b*x)**p, x), x, x**n), x)
def replacement762(a1, a2, b1, b2, m, n, p, x):
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a1 + b1*x)**p*(a2 + b2*x)**p, x), x, x**n), x)
def replacement763(a, b, c, m, n, p, x):
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a + b*x**n)**p, x), x)
def replacement764(a1, a2, b1, b2, c, m, n, p, x):
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x)
def replacement765(a, b, c, m, n, p, x):
return Int(ExpandIntegrand((c*x)**m*(a + b*x**n)**p, x), x)
def replacement766(a, b, m, n, p, x):
return -Dist(b*(m + n*(p + S(1)) + S(1))/(a*(m + S(1))), Int(x**(m + n)*(a + b*x**n)**p, x), x) + Simp(x**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*(m + S(1))), x)
def replacement767(a1, a2, b1, b2, m, n, p, x):
return -Dist(b1*b2*(m + S(2)*n*(p + S(1)) + S(1))/(a1*a2*(m + S(1))), Int(x**(m + S(2)*n)*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x) + Simp(x**(m + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(a1*a2*(m + S(1))), x)
def replacement768(a, b, c, m, n, p, x):
return Dist((m + n*(p + S(1)) + S(1))/(a*n*(p + S(1))), Int((c*x)**m*(a + b*x**n)**(p + S(1)), x), x) - Simp((c*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*c*n*(p + S(1))), x)
def replacement769(a1, a2, b1, b2, c, m, n, p, x):
return Dist((m + S(2)*n*(p + S(1)) + S(1))/(S(2)*a1*a2*n*(p + S(1))), Int((c*x)**m*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1)), x), x) - Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(S(2)*a1*a2*c*n*(p + S(1))), x)
def With770(a, b, m, n, p, x):
if isinstance(x, (int, Integer, float, Float)):
return False
k = GCD(m + S(1), n)
if Unequal(k, S(1)):
return True
return False
def replacement770(a, b, m, n, p, x):
k = GCD(m + S(1), n)
return Dist(S(1)/k, Subst(Int(x**(S(-1) + (m + S(1))/k)*(a + b*x**(n/k))**p, x), x, x**k), x)
def With771(a1, a2, b1, b2, m, n, p, x):
if isinstance(x, (int, Integer, float, Float)):
return False
k = GCD(m + S(1), S(2)*n)
if Unequal(k, S(1)):
return True
return False
def replacement771(a1, a2, b1, b2, m, n, p, x):
k = GCD(m + S(1), S(2)*n)
return Dist(S(1)/k, Subst(Int(x**(S(-1) + (m + S(1))/k)*(a1 + b1*x**(n/k))**p*(a2 + b2*x**(n/k))**p, x), x, x**k), x)
def replacement772(a, b, c, m, n, p, x):
return -Dist(b*c**(-n)*n*p/(m + S(1)), Int((c*x)**(m + n)*(a + b*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a + b*x**n)**p/(c*(m + S(1))), x)
def replacement773(a1, a2, b1, b2, c, m, n, p, x):
return Dist(S(2)*a1*a2*n*p/(m + S(2)*n*p + S(1)), Int((c*x)**m*(a1 + b1*x**n)**(p + S(-1))*(a2 + b2*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p/(c*(m + S(2)*n*p + S(1))), x)
def replacement774(a, b, c, m, n, p, x):
return Dist(a*n*p/(m + n*p + S(1)), Int((c*x)**m*(a + b*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a + b*x**n)**p/(c*(m + n*p + S(1))), x)
def replacement775(a, b, x):
return Dist(x*(a/(b*x**S(4)) + S(1))**(S(1)/4)/(b*(a + b*x**S(4))**(S(1)/4)), Int(S(1)/(x**S(3)*(a/(b*x**S(4)) + S(1))**(S(5)/4)), x), x)
def replacement776(a, b, m, x):
return -Dist(a*(m + S(-3))/(b*(m + S(-4))), Int(x**(m + S(-4))/(a + b*x**S(4))**(S(5)/4), x), x) + Simp(x**(m + S(-3))/(b*(a + b*x**S(4))**(S(1)/4)*(m + S(-4))), x)
def replacement777(a, b, m, x):
return -Dist(b*m/(a*(m + S(1))), Int(x**(m + S(4))/(a + b*x**S(4))**(S(5)/4), x), x) + Simp(x**(m + S(1))/(a*(a + b*x**S(4))**(S(1)/4)*(m + S(1))), x)
def replacement778(a, b, c, x):
return Dist(sqrt(c*x)*(a/(b*x**S(2)) + S(1))**(S(1)/4)/(b*(a + b*x**S(2))**(S(1)/4)), Int(S(1)/(x**S(2)*(a/(b*x**S(2)) + S(1))**(S(5)/4)), x), x)
def replacement779(a, b, c, m, x):
return -Dist(S(2)*a*c**S(2)*(m + S(-1))/(b*(S(2)*m + S(-3))), Int((c*x)**(m + S(-2))/(a + b*x**S(2))**(S(5)/4), x), x) + Simp(S(2)*c*(c*x)**(m + S(-1))/(b*(a + b*x**S(2))**(S(1)/4)*(S(2)*m + S(-3))), x)
def replacement780(a, b, c, m, x):
return -Dist(b*(S(2)*m + S(1))/(S(2)*a*c**S(2)*(m + S(1))), Int((c*x)**(m + S(2))/(a + b*x**S(2))**(S(5)/4), x), x) + Simp((c*x)**(m + S(1))/(a*c*(a + b*x**S(2))**(S(1)/4)*(m + S(1))), x)
def replacement781(a, b, x):
return -Dist(S(1)/b, Int(S(1)/(x**S(2)*(a + b*x**S(4))**(S(1)/4)), x), x) - Simp(S(1)/(b*x*(a + b*x**S(4))**(S(1)/4)), x)
def replacement782(a, b, c, m, n, p, x):
return -Dist(c**n*(m - n + S(1))/(b*n*(p + S(1))), Int((c*x)**(m - n)*(a + b*x**n)**(p + S(1)), x), x) + Simp(c**(n + S(-1))*(c*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x)
def replacement783(a1, a2, b1, b2, c, m, n, p, x):
return -Dist(c**(S(2)*n)*(m - S(2)*n + S(1))/(S(2)*b1*b2*n*(p + S(1))), Int((c*x)**(m - S(2)*n)*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1)), x), x) + Simp(c**(S(2)*n + S(-1))*(c*x)**(m - S(2)*n + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(S(2)*b1*b2*n*(p + S(1))), x)
def replacement784(a, b, c, m, n, p, x):
return Dist((m + n*(p + S(1)) + S(1))/(a*n*(p + S(1))), Int((c*x)**m*(a + b*x**n)**(p + S(1)), x), x) - Simp((c*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*c*n*(p + S(1))), x)
def replacement785(a1, a2, b1, b2, c, m, n, p, x):
return Dist((m + S(2)*n*(p + S(1)) + S(1))/(S(2)*a1*a2*n*(p + S(1))), Int((c*x)**m*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1)), x), x) - Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(S(2)*a1*a2*c*n*(p + S(1))), x)
def replacement786(a, b, x):
return Dist(S(1)/(S(3)*Rt(a, S(3))*Rt(b, S(3))), Int((x*Rt(b, S(3)) + Rt(a, S(3)))/(x**S(2)*Rt(b, S(3))**S(2) - x*Rt(a, S(3))*Rt(b, S(3)) + Rt(a, S(3))**S(2)), x), x) - Dist(S(1)/(S(3)*Rt(a, S(3))*Rt(b, S(3))), Int(S(1)/(x*Rt(b, S(3)) + Rt(a, S(3))), x), x)
def With787(a, b, m, n, x):
r = Numerator(Rt(a/b, n))
s = Denominator(Rt(a/b, n))
k = Symbol('k')
u = Symbol('u')
u = Int((r*cos(Pi*m*(S(2)*k + S(-1))/n) - s*x*cos(Pi*(S(2)*k + S(-1))*(m + S(1))/n))/(r**S(2) - S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x)
u = Int((r*cos(Pi*m*(2*k - 1)/n) - s*x*cos(Pi*(2*k - 1)*(m + 1)/n))/(r**2 - 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x)
return Simp(Dist(2*r**(m + 1)*s**(-m)/(a*n), Sum_doit(u, List(k, 1, n/2 - 1/2)), x) - s**(-m)*(-r)**(m + 1)*Int(1/(r + s*x), x)/(a*n), x)
def With788(a, b, m, n, x):
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
k = Symbol('k')
u = Symbol('u')
u = Int((r*cos(Pi*m*(S(2)*k + S(-1))/n) + s*x*cos(Pi*(S(2)*k + S(-1))*(m + S(1))/n))/(r**S(2) + S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x)
u = Int((r*cos(Pi*m*(2*k - 1)/n) + s*x*cos(Pi*(2*k - 1)*(m + 1)/n))/(r**2 + 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x)
return Simp(-Dist(2*s**(-m)*(-r)**(m + 1)/(a*n), Sum_doit(u, List(k, 1, n/2 - 1/2)), x) + r**(m + 1)*s**(-m)*Int(1/(r - s*x), x)/(a*n), x)
def With789(a, b, m, n, x):
r = Numerator(Rt(a/b, n))
s = Denominator(Rt(a/b, n))
k = Symbol('k')
u = Symbol('u')
u = Int((r*cos(Pi*m*(S(2)*k + S(-1))/n) - s*x*cos(Pi*(S(2)*k + S(-1))*(m + S(1))/n))/(r**S(2) - S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x) + Int((r*cos(Pi*m*(S(2)*k + S(-1))/n) + s*x*cos(Pi*(S(2)*k + S(-1))*(m + S(1))/n))/(r**S(2) + S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x)
u = Int((r*cos(Pi*m*(2*k - 1)/n) - s*x*cos(Pi*(2*k - 1)*(m + 1)/n))/(r**2 - 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x) + Int((r*cos(Pi*m*(2*k - 1)/n) + s*x*cos(Pi*(2*k - 1)*(m + 1)/n))/(r**2 + 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x)
return Simp(2*(-1)**(m/2)*r**(m + 2)*s**(-m)*Int(1/(r**2 + s**2*x**2), x)/(a*n) + Dist(2*r**(m + 1)*s**(-m)/(a*n), Sum_doit(u, List(k, 1, n/4 - 1/2)), x), x)
def With790(a, b, m, n, x):
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
k = Symbol('k')
u = Symbol('u')
u = Int((r*cos(S(2)*Pi*k*m/n) - s*x*cos(S(2)*Pi*k*(m + S(1))/n))/(r**S(2) - S(2)*r*s*x*cos(S(2)*Pi*k/n) + s**S(2)*x**S(2)), x) + Int((r*cos(S(2)*Pi*k*m/n) + s*x*cos(S(2)*Pi*k*(m + S(1))/n))/(r**S(2) + S(2)*r*s*x*cos(S(2)*Pi*k/n) + s**S(2)*x**S(2)), x)
u = Int((r*cos(2*Pi*k*m/n) - s*x*cos(2*Pi*k*(m + 1)/n))/(r**2 - 2*r*s*x*cos(2*Pi*k/n) + s**2*x**2), x) + Int((r*cos(2*Pi*k*m/n) + s*x*cos(2*Pi*k*(m + 1)/n))/(r**2 + 2*r*s*x*cos(2*Pi*k/n) + s**2*x**2), x)
return Simp(Dist(2*r**(m + 1)*s**(-m)/(a*n), Sum_doit(u, List(k, 1, n/4 - 1/2)), x) + 2*r**(m + 2)*s**(-m)*Int(1/(r**2 - s**2*x**2), x)/(a*n), x)
def With791(a, b, x):
r = Numerator(Rt(a/b, S(2)))
s = Denominator(Rt(a/b, S(2)))
return -Dist(S(1)/(S(2)*s), Int((r - s*x**S(2))/(a + b*x**S(4)), x), x) + Dist(S(1)/(S(2)*s), Int((r + s*x**S(2))/(a + b*x**S(4)), x), x)
def With792(a, b, x):
r = Numerator(Rt(-a/b, S(2)))
s = Denominator(Rt(-a/b, S(2)))
return -Dist(s/(S(2)*b), Int(S(1)/(r - s*x**S(2)), x), x) + Dist(s/(S(2)*b), Int(S(1)/(r + s*x**S(2)), x), x)
def With793(a, b, m, n, x):
r = Numerator(Rt(a/b, S(4)))
s = Denominator(Rt(a/b, S(4)))
return Dist(sqrt(S(2))*s**S(3)/(S(4)*b*r), Int(x**(m - n/S(4))/(r**S(2) - sqrt(S(2))*r*s*x**(n/S(4)) + s**S(2)*x**(n/S(2))), x), x) - Dist(sqrt(S(2))*s**S(3)/(S(4)*b*r), Int(x**(m - n/S(4))/(r**S(2) + sqrt(S(2))*r*s*x**(n/S(4)) + s**S(2)*x**(n/S(2))), x), x)
def With794(a, b, m, n, x):
r = Numerator(Rt(-a/b, S(2)))
s = Denominator(Rt(-a/b, S(2)))
return Dist(r/(S(2)*a), Int(x**m/(r - s*x**(n/S(2))), x), x) + Dist(r/(S(2)*a), Int(x**m/(r + s*x**(n/S(2))), x), x)
def With795(a, b, m, n, x):
r = Numerator(Rt(-a/b, S(2)))
s = Denominator(Rt(-a/b, S(2)))
return -Dist(s/(S(2)*b), Int(x**(m - n/S(2))/(r - s*x**(n/S(2))), x), x) + Dist(s/(S(2)*b), Int(x**(m - n/S(2))/(r + s*x**(n/S(2))), x), x)
def replacement796(a, b, m, n, x):
return Int(PolynomialDivide(x**m, a + b*x**n, x), x)
def With797(a, b, x):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
return Dist(S(1)/r, Int((r*x + s*(S(1) - sqrt(S(3))))/sqrt(a + b*x**S(3)), x), x) + Dist(sqrt(S(2))*s/(r*sqrt(sqrt(S(3)) + S(2))), Int(S(1)/sqrt(a + b*x**S(3)), x), x)
def With798(a, b, x):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
return Dist(S(1)/r, Int((r*x + s*(S(1) + sqrt(S(3))))/sqrt(a + b*x**S(3)), x), x) - Dist(sqrt(S(2))*s/(r*sqrt(S(2) - sqrt(S(3)))), Int(S(1)/sqrt(a + b*x**S(3)), x), x)
def With799(a, b, x):
q = Rt(b/a, S(2))
return -Dist(S(1)/q, Int((-q*x**S(2) + S(1))/sqrt(a + b*x**S(4)), x), x) + Dist(S(1)/q, Int(S(1)/sqrt(a + b*x**S(4)), x), x)
def With800(a, b, x):
q = Rt(-b/a, S(2))
return -Dist(S(1)/q, Int((-q*x**S(2) + S(1))/sqrt(a + b*x**S(4)), x), x) + Dist(S(1)/q, Int(S(1)/sqrt(a + b*x**S(4)), x), x)
def With801(a, b, x):
q = Rt(-b/a, S(2))
return Dist(S(1)/q, Int((q*x**S(2) + S(1))/sqrt(a + b*x**S(4)), x), x) - Dist(S(1)/q, Int(S(1)/sqrt(a + b*x**S(4)), x), x)
def With802(a, b, x):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
return -Dist(S(1)/(S(2)*r**S(2)), Int((-S(2)*r**S(2)*x**S(4) + s**S(2)*(S(-1) + sqrt(S(3))))/sqrt(a + b*x**S(6)), x), x) + Dist(s**S(2)*(S(-1) + sqrt(S(3)))/(S(2)*r**S(2)), Int(S(1)/sqrt(a + b*x**S(6)), x), x)
def replacement803(a, b, x):
return -Dist(S(1)/(S(2)*Rt(b/a, S(4))), Int((-x**S(2)*Rt(b/a, S(4)) + S(1))/sqrt(a + b*x**S(8)), x), x) + Dist(S(1)/(S(2)*Rt(b/a, S(4))), Int((x**S(2)*Rt(b/a, S(4)) + S(1))/sqrt(a + b*x**S(8)), x), x)
def replacement804(a, b, x):
return -Dist(a/S(2), Int(x**S(2)/(a + b*x**S(4))**(S(5)/4), x), x) + Simp(x**S(3)/(S(2)*(a + b*x**S(4))**(S(1)/4)), x)
def replacement805(a, b, x):
return Dist(a/(S(2)*b), Int(S(1)/(x**S(2)*(a + b*x**S(4))**(S(1)/4)), x), x) + Simp((a + b*x**S(4))**(S(3)/4)/(S(2)*b*x), x)
def replacement806(a, b, x):
return -Dist(b, Int(x**S(2)/(a + b*x**S(4))**(S(5)/4), x), x) - Simp(S(1)/(x*(a + b*x**S(4))**(S(1)/4)), x)
def replacement807(a, b, x):
return Dist(x*(a/(b*x**S(4)) + S(1))**(S(1)/4)/(a + b*x**S(4))**(S(1)/4), Int(S(1)/(x**S(3)*(a/(b*x**S(4)) + S(1))**(S(1)/4)), x), x)
def replacement808(a, b, c, x):
return -Dist(a/S(2), Int(sqrt(c*x)/(a + b*x**S(2))**(S(5)/4), x), x) + Simp(x*sqrt(c*x)/(a + b*x**S(2))**(S(1)/4), x)
def replacement809(a, b, c, x):
return Dist(a*c**S(2)/(S(2)*b), Int(S(1)/((c*x)**(S(3)/2)*(a + b*x**S(2))**(S(1)/4)), x), x) + Simp(c*(a + b*x**S(2))**(S(3)/4)/(b*sqrt(c*x)), x)
def replacement810(a, b, c, x):
return -Dist(b/c**S(2), Int(sqrt(c*x)/(a + b*x**S(2))**(S(5)/4), x), x) + Simp(-S(2)/(c*sqrt(c*x)*(a + b*x**S(2))**(S(1)/4)), x)
def replacement811(a, b, c, x):
return Dist(sqrt(c*x)*(a/(b*x**S(2)) + S(1))**(S(1)/4)/(c**S(2)*(a + b*x**S(2))**(S(1)/4)), Int(S(1)/(x**S(2)*(a/(b*x**S(2)) + S(1))**(S(1)/4)), x), x)
def replacement812(a, b, c, m, n, p, x):
return -Dist(a*c**n*(m - n + S(1))/(b*(m + n*p + S(1))), Int((c*x)**(m - n)*(a + b*x**n)**p, x), x) + Simp(c**(n + S(-1))*(c*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))/(b*(m + n*p + S(1))), x)
def replacement813(a, b, c, m, n, p, x):
return -Dist(a*c**n*(m - n + S(1))/(b*(m + n*p + S(1))), Int((c*x)**(m - n)*(a + b*x**n)**p, x), x) + Simp(c**(n + S(-1))*(c*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))/(b*(m + n*p + S(1))), x)
def replacement814(a1, a2, b1, b2, c, m, n, p, x):
return -Dist(a1*a2*c**(S(2)*n)*(m - S(2)*n + S(1))/(b1*b2*(m + S(2)*n*p + S(1))), Int((c*x)**(m - S(2)*n)*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x) + Simp(c**(S(2)*n + S(-1))*(c*x)**(m - S(2)*n + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(b1*b2*(m + S(2)*n*p + S(1))), x)
def replacement815(a1, a2, b1, b2, c, m, n, p, x):
return -Dist(a1*a2*c**(S(2)*n)*(m - S(2)*n + S(1))/(b1*b2*(m + S(2)*n*p + S(1))), Int((c*x)**(m - S(2)*n)*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x) + Simp(c**(S(2)*n + S(-1))*(c*x)**(m - S(2)*n + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(b1*b2*(m + S(2)*n*p + S(1))), x)
def replacement816(a, b, c, m, n, p, x):
return -Dist(b*c**(-n)*(m + n*(p + S(1)) + S(1))/(a*(m + S(1))), Int((c*x)**(m + n)*(a + b*x**n)**p, x), x) + Simp((c*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*c*(m + S(1))), x)
def replacement817(a, b, c, m, n, p, x):
return -Dist(b*c**(-n)*(m + n*(p + S(1)) + S(1))/(a*(m + S(1))), Int((c*x)**(m + n)*(a + b*x**n)**p, x), x) + Simp((c*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*c*(m + S(1))), x)
def replacement818(a1, a2, b1, b2, c, m, n, p, x):
return -Dist(b1*b2*c**(-S(2)*n)*(m + S(2)*n*(p + S(1)) + S(1))/(a1*a2*(m + S(1))), Int((c*x)**(m + S(2)*n)*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x) + Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(a1*a2*c*(m + S(1))), x)
def replacement819(a1, a2, b1, b2, c, m, n, p, x):
return -Dist(b1*b2*c**(-S(2)*n)*(m + S(2)*n*(p + S(1)) + S(1))/(a1*a2*(m + S(1))), Int((c*x)**(m + S(2)*n)*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x) + Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(a1*a2*c*(m + S(1))), x)
def With820(a, b, c, m, n, p, x):
k = Denominator(m)
return Dist(k/c, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*c**(-n)*x**(k*n))**p, x), x, (c*x)**(S(1)/k)), x)
def With821(a1, a2, b1, b2, c, m, n, p, x):
k = Denominator(m)
return Dist(k/c, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a1 + b1*c**(-n)*x**(k*n))**p*(a2 + b2*c**(-n)*x**(k*n))**p, x), x, (c*x)**(S(1)/k)), x)
def replacement822(a, b, m, n, p, x):
return Dist(a**(p + (m + S(1))/n), Subst(Int(x**m*(-b*x**n + S(1))**(-p + S(-1) - (m + S(1))/n), x), x, x*(a + b*x**n)**(-S(1)/n)), x)
def replacement823(a1, a2, b1, b2, m, n, p, x):
return Dist((a1*a2)**(p + (m + S(1))/(S(2)*n)), Subst(Int(x**m*(-b1*x**n + S(1))**(-p + S(-1) - (m + S(1))/(S(2)*n))*(-b2*x**n + S(1))**(-p + S(-1) - (m + S(1))/(S(2)*n)), x), x, x*(a1 + b1*x**n)**(-S(1)/(S(2)*n))*(a2 + b2*x**n)**(-S(1)/(S(2)*n))), x)
def replacement824(a, b, m, n, p, x):
return Dist((a/(a + b*x**n))**(p + (m + S(1))/n)*(a + b*x**n)**(p + (m + S(1))/n), Subst(Int(x**m*(-b*x**n + S(1))**(-p + S(-1) - (m + S(1))/n), x), x, x*(a + b*x**n)**(-S(1)/n)), x)
def replacement825(a1, a2, b1, b2, m, n, p, x):
return Dist((a1/(a1 + b1*x**n))**(p + (m + S(1))/(S(2)*n))*(a2/(a2 + b2*x**n))**(p + (m + S(1))/(S(2)*n))*(a1 + b1*x**n)**(p + (m + S(1))/(S(2)*n))*(a2 + b2*x**n)**(p + (m + S(1))/(S(2)*n)), Subst(Int(x**m*(-b1*x**n + S(1))**(-p + S(-1) - (m + S(1))/(S(2)*n))*(-b2*x**n + S(1))**(-p + S(-1) - (m + S(1))/(S(2)*n)), x), x, x*(a1 + b1*x**n)**(-S(1)/(S(2)*n))*(a2 + b2*x**n)**(-S(1)/(S(2)*n))), x)
def replacement826(a, b, m, n, p, x):
return -Subst(Int(x**(-m + S(-2))*(a + b*x**(-n))**p, x), x, S(1)/x)
def replacement827(a1, a2, b1, b2, m, n, p, x):
return -Subst(Int(x**(-m + S(-2))*(a1 + b1*x**(-n))**p*(a2 + b2*x**(-n))**p, x), x, S(1)/x)
def With828(a, b, c, m, n, p, x):
k = Denominator(m)
return -Dist(k/c, Subst(Int(x**(-k*(m + S(1)) + S(-1))*(a + b*c**(-n)*x**(-k*n))**p, x), x, (c*x)**(-S(1)/k)), x)
def With829(a1, a2, b1, b2, c, m, n, p, x):
k = Denominator(m)
return -Dist(k/c, Subst(Int(x**(-k*(m + S(1)) + S(-1))*(a1 + b1*c**(-n)*x**(-k*n))**p*(a2 + b2*c**(-n)*x**(-k*n))**p, x), x, (c*x)**(-S(1)/k)), x)
def replacement830(a, b, c, m, n, p, x):
return -Dist((c*x)**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(a + b*x**(-n))**p, x), x, S(1)/x), x)
def replacement831(a1, a2, b1, b2, c, m, n, p, x):
return -Dist((c*x)**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(a1 + b1*x**(-n))**p*(a2 + b2*x**(-n))**p, x), x, S(1)/x), x)
def With832(a, b, m, n, p, x):
k = Denominator(n)
return Dist(k, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*x**(k*n))**p, x), x, x**(S(1)/k)), x)
def With833(a1, a2, b1, b2, m, n, p, x):
k = Denominator(S(2)*n)
return Dist(k, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a1 + b1*x**(k*n))**p*(a2 + b2*x**(k*n))**p, x), x, x**(S(1)/k)), x)
def replacement834(a, b, c, m, n, p, x):
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a + b*x**n)**p, x), x)
def replacement835(a1, a2, b1, b2, c, m, n, p, x):
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x)
def replacement836(a, b, m, n, p, x):
return Dist(S(1)/(m + S(1)), Subst(Int((a + b*x**(n/(m + S(1))))**p, x), x, x**(m + S(1))), x)
def replacement837(a1, a2, b1, b2, m, n, p, x):
return Dist(S(1)/(m + S(1)), Subst(Int((a1 + b1*x**(n/(m + S(1))))**p*(a2 + b2*x**(n/(m + S(1))))**p, x), x, x**(m + S(1))), x)
def replacement838(a, b, c, m, n, p, x):
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a + b*x**n)**p, x), x)
def replacement839(a1, a2, b1, b2, c, m, n, p, x):
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x)
def replacement840(a, b, m, n, p, x):
return -Dist(b*n*p/(m + S(1)), Int(x**(m + n)*(a + b*x**n)**(p + S(-1)), x), x) + Simp(x**(m + S(1))*(a + b*x**n)**p/(m + S(1)), x)
def replacement841(a1, a2, b1, b2, m, n, p, x):
return -Dist(S(2)*b1*b2*n*p/(m + S(1)), Int(x**(m + n)*(a1 + b1*x**n)**(p + S(-1))*(a2 + b2*x**n)**(p + S(-1)), x), x) + Simp(x**(m + S(1))*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p/(m + S(1)), x)
def replacement842(a, b, c, m, n, p, x):
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a + b*x**n)**p, x), x)
def replacement843(a1, a2, b1, b2, c, m, n, p, x):
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x)
def replacement844(a, b, c, m, n, p, x):
return Dist(a*n*p/(m + n*p + S(1)), Int((c*x)**m*(a + b*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a + b*x**n)**p/(c*(m + n*p + S(1))), x)
def replacement845(a1, a2, b1, b2, c, m, n, p, x):
return Dist(S(2)*a1*a2*n*p/(m + S(2)*n*p + S(1)), Int((c*x)**m*(a1 + b1*x**n)**(p + S(-1))*(a2 + b2*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p/(c*(m + S(2)*n*p + S(1))), x)
def With846(a, b, m, n, p, x):
k = Denominator(p)
return Dist(a**(p + (m + S(1))/n)*k/n, Subst(Int(x**(k*(m + S(1))/n + S(-1))*(-b*x**k + S(1))**(-p + S(-1) - (m + S(1))/n), x), x, x**(n/k)*(a + b*x**n)**(-S(1)/k)), x)
def With847(a1, a2, b1, b2, m, n, p, x):
k = Denominator(p)
return Dist(k*(a1*a2)**(p + (m + S(1))/(S(2)*n))/(S(2)*n), Subst(Int(x**(k*(m + S(1))/(S(2)*n) + S(-1))*(-b1*x**k + S(1))**(-p + S(-1) - (m + S(1))/(S(2)*n))*(-b2*x**k + S(1))**(-p + S(-1) - (m + S(1))/(S(2)*n)), x), x, x**(S(2)*n/k)*(a1 + b1*x**n)**(-S(1)/k)*(a2 + b2*x**n)**(-S(1)/k)), x)
def replacement848(a, b, c, m, n, p, x):
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a + b*x**n)**p, x), x)
def replacement849(a1, a2, b1, b2, c, m, n, p, x):
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x)
def replacement850(a, b, c, m, n, p, x):
return Dist((m + n*(p + S(1)) + S(1))/(a*n*(p + S(1))), Int((c*x)**m*(a + b*x**n)**(p + S(1)), x), x) - Simp((c*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*c*n*(p + S(1))), x)
def replacement851(a1, a2, b1, b2, c, m, n, p, x):
return Dist((m + S(2)*n*(p + S(1)) + S(1))/(S(2)*a1*a2*n*(p + S(1))), Int((c*x)**m*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1)), x), x) - Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(S(2)*a1*a2*c*n*(p + S(1))), x)
def With852(a, b, m, n, x):
mn = m - n
return -Dist(a/b, Int(x**mn/(a + b*x**n), x), x) + Simp(x**(mn + S(1))/(b*(mn + S(1))), x)
def replacement853(a, b, m, n, x):
return -Dist(b/a, Int(x**(m + n)/(a + b*x**n), x), x) + Simp(x**(m + S(1))/(a*(m + S(1))), x)
def replacement854(a, b, c, m, n, x):
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m/(a + b*x**n), x), x)
def replacement855(a, b, c, m, n, p, x):
return Simp(a**p*(c*x)**(m + S(1))*Hypergeometric2F1(-p, (m + S(1))/n, S(1) + (m + S(1))/n, -b*x**n/a)/(c*(m + S(1))), x)
def replacement856(a, b, c, m, n, p, x):
return Dist(a**IntPart(p)*(S(1) + b*x**n/a)**(-FracPart(p))*(a + b*x**n)**FracPart(p), Int((c*x)**m*(S(1) + b*x**n/a)**p, x), x)
def replacement857(a, b, m, n, p, v, x):
return Dist(Coefficient(v, x, S(1))**(-m + S(-1)), Subst(Int(SimplifyIntegrand((a + b*x**n)**p*(x - Coefficient(v, x, S(0)))**m, x), x), x, v), x)
def replacement858(a, b, m, n, p, u, v, x):
return Dist(u**m*v**(-m)/Coefficient(v, x, S(1)), Subst(Int(x**m*(a + b*x**n)**p, x), x, v), x)
def replacement859(a1, a2, b1, b2, c, m, n, p, x):
return Dist((a1 + b1*x**n)**FracPart(p)*(a2 + b2*x**n)**FracPart(p)*(a1*a2 + b1*b2*x**(S(2)*n))**(-FracPart(p)), Int((c*x)**m*(a1*a2 + b1*b2*x**(S(2)*n))**p, x), x)
def replacement860(a, b, c, d, n, p, q, x):
return Int(ExpandIntegrand((a + b*x**n)**p*(c + d*x**n)**q, x), x)
def replacement861(a, b, c, d, n, p, q, x):
return Int(x**(n*(p + q))*(a*x**(-n) + b)**p*(c*x**(-n) + d)**q, x)
def replacement862(a, b, c, d, n, p, q, x):
return -Subst(Int((a + b*x**(-n))**p*(c + d*x**(-n))**q/x**S(2), x), x, S(1)/x)
def With863(a, b, c, d, n, p, q, x):
g = Denominator(n)
return Dist(g, Subst(Int(x**(g + S(-1))*(a + b*x**(g*n))**p*(c + d*x**(g*n))**q, x), x, x**(S(1)/g)), x)
def replacement864(a, b, c, d, n, p, x):
return Subst(Int(S(1)/(c - x**n*(-a*d + b*c)), x), x, x*(a + b*x**n)**(-S(1)/n))
def replacement865(a, b, c, d, n, p, q, x):
return -Dist(c*q/(a*(p + S(1))), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1)), x), x) - Simp(x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(a*n*(p + S(1))), x)
def replacement866(a, b, c, d, n, p, q, x):
return Simp(a**p*c**(-p + S(-1))*x*(c + d*x**n)**(-S(1)/n)*Hypergeometric2F1(S(1)/n, -p, S(1) + S(1)/n, -x**n*(-a*d + b*c)/(a*(c + d*x**n))), x)
def replacement867(a, b, c, d, n, p, q, x):
return Simp(x*(c*(a + b*x**n)/(a*(c + d*x**n)))**(-p)*(a + b*x**n)**p*(c + d*x**n)**(-p - S(1)/n)*Hypergeometric2F1(S(1)/n, -p, S(1) + S(1)/n, -x**n*(-a*d + b*c)/(a*(c + d*x**n)))/c, x)
def replacement868(a, b, c, d, n, p, q, x):
return Simp(x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(a*c), x)
def replacement869(a, b, c, d, n, p, q, x):
return Dist((b*c + n*(p + S(1))*(-a*d + b*c))/(a*n*(p + S(1))*(-a*d + b*c)), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**q, x), x) - Simp(b*x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(a*n*(p + S(1))*(-a*d + b*c)), x)
def replacement870(a, b, c, d, n, p, x):
return Simp(c*x*(a + b*x**n)**(p + S(1))/a, x)
def replacement871(a, b, c, d, n, p, x):
return -Dist((a*d - b*c*(n*(p + S(1)) + S(1)))/(a*b*n*(p + S(1))), Int((a + b*x**n)**(p + S(1)), x), x) - Simp(x*(a + b*x**n)**(p + S(1))*(-a*d + b*c)/(a*b*n*(p + S(1))), x)
def replacement872(a, b, c, d, n, x):
return -Dist((-a*d + b*c)/a, Int(S(1)/(a*x**(-n) + b), x), x) + Simp(c*x/a, x)
def replacement873(a, b, c, d, n, p, x):
return -Dist((a*d - b*c*(n*(p + S(1)) + S(1)))/(b*(n*(p + S(1)) + S(1))), Int((a + b*x**n)**p, x), x) + Simp(d*x*(a + b*x**n)**(p + S(1))/(b*(n*(p + S(1)) + S(1))), x)
def replacement874(a, b, c, d, n, p, q, x):
return Int(PolynomialDivide((a + b*x**n)**p, (c + d*x**n)**(-q), x), x)
def replacement875(a, b, c, d, n, x):
return Dist(b/(-a*d + b*c), Int(S(1)/(a + b*x**n), x), x) - Dist(d/(-a*d + b*c), Int(S(1)/(c + d*x**n), x), x)
def replacement876(a, b, c, d, x):
return Dist(sqrt(S(3))/(S(2)*c), Int(S(1)/((a + b*x**S(2))**(S(1)/3)*(-x*Rt(b/a, S(2)) + sqrt(S(3)))), x), x) + Dist(sqrt(S(3))/(S(2)*c), Int(S(1)/((a + b*x**S(2))**(S(1)/3)*(x*Rt(b/a, S(2)) + sqrt(S(3)))), x), x)
def replacement877(a, b, c, d, x):
return Dist(S(1)/6, Int((-x*Rt(-b/a, S(2)) + S(3))/((a + b*x**S(2))**(S(1)/3)*(c + d*x**S(2))), x), x) + Dist(S(1)/6, Int((x*Rt(-b/a, S(2)) + S(3))/((a + b*x**S(2))**(S(1)/3)*(c + d*x**S(2))), x), x)
def replacement878(a, b, c, d, x):
return Dist(b/d, Int((a + b*x**S(2))**(S(-1)/3), x), x) - Dist((-a*d + b*c)/d, Int(S(1)/((a + b*x**S(2))**(S(1)/3)*(c + d*x**S(2))), x), x)
def replacement879(a, b, c, d, x):
return Dist(sqrt(-b*x**S(2)/a)/(S(2)*x), Subst(Int(S(1)/(sqrt(-b*x/a)*(a + b*x)**(S(1)/4)*(c + d*x)), x), x, x**S(2)), x)
def replacement880(a, b, c, d, x):
return Dist(sqrt(-b*x**S(2)/a)/(S(2)*x), Subst(Int(S(1)/(sqrt(-b*x/a)*(a + b*x)**(S(3)/4)*(c + d*x)), x), x, x**S(2)), x)
def replacement881(a, b, c, d, p, x):
return Dist(b/d, Int((a + b*x**S(2))**(p + S(-1)), x), x) - Dist((-a*d + b*c)/d, Int((a + b*x**S(2))**(p + S(-1))/(c + d*x**S(2)), x), x)
def replacement882(a, b, c, d, p, x):
return Dist(b/(-a*d + b*c), Int((a + b*x**S(2))**p, x), x) - Dist(d/(-a*d + b*c), Int((a + b*x**S(2))**(p + S(1))/(c + d*x**S(2)), x), x)
def replacement883(a, b, c, d, x):
return Dist(a/c, Subst(Int(S(1)/(-S(4)*a*b*x**S(4) + S(1)), x), x, x/sqrt(a + b*x**S(4))), x)
def With884(a, b, c, d, x):
q = Rt(-a*b, S(4))
return Simp(a*ArcTan(q*x*(a + q**S(2)*x**S(2))/(a*sqrt(a + b*x**S(4))))/(S(2)*c*q), x) + Simp(a*atanh(q*x*(a - q**S(2)*x**S(2))/(a*sqrt(a + b*x**S(4))))/(S(2)*c*q), x)
def replacement885(a, b, c, d, x):
return Dist(b/d, Int(S(1)/sqrt(a + b*x**S(4)), x), x) - Dist((-a*d + b*c)/d, Int(S(1)/(sqrt(a + b*x**S(4))*(c + d*x**S(4))), x), x)
def replacement886(a, b, c, d, x):
return Dist(sqrt(a/(a + b*x**S(4)))*sqrt(a + b*x**S(4)), Subst(Int(S(1)/((c - x**S(4)*(-a*d + b*c))*sqrt(-b*x**S(4) + S(1))), x), x, x/(a + b*x**S(4))**(S(1)/4)), x)
def replacement887(a, b, c, d, p, x):
return Dist(b/d, Int((a + b*x**S(4))**(p + S(-1)), x), x) - Dist((-a*d + b*c)/d, Int((a + b*x**S(4))**(p + S(-1))/(c + d*x**S(4)), x), x)
def replacement888(a, b, c, d, x):
return Dist(S(1)/(S(2)*c), Int(S(1)/(sqrt(a + b*x**S(4))*(-x**S(2)*Rt(-d/c, S(2)) + S(1))), x), x) + Dist(S(1)/(S(2)*c), Int(S(1)/(sqrt(a + b*x**S(4))*(x**S(2)*Rt(-d/c, S(2)) + S(1))), x), x)
def replacement889(a, b, c, d, x):
return Dist(b/(-a*d + b*c), Int((a + b*x**S(4))**(S(-3)/4), x), x) - Dist(d/(-a*d + b*c), Int((a + b*x**S(4))**(S(1)/4)/(c + d*x**S(4)), x), x)
def replacement890(a, b, c, d, x):
return Simp(sqrt(a + b*x**S(2))*EllipticE(ArcTan(x*Rt(d/c, S(2))), S(1) - b*c/(a*d))/(c*sqrt(c*(a + b*x**S(2))/(a*(c + d*x**S(2))))*sqrt(c + d*x**S(2))*Rt(d/c, S(2))), x)
def replacement891(a, b, c, d, n, p, q, x):
return Dist(S(1)/(a*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*Simp(c*(n*(p + S(1)) + S(1)) + d*x**n*(n*(p + q + S(1)) + S(1)), x), x), x) - Simp(x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(a*n*(p + S(1))), x)
def replacement892(a, b, c, d, n, p, q, x):
return -Dist(S(1)/(a*b*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-2))*Simp(c*(a*d - b*c*(n*(p + S(1)) + S(1))) + d*x**n*(a*d*(n*(q + S(-1)) + S(1)) - b*c*(n*(p + q) + S(1))), x), x), x) + Simp(x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*(a*d - b*c)/(a*b*n*(p + S(1))), x)
def replacement893(a, b, c, d, n, p, q, x):
return Dist(S(1)/(a*n*(p + S(1))*(-a*d + b*c)), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(b*c + b*d*x**n*(n*(p + q + S(2)) + S(1)) + n*(p + S(1))*(-a*d + b*c), x), x), x) - Simp(b*x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(a*n*(p + S(1))*(-a*d + b*c)), x)
def replacement894(a, b, c, d, n, p, q, x):
return Int(ExpandIntegrand((a + b*x**n)**p*(c + d*x**n)**q, x), x)
def replacement895(a, b, c, d, n, p, q, x):
return Dist(S(1)/(b*(n*(p + q) + S(1))), Int((a + b*x**n)**p*(c + d*x**n)**(q + S(-2))*Simp(c*(-a*d + b*c*(n*(p + q) + S(1))) + d*x**n*(-a*d*(n*(q + S(-1)) + S(1)) + b*c*(n*(p + S(2)*q + S(-1)) + S(1))), x), x), x) + Simp(d*x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))/(b*(n*(p + q) + S(1))), x)
def replacement896(a, b, c, d, n, p, q, x):
return Dist(n/(n*(p + q) + S(1)), Int((a + b*x**n)**(p + S(-1))*(c + d*x**n)**(q + S(-1))*Simp(a*c*(p + q) + x**n*(a*d*(p + q) + q*(-a*d + b*c)), x), x), x) + Simp(x*(a + b*x**n)**p*(c + d*x**n)**q/(n*(p + q) + S(1)), x)
def replacement897(a, b, c, d, x):
return Simp(sqrt(a + b*x**S(2))*EllipticF(ArcTan(x*Rt(d/c, S(2))), S(1) - b*c/(a*d))/(a*sqrt(c*(a + b*x**S(2))/(a*(c + d*x**S(2))))*sqrt(c + d*x**S(2))*Rt(d/c, S(2))), x)
def replacement898(a, b, c, d, x):
return Simp(EllipticF(asin(x*Rt(-d/c, S(2))), b*c/(a*d))/(sqrt(a)*sqrt(c)*Rt(-d/c, S(2))), x)
def replacement899(a, b, c, d, x):
return -Simp(EllipticF(acos(x*Rt(-d/c, S(2))), b*c/(-a*d + b*c))/(sqrt(c)*sqrt(a - b*c/d)*Rt(-d/c, S(2))), x)
def replacement900(a, b, c, d, x):
return Dist(sqrt(S(1) + d*x**S(2)/c)/sqrt(c + d*x**S(2)), Int(S(1)/(sqrt(S(1) + d*x**S(2)/c)*sqrt(a + b*x**S(2))), x), x)
def replacement901(a, b, c, d, x):
return Dist(a, Int(S(1)/(sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x) + Dist(b, Int(x**S(2)/(sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x)
def replacement902(a, b, c, d, x):
return Dist(b/d, Int(sqrt(c + d*x**S(2))/sqrt(a + b*x**S(2)), x), x) - Dist((-a*d + b*c)/d, Int(S(1)/(sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x)
def replacement903(a, b, c, d, x):
return Simp(sqrt(a)*EllipticE(asin(x*Rt(-d/c, S(2))), b*c/(a*d))/(sqrt(c)*Rt(-d/c, S(2))), x)
def replacement904(a, b, c, d, x):
return -Simp(sqrt(a - b*c/d)*EllipticE(acos(x*Rt(-d/c, S(2))), b*c/(-a*d + b*c))/(sqrt(c)*Rt(-d/c, S(2))), x)
def replacement905(a, b, c, d, x):
return Dist(sqrt(a + b*x**S(2))/sqrt(S(1) + b*x**S(2)/a), Int(sqrt(S(1) + b*x**S(2)/a)/sqrt(c + d*x**S(2)), x), x)
def replacement906(a, b, c, d, x):
return Dist(sqrt(S(1) + d*x**S(2)/c)/sqrt(c + d*x**S(2)), Int(sqrt(a + b*x**S(2))/sqrt(S(1) + d*x**S(2)/c), x), x)
def replacement907(a, b, c, d, n, p, q, x):
return Int(ExpandIntegrand((a + b*x**n)**p*(c + d*x**n)**q, x), x)
def replacement908(a, b, c, d, n, p, q, x):
return Simp(a**p*c**q*x*AppellF1(S(1)/n, -p, -q, S(1) + S(1)/n, -b*x**n/a, -d*x**n/c), x)
def replacement909(a, b, c, d, n, p, q, x):
return Dist(a**IntPart(p)*(S(1) + b*x**n/a)**(-FracPart(p))*(a + b*x**n)**FracPart(p), Int((S(1) + b*x**n/a)**p*(c + d*x**n)**q, x), x)
def replacement910(a, b, c, d, mn, n, p, q, x):
return Int(x**(-n*q)*(a + b*x**n)**p*(c*x**n + d)**q, x)
def replacement911(a, b, c, d, mn, n, p, q, x):
return Dist(x**(n*FracPart(q))*(c + d*x**(-n))**FracPart(q)*(c*x**n + d)**(-FracPart(q)), Int(x**(-n*q)*(a + b*x**n)**p*(c*x**n + d)**q, x), x)
def replacement912(a, b, c, d, n, p, q, u, x):
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*x**n)**p*(c + d*x**n)**q, x), x, u), x)
def replacement913(p, q, u, v, x):
return Int(NormalizePseudoBinomial(u, x)**p*NormalizePseudoBinomial(v, x)**q, x)
def replacement914(m, p, q, u, v, x):
return Int(NormalizePseudoBinomial(v, x)**q*NormalizePseudoBinomial(u*x**(m/p), x)**p, x)
def replacement915(b, c, d, e, m, n, p, q, x):
return Dist(b**(S(1) - (m + S(1))/n)*e**m/n, Subst(Int((b*x)**(p + S(-1) + (m + S(1))/n)*(c + d*x)**q, x), x, x**n), x)
def replacement916(b, c, d, e, m, n, p, q, x):
return Dist(b**IntPart(p)*e**m*x**(-n*FracPart(p))*(b*x**n)**FracPart(p), Int(x**(m + n*p)*(c + d*x**n)**q, x), x)
def replacement917(b, c, d, e, m, n, p, q, x):
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(b*x**n)**p*(c + d*x**n)**q, x), x)
def replacement918(a, b, c, d, m, n, p, q, x):
return Dist(S(1)/n, Subst(Int((a + b*x)**p*(c + d*x)**q, x), x, x**n), x)
def replacement919(a, b, c, d, m, n, p, q, x):
return Int(x**(m + n*(p + q))*(a*x**(-n) + b)**p*(c*x**(-n) + d)**q, x)
def replacement920(a, b, c, d, m, n, p, q, x):
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b*x)**p*(c + d*x)**q, x), x, x**n), x)
def replacement921(a, b, c, d, e, m, n, p, q, x):
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
def replacement922(a, b, c, d, e, m, n, p, q, x):
return Int(ExpandIntegrand((e*x)**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
def replacement923(a, b, c, d, e, m, n, p, x):
return Simp(c*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*e*(m + S(1))), x)
def replacement924(a1, a2, b1, b2, c, d, e, m, n, non2, p, x):
return Simp(c*(e*x)**(m + S(1))*(a1 + b1*x**(n/S(2)))**(p + S(1))*(a2 + b2*x**(n/S(2)))**(p + S(1))/(a1*a2*e*(m + S(1))), x)
def replacement925(a, b, c, d, e, m, n, p, x):
return Dist(d*e**(-n), Int((e*x)**(m + n)*(a + b*x**n)**p, x), x) + Simp(c*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*e*(m + S(1))), x)
def replacement926(a, b, c, d, e, m, n, p, x):
return Dist(d/b, Int((e*x)**m*(a + b*x**n)**(p + S(1)), x), x) + Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(-a*d + b*c)/(a*b*e*(m + S(1))), x)
def replacement927(a, b, c, d, e, m, n, p, x):
return Dist(e**(-n)*(a*d*(m + S(1)) - b*c*(m + n*(p + S(1)) + S(1)))/(a*(m + S(1))), Int((e*x)**(m + n)*(a + b*x**n)**p, x), x) + Simp(c*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*e*(m + S(1))), x)
def replacement928(a1, a2, b1, b2, c, d, e, m, n, non2, p, x):
return Dist(e**(-n)*(a1*a2*d*(m + S(1)) - b1*b2*c*(m + n*(p + S(1)) + S(1)))/(a1*a2*(m + S(1))), Int((e*x)**(m + n)*(a1 + b1*x**(n/S(2)))**p*(a2 + b2*x**(n/S(2)))**p, x), x) + Simp(c*(e*x)**(m + S(1))*(a1 + b1*x**(n/S(2)))**(p + S(1))*(a2 + b2*x**(n/S(2)))**(p + S(1))/(a1*a2*e*(m + S(1))), x)
def replacement929(a, b, c, d, m, p, x):
return Dist(b**(-m/S(2) + S(-1))/(S(2)*(p + S(1))), Int((a + b*x**S(2))**(p + S(1))*ExpandToSum(S(2)*b*x**S(2)*(p + S(1))*Together((b**(m/S(2))*x**(m + S(-2))*(c + d*x**S(2)) - (-a)**(m/S(2) + S(-1))*(-a*d + b*c))/(a + b*x**S(2))) - (-a)**(m/S(2) + S(-1))*(-a*d + b*c), x), x), x) + Simp(b**(-m/S(2) + S(-1))*x*(-a)**(m/S(2) + S(-1))*(a + b*x**S(2))**(p + S(1))*(-a*d + b*c)/(S(2)*(p + S(1))), x)
def replacement930(a, b, c, d, m, p, x):
return Dist(b**(-m/S(2) + S(-1))/(S(2)*(p + S(1))), Int(x**m*(a + b*x**S(2))**(p + S(1))*ExpandToSum(S(2)*b*(p + S(1))*Together((b**(m/S(2))*(c + d*x**S(2)) - x**(S(2) - m)*(-a)**(m/S(2) + S(-1))*(-a*d + b*c))/(a + b*x**S(2))) - x**(-m)*(-a)**(m/S(2) + S(-1))*(-a*d + b*c), x), x), x) + Simp(b**(-m/S(2) + S(-1))*x*(-a)**(m/S(2) + S(-1))*(a + b*x**S(2))**(p + S(1))*(-a*d + b*c)/(S(2)*(p + S(1))), x)
def replacement931(a, b, c, d, e, m, n, p, x):
return -Dist((a*d*(m + S(1)) - b*c*(m + n*(p + S(1)) + S(1)))/(a*b*n*(p + S(1))), Int((e*x)**m*(a + b*x**n)**(p + S(1)), x), x) - Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(-a*d + b*c)/(a*b*e*n*(p + S(1))), x)
def replacement932(a1, a2, b1, b2, c, d, e, m, n, non2, p, x):
return -Dist((a1*a2*d*(m + S(1)) - b1*b2*c*(m + n*(p + S(1)) + S(1)))/(a1*a2*b1*b2*n*(p + S(1))), Int((e*x)**m*(a1 + b1*x**(n/S(2)))**(p + S(1))*(a2 + b2*x**(n/S(2)))**(p + S(1)), x), x) - Simp((e*x)**(m + S(1))*(a1 + b1*x**(n/S(2)))**(p + S(1))*(a2 + b2*x**(n/S(2)))**(p + S(1))*(-a1*a2*d + b1*b2*c)/(a1*a2*b1*b2*e*n*(p + S(1))), x)
def replacement933(a, b, c, d, e, m, n, p, x):
return -Dist((a*d*(m + S(1)) - b*c*(m + n*(p + S(1)) + S(1)))/(b*(m + n*(p + S(1)) + S(1))), Int((e*x)**m*(a + b*x**n)**p, x), x) + Simp(d*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(b*e*(m + n*(p + S(1)) + S(1))), x)
def replacement934(a1, a2, b1, b2, c, d, e, m, n, non2, p, x):
return -Dist((a1*a2*d*(m + S(1)) - b1*b2*c*(m + n*(p + S(1)) + S(1)))/(b1*b2*(m + n*(p + S(1)) + S(1))), Int((e*x)**m*(a1 + b1*x**(n/S(2)))**p*(a2 + b2*x**(n/S(2)))**p, x), x) + Simp(d*(e*x)**(m + S(1))*(a1 + b1*x**(n/S(2)))**(p + S(1))*(a2 + b2*x**(n/S(2)))**(p + S(1))/(b1*b2*e*(m + n*(p + S(1)) + S(1))), x)
def replacement935(a, b, c, d, e, m, n, p, x):
return Int(ExpandIntegrand((e*x)**m*(a + b*x**n)**p/(c + d*x**n), x), x)
def replacement936(a, b, c, d, e, m, n, p, x):
return -Dist(e**(-n)/(a*(m + S(1))), Int((e*x)**(m + n)*(a + b*x**n)**p*Simp(-a*d**S(2)*x**n*(m + S(1)) + b*c**S(2)*n*(p + S(1)) + c*(m + S(1))*(-S(2)*a*d + b*c), x), x), x) + Simp(c**S(2)*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*e*(m + S(1))), x)
def replacement937(a, b, c, d, e, m, n, p, x):
return Dist(S(1)/(a*b**S(2)*n*(p + S(1))), Int((e*x)**m*(a + b*x**n)**(p + S(1))*Simp(a*b*d**S(2)*n*x**n*(p + S(1)) + b**S(2)*c**S(2)*n*(p + S(1)) + (m + S(1))*(-a*d + b*c)**S(2), x), x), x) - Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(-a*d + b*c)**S(2)/(a*b**S(2)*e*n*(p + S(1))), x)
def replacement938(a, b, c, d, e, m, n, p, x):
return Dist(S(1)/(b*(m + n*(p + S(2)) + S(1))), Int((e*x)**m*(a + b*x**n)**p*Simp(b*c**S(2)*(m + n*(p + S(2)) + S(1)) + d*x**n*(S(2)*b*c*n*(p + S(1)) + (-a*d + S(2)*b*c)*(m + n + S(1))), x), x), x) + Simp(d**S(2)*e**(-n + S(-1))*(e*x)**(m + n + S(1))*(a + b*x**n)**(p + S(1))/(b*(m + n*(p + S(2)) + S(1))), x)
def With939(a, b, c, d, m, n, p, q, x):
if isinstance(x, (int, Integer, float, Float)):
return False
k = GCD(m + S(1), n)
if Unequal(k, S(1)):
return True
return False
def replacement939(a, b, c, d, m, n, p, q, x):
k = GCD(m + S(1), n)
return Dist(S(1)/k, Subst(Int(x**(S(-1) + (m + S(1))/k)*(a + b*x**(n/k))**p*(c + d*x**(n/k))**q, x), x, x**k), x)
def With940(a, b, c, d, e, m, n, p, q, x):
k = Denominator(m)
return Dist(k/e, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*e**(-n)*x**(k*n))**p*(c + d*e**(-n)*x**(k*n))**q, x), x, (e*x)**(S(1)/k)), x)
def replacement941(a, b, c, d, e, m, n, p, q, x):
return -Dist(e**n/(b*n*(p + S(1))), Int((e*x)**(m - n)*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*Simp(c*(m - n + S(1)) + d*x**n*(m + n*(q + S(-1)) + S(1)), x), x), x) + Simp(e**(n + S(-1))*(e*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(b*n*(p + S(1))), x)
def replacement942(a, b, c, d, e, m, n, p, q, x):
return Dist(S(1)/(a*b*n*(p + S(1))), Int((e*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-2))*Simp(c*(b*c*n*(p + S(1)) + (m + S(1))*(-a*d + b*c)) + d*x**n*(b*c*n*(p + S(1)) + (-a*d + b*c)*(m + n*(q + S(-1)) + S(1))), x), x), x) - Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*(-a*d + b*c)/(a*b*e*n*(p + S(1))), x)
def replacement943(a, b, c, d, e, m, n, p, q, x):
return Dist(S(1)/(a*n*(p + S(1))), Int((e*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*Simp(c*(m + n*(p + S(1)) + S(1)) + d*x**n*(m + n*(p + q + S(1)) + S(1)), x), x), x) - Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(a*e*n*(p + S(1))), x)
def replacement944(a, b, c, d, e, m, n, p, q, x):
return Dist(e**(S(2)*n)/(b*n*(p + S(1))*(-a*d + b*c)), Int((e*x)**(m - S(2)*n)*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(a*c*(m - S(2)*n + S(1)) + x**n*(a*d*(m + n*q - n + S(1)) + b*c*n*(p + S(1))), x), x), x) - Simp(a*e**(S(2)*n + S(-1))*(e*x)**(m - S(2)*n + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(b*n*(p + S(1))*(-a*d + b*c)), x)
def replacement945(a, b, c, d, e, m, n, p, q, x):
return -Dist(e**n/(n*(p + S(1))*(-a*d + b*c)), Int((e*x)**(m - n)*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(c*(m - n + S(1)) + d*x**n*(m + n*(p + q + S(1)) + S(1)), x), x), x) + Simp(e**(n + S(-1))*(e*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(n*(p + S(1))*(-a*d + b*c)), x)
def replacement946(a, b, c, d, e, m, n, p, q, x):
return Dist(S(1)/(a*n*(p + S(1))*(-a*d + b*c)), Int((e*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(b*c*(m + S(1)) + b*d*x**n*(m + n*(p + q + S(2)) + S(1)) + n*(p + S(1))*(-a*d + b*c), x), x), x) - Simp(b*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(a*e*n*(p + S(1))*(-a*d + b*c)), x)
def replacement947(a, b, c, d, e, m, n, p, q, x):
return -Dist(e**(-n)*n/(m + S(1)), Int((e*x)**(m + n)*(a + b*x**n)**(p + S(-1))*(c + d*x**n)**(q + S(-1))*Simp(a*d*q + b*c*p + b*d*x**n*(p + q), x), x), x) + Simp((e*x)**(m + S(1))*(a + b*x**n)**p*(c + d*x**n)**q/(e*(m + S(1))), x)
def replacement948(a, b, c, d, e, m, n, p, q, x):
return -Dist(e**(-n)/(a*(m + S(1))), Int((e*x)**(m + n)*(a + b*x**n)**p*(c + d*x**n)**(q + S(-2))*Simp(c*n*(a*d*(q + S(-1)) + b*c*(p + S(1))) + c*(m + S(1))*(-a*d + b*c) + d*x**n*(b*c*n*(p + q) + (m + S(1))*(-a*d + b*c)), x), x), x) + Simp(c*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))/(a*e*(m + S(1))), x)
def replacement949(a, b, c, d, e, m, n, p, q, x):
return -Dist(e**(-n)/(a*(m + S(1))), Int((e*x)**(m + n)*(a + b*x**n)**p*(c + d*x**n)**(q + S(-1))*Simp(b*c*(m + S(1)) + d*x**n*(b*n*(p + q + S(1)) + b*(m + S(1))) + n*(a*d*q + b*c*(p + S(1))), x), x), x) + Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(a*e*(m + S(1))), x)
def replacement950(a, b, c, d, e, m, n, p, q, x):
return Dist(n/(m + n*(p + q) + S(1)), Int((e*x)**m*(a + b*x**n)**(p + S(-1))*(c + d*x**n)**(q + S(-1))*Simp(a*c*(p + q) + x**n*(a*d*(p + q) + q*(-a*d + b*c)), x), x), x) + Simp((e*x)**(m + S(1))*(a + b*x**n)**p*(c + d*x**n)**q/(e*(m + n*(p + q) + S(1))), x)
def replacement951(a, b, c, d, e, m, n, p, q, x):
return Dist(S(1)/(b*(m + n*(p + q) + S(1))), Int((e*x)**m*(a + b*x**n)**p*(c + d*x**n)**(q + S(-2))*Simp(c*(b*c*n*(p + q) + (m + S(1))*(-a*d + b*c)) + x**n*(b*c*d*n*(p + q) + d*n*(q + S(-1))*(-a*d + b*c) + d*(m + S(1))*(-a*d + b*c)), x), x), x) + Simp(d*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))/(b*e*(m + n*(p + q) + S(1))), x)
def replacement952(a, b, c, d, e, m, n, p, q, x):
return -Dist(e**n/(b*(m + n*(p + q) + S(1))), Int((e*x)**(m - n)*(a + b*x**n)**p*(c + d*x**n)**(q + S(-1))*Simp(a*c*(m - n + S(1)) + x**n*(a*d*(m - n + S(1)) - n*q*(-a*d + b*c)), x), x), x) + Simp(e**(n + S(-1))*(e*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(b*(m + n*(p + q) + S(1))), x)
def replacement953(a, b, c, d, e, m, n, p, q, x):
return -Dist(e**(S(2)*n)/(b*d*(m + n*(p + q) + S(1))), Int((e*x)**(m - S(2)*n)*(a + b*x**n)**p*(c + d*x**n)**q*Simp(a*c*(m - S(2)*n + S(1)) + x**n*(a*d*(m + n*(q + S(-1)) + S(1)) + b*c*(m + n*(p + S(-1)) + S(1))), x), x), x) + Simp(e**(S(2)*n + S(-1))*(e*x)**(m - S(2)*n + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(b*d*(m + n*(p + q) + S(1))), x)
def replacement954(a, b, c, d, e, m, n, p, q, x):
return -Dist(e**(-n)/(a*c*(m + S(1))), Int((e*x)**(m + n)*(a + b*x**n)**p*(c + d*x**n)**q*Simp(b*d*x**n*(m + n*(p + q + S(2)) + S(1)) + n*(a*d*q + b*c*p) + (a*d + b*c)*(m + n + S(1)), x), x), x) + Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(a*c*e*(m + S(1))), x)
def replacement955(a, b, c, d, e, m, n, x):
return -Dist(a*e**n/(-a*d + b*c), Int((e*x)**(m - n)/(a + b*x**n), x), x) + Dist(c*e**n/(-a*d + b*c), Int((e*x)**(m - n)/(c + d*x**n), x), x)
def replacement956(a, b, c, d, e, m, n, x):
return Dist(b/(-a*d + b*c), Int((e*x)**m/(a + b*x**n), x), x) - Dist(d/(-a*d + b*c), Int((e*x)**m/(c + d*x**n), x), x)
def replacement957(a, b, c, d, m, n, x):
return Dist(S(1)/b, Int(x**(m - n)/sqrt(c + d*x**n), x), x) - Dist(a/b, Int(x**(m - n)/((a + b*x**n)*sqrt(c + d*x**n)), x), x)
def With958(a, b, c, d, x):
r = Numerator(Rt(-a/b, S(2)))
s = Denominator(Rt(-a/b, S(2)))
return -Dist(s/(S(2)*b), Int(S(1)/(sqrt(c + d*x**S(4))*(r - s*x**S(2))), x), x) + Dist(s/(S(2)*b), Int(S(1)/(sqrt(c + d*x**S(4))*(r + s*x**S(2))), x), x)
def With959(a, b, c, d, x):
q = Rt(d/c, S(3))
return Simp(S(2)**(S(1)/3)*q*log(-S(2)**(S(1)/3)*q*x + S(1) - sqrt(c + d*x**S(3))/sqrt(c))/(S(12)*b*sqrt(c)), x) - Simp(S(2)**(S(1)/3)*q*log(-S(2)**(S(1)/3)*q*x + S(1) + sqrt(c + d*x**S(3))/sqrt(c))/(S(12)*b*sqrt(c)), x) + Simp(S(2)**(S(1)/3)*q*atanh(sqrt(c + d*x**S(3))/sqrt(c))/(S(18)*b*sqrt(c)), x) - Simp(S(2)**(S(1)/3)*sqrt(S(3))*q*ArcTan(sqrt(S(3))/S(3) + S(2)**(S(2)/3)*sqrt(S(3))*(sqrt(c) - sqrt(c + d*x**S(3)))/(S(3)*sqrt(c)*q*x))/(S(18)*b*sqrt(c)), x) + Simp(S(2)**(S(1)/3)*sqrt(S(3))*q*ArcTan(sqrt(S(3))/S(3) + S(2)**(S(2)/3)*sqrt(S(3))*(sqrt(c) + sqrt(c + d*x**S(3)))/(S(3)*sqrt(c)*q*x))/(S(18)*b*sqrt(c)), x)
def replacement960(a, b, c, d, m, x):
return Dist(S(1)/b, Int(x**(m + S(-3))/sqrt(c + d*x**S(3)), x), x) - Dist(a/b, Int(x**(m + S(-3))/((a + b*x**S(3))*sqrt(c + d*x**S(3))), x), x)
def replacement961(a, b, c, d, m, x):
return Dist(S(1)/a, Int(x**m/sqrt(c + d*x**S(3)), x), x) - Dist(b/a, Int(x**(m + S(3))/((a + b*x**S(3))*sqrt(c + d*x**S(3))), x), x)
def replacement962(a, b, c, d, x):
return Dist(d/b, Int(x**S(2)/sqrt(c + d*x**S(4)), x), x) + Dist((-a*d + b*c)/b, Int(x**S(2)/((a + b*x**S(4))*sqrt(c + d*x**S(4))), x), x)
def replacement963(a, b, c, d, m, x):
return Dist(d/b, Int(x**m/sqrt(c + d*x**S(3)), x), x) + Dist((-a*d + b*c)/b, Int(x**m/((a + b*x**S(3))*sqrt(c + d*x**S(3))), x), x)
def replacement964(a, b, c, d, x):
return -Dist(c/b, Int(sqrt(a + b*x**S(2))/(c + d*x**S(2))**(S(3)/2), x), x) + Simp(x*sqrt(a + b*x**S(2))/(b*sqrt(c + d*x**S(2))), x)
def replacement965(a, b, c, d, n, x):
return Dist(S(1)/b, Int(sqrt(a + b*x**n)/sqrt(c + d*x**n), x), x) - Dist(a/b, Int(S(1)/(sqrt(a + b*x**n)*sqrt(c + d*x**n)), x), x)
def With966(a, b, c, d, m, n, p, q, x):
k = Denominator(p)
return Dist(a**(p + (m + S(1))/n)*k/n, Subst(Int(x**(k*(m + S(1))/n + S(-1))*(c - x**k*(-a*d + b*c))**q*(-b*x**k + S(1))**(-p - q + S(-1) - (m + S(1))/n), x), x, x**(n/k)*(a + b*x**n)**(-S(1)/k)), x)
def replacement967(a, b, c, d, m, n, p, q, x):
return -Subst(Int(x**(-m + S(-2))*(a + b*x**(-n))**p*(c + d*x**(-n))**q, x), x, S(1)/x)
def With968(a, b, c, d, e, m, n, p, q, x):
g = Denominator(m)
return -Dist(g/e, Subst(Int(x**(-g*(m + S(1)) + S(-1))*(a + b*e**(-n)*x**(-g*n))**p*(c + d*e**(-n)*x**(-g*n))**q, x), x, (e*x)**(-S(1)/g)), x)
def replacement969(a, b, c, d, e, m, n, p, q, x):
return -Dist((e*x)**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(a + b*x**(-n))**p*(c + d*x**(-n))**q, x), x, S(1)/x), x)
def With970(a, b, c, d, m, n, p, q, x):
g = Denominator(n)
return Dist(g, Subst(Int(x**(g*(m + S(1)) + S(-1))*(a + b*x**(g*n))**p*(c + d*x**(g*n))**q, x), x, x**(S(1)/g)), x)
def replacement971(a, b, c, d, e, m, n, p, q, x):
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
def replacement972(a, b, c, d, m, n, p, q, x):
return Dist(S(1)/(m + S(1)), Subst(Int((a + b*x**(n/(m + S(1))))**p*(c + d*x**(n/(m + S(1))))**q, x), x, x**(m + S(1))), x)
def replacement973(a, b, c, d, e, m, n, p, q, x):
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
def replacement974(a, b, c, d, e, m, n, p, q, x):
return Dist(S(1)/(a*b*n*(p + S(1))), Int((e*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-2))*Simp(c*(b*c*n*(p + S(1)) + (m + S(1))*(-a*d + b*c)) + d*x**n*(b*c*n*(p + S(1)) + (-a*d + b*c)*(m + n*(q + S(-1)) + S(1))), x), x), x) - Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*(-a*d + b*c)/(a*b*e*n*(p + S(1))), x)
def replacement975(a, b, c, d, e, m, n, p, q, x):
return Dist(S(1)/(a*n*(p + S(1))), Int((e*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*Simp(c*(m + n*(p + S(1)) + S(1)) + d*x**n*(m + n*(p + q + S(1)) + S(1)), x), x), x) - Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(a*e*n*(p + S(1))), x)
def replacement976(a, b, c, d, e, m, n, p, q, x):
return Dist(S(1)/(a*n*(p + S(1))*(-a*d + b*c)), Int((e*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(b*c*(m + S(1)) + b*d*x**n*(m + n*(p + q + S(2)) + S(1)) + n*(p + S(1))*(-a*d + b*c), x), x), x) - Simp(b*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(a*e*n*(p + S(1))*(-a*d + b*c)), x)
def replacement977(a, b, c, d, e, m, n, p, q, x):
return Dist(n/(m + n*(p + q) + S(1)), Int((e*x)**m*(a + b*x**n)**(p + S(-1))*(c + d*x**n)**(q + S(-1))*Simp(a*c*(p + q) + x**n*(a*d*(p + q) + q*(-a*d + b*c)), x), x), x) + Simp((e*x)**(m + S(1))*(a + b*x**n)**p*(c + d*x**n)**q/(e*(m + n*(p + q) + S(1))), x)
def replacement978(a, b, c, d, e, m, n, p, q, x):
return Dist(S(1)/(b*(m + n*(p + q) + S(1))), Int((e*x)**m*(a + b*x**n)**p*(c + d*x**n)**(q + S(-2))*Simp(c*(b*c*n*(p + q) + (m + S(1))*(-a*d + b*c)) + x**n*(b*c*d*n*(p + q) + d*n*(q + S(-1))*(-a*d + b*c) + d*(m + S(1))*(-a*d + b*c)), x), x), x) + Simp(d*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))/(b*e*(m + n*(p + q) + S(1))), x)
def replacement979(a, b, c, d, m, n, x):
return -Dist(a/(-a*d + b*c), Int(x**(m - n)/(a + b*x**n), x), x) + Dist(c/(-a*d + b*c), Int(x**(m - n)/(c + d*x**n), x), x)
def replacement980(a, b, c, d, e, m, n, x):
return Dist(b/(-a*d + b*c), Int((e*x)**m/(a + b*x**n), x), x) - Dist(d/(-a*d + b*c), Int((e*x)**m/(c + d*x**n), x), x)
def replacement981(a, b, c, d, e, m, n, p, q, x):
return Int(ExpandIntegrand((e*x)**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
def replacement982(a, b, c, d, m, mn, n, p, q, x):
return Int(x**(m - n*q)*(a + b*x**n)**p*(c*x**n + d)**q, x)
def replacement983(a, b, c, d, m, mn, n, p, q, x):
return Dist(x**(n*FracPart(q))*(c + d*x**(-n))**FracPart(q)*(c*x**n + d)**(-FracPart(q)), Int(x**(m - n*q)*(a + b*x**n)**p*(c*x**n + d)**q, x), x)
def replacement984(a, b, c, d, e, m, mn, n, p, q, x):
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**(-n))**q, x), x)
def replacement985(a, b, c, d, e, m, n, p, q, x):
return Simp(a**p*c**q*(e*x)**(m + S(1))*AppellF1((m + S(1))/n, -p, -q, S(1) + (m + S(1))/n, -b*x**n/a, -d*x**n/c)/(e*(m + S(1))), x)
def replacement986(a, b, c, d, e, m, n, p, q, x):
return Dist(a**IntPart(p)*(S(1) + b*x**n/a)**(-FracPart(p))*(a + b*x**n)**FracPart(p), Int((e*x)**m*(S(1) + b*x**n/a)**p*(c + d*x**n)**q, x), x)
def replacement987(a, b, c, d, m, n, p, q, v, x):
return Dist(Coefficient(v, x, S(1))**(-m + S(-1)), Subst(Int(SimplifyIntegrand((a + b*x**n)**p*(c + d*x**n)**q*(x - Coefficient(v, x, S(0)))**m, x), x), x, v), x)
def replacement988(a, b, c, d, m, n, p, q, u, v, x):
return Dist(u**m*v**(-m)/Coefficient(v, x, S(1)), Subst(Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x, v), x)
def replacement989(a1, a2, b1, b2, c, d, n, non2, p, q, u, x):
return Int(u*(c + d*x**n)**q*(a1*a2 + b1*b2*x**n)**p, x)
def replacement990(a1, a2, b1, b2, c, d, e, n, n2, non2, p, q, u, x):
return Int(u*(a1*a2 + b1*b2*x**n)**p*(c + d*x**n + e*x**(S(2)*n))**q, x)
def replacement991(a1, a2, b1, b2, c, d, n, non2, p, q, u, x):
return Dist((a1 + b1*x**(n/S(2)))**FracPart(p)*(a2 + b2*x**(n/S(2)))**FracPart(p)*(a1*a2 + b1*b2*x**n)**(-FracPart(p)), Int(u*(c + d*x**n)**q*(a1*a2 + b1*b2*x**n)**p, x), x)
def replacement992(a1, a2, b1, b2, c, d, e, n, n2, non2, p, q, u, x):
return Dist((a1 + b1*x**(n/S(2)))**FracPart(p)*(a2 + b2*x**(n/S(2)))**FracPart(p)*(a1*a2 + b1*b2*x**n)**(-FracPart(p)), Int(u*(a1*a2 + b1*b2*x**n)**p*(c + d*x**n + e*x**(S(2)*n))**q, x), x)
def replacement993(a, b, c, d, e, f, n, p, q, r, x):
return Int(ExpandIntegrand((a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
def replacement994(a, b, c, d, e, f, n, x):
return Dist((-a*f + b*e)/(-a*d + b*c), Int(S(1)/(a + b*x**n), x), x) - Dist((-c*f + d*e)/(-a*d + b*c), Int(S(1)/(c + d*x**n), x), x)
def replacement995(a, b, c, d, e, f, n, x):
return Dist(f/b, Int(S(1)/sqrt(c + d*x**n), x), x) + Dist((-a*f + b*e)/b, Int(S(1)/((a + b*x**n)*sqrt(c + d*x**n)), x), x)
def replacement996(a, b, c, d, e, f, n, x):
return Dist(f/b, Int(sqrt(a + b*x**n)/sqrt(c + d*x**n), x), x) + Dist((-a*f + b*e)/b, Int(S(1)/(sqrt(a + b*x**n)*sqrt(c + d*x**n)), x), x)
def replacement997(a, b, c, d, e, f, x):
return Dist((-a*f + b*e)/(-a*d + b*c), Int(S(1)/(sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x) - Dist((-c*f + d*e)/(-a*d + b*c), Int(sqrt(a + b*x**S(2))/(c + d*x**S(2))**(S(3)/2), x), x)
def replacement998(a, b, c, d, e, f, n, p, q, x):
return Dist(S(1)/(a*b*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*Simp(c*(-a*f + b*e*n*(p + S(1)) + b*e) + d*x**n*(b*e*n*(p + S(1)) + (-a*f + b*e)*(n*q + S(1))), x), x), x) - Simp(x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*(-a*f + b*e)/(a*b*n*(p + S(1))), x)
def replacement999(a, b, c, d, e, f, n, p, q, x):
return Dist(S(1)/(a*n*(p + S(1))*(-a*d + b*c)), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(c*(-a*f + b*e) + d*x**n*(-a*f + b*e)*(n*(p + q + S(2)) + S(1)) + e*n*(p + S(1))*(-a*d + b*c), x), x), x) - Simp(x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))*(-a*f + b*e)/(a*n*(p + S(1))*(-a*d + b*c)), x)
def replacement1000(a, b, c, d, e, f, n, p, q, x):
return Dist(S(1)/(b*(n*(p + q + S(1)) + S(1))), Int((a + b*x**n)**p*(c + d*x**n)**(q + S(-1))*Simp(c*(-a*f + b*e*n*(p + q + S(1)) + b*e) + x**n*(b*d*e*n*(p + q + S(1)) + d*(-a*f + b*e) + f*n*q*(-a*d + b*c)), x), x), x) + Simp(f*x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(b*(n*(p + q + S(1)) + S(1))), x)
def replacement1001(a, b, c, d, e, f, x):
return Dist((-a*f + b*e)/(-a*d + b*c), Int((a + b*x**S(4))**(S(-3)/4), x), x) - Dist((-c*f + d*e)/(-a*d + b*c), Int((a + b*x**S(4))**(S(1)/4)/(c + d*x**S(4)), x), x)
def replacement1002(a, b, c, d, e, f, n, p, x):
return Dist(f/d, Int((a + b*x**n)**p, x), x) + Dist((-c*f + d*e)/d, Int((a + b*x**n)**p/(c + d*x**n), x), x)
def replacement1003(a, b, c, d, e, f, n, p, q, x):
return Dist(e, Int((a + b*x**n)**p*(c + d*x**n)**q, x), x) + Dist(f, Int(x**n*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
def replacement1004(a, b, c, d, e, f, x):
return Dist(b/(-a*d + b*c), Int(S(1)/((a + b*x**S(2))*sqrt(e + f*x**S(2))), x), x) - Dist(d/(-a*d + b*c), Int(S(1)/((c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x)
def replacement1005(c, d, e, f, x):
return Dist(S(1)/c, Int(S(1)/(x**S(2)*sqrt(e + f*x**S(2))), x), x) - Dist(d/c, Int(S(1)/((c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x)
def replacement1006(a, b, c, d, e, f, x):
return Dist(d/b, Int(sqrt(e + f*x**S(2))/sqrt(c + d*x**S(2)), x), x) + Dist((-a*d + b*c)/b, Int(sqrt(e + f*x**S(2))/((a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x)
def replacement1007(a, b, c, d, e, f, x):
return Dist(d/b, Int(sqrt(e + f*x**S(2))/sqrt(c + d*x**S(2)), x), x) + Dist((-a*d + b*c)/b, Int(sqrt(e + f*x**S(2))/((a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x)
def replacement1008(a, b, c, d, e, f, x):
return Dist(b/(-a*f + b*e), Int(sqrt(e + f*x**S(2))/((a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x) - Dist(f/(-a*f + b*e), Int(S(1)/(sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x)
def replacement1009(a, b, c, d, e, f, x):
return Simp(EllipticPi(b*c/(a*d), asin(x*Rt(-d/c, S(2))), c*f/(d*e))/(a*sqrt(c)*sqrt(e)*Rt(-d/c, S(2))), x)
def replacement1010(a, b, c, d, e, f, x):
return Dist(sqrt(S(1) + d*x**S(2)/c)/sqrt(c + d*x**S(2)), Int(S(1)/(sqrt(S(1) + d*x**S(2)/c)*(a + b*x**S(2))*sqrt(e + f*x**S(2))), x), x)
def replacement1011(a, b, c, d, e, f, x):
return Simp(c*sqrt(e + f*x**S(2))*EllipticPi(S(1) - b*c/(a*d), ArcTan(x*Rt(d/c, S(2))), -c*f/(d*e) + S(1))/(a*e*sqrt(c*(e + f*x**S(2))/(e*(c + d*x**S(2))))*sqrt(c + d*x**S(2))*Rt(d/c, S(2))), x)
def replacement1012(a, b, c, d, e, f, x):
return Dist(d/b, Int(S(1)/(sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) + Dist((-a*d + b*c)/b, Int(S(1)/((a + b*x**S(2))*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x)
def replacement1013(a, b, c, d, e, f, x):
return Dist(b/(-a*d + b*c), Int(sqrt(e + f*x**S(2))/((a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x) - Dist(d/(-a*d + b*c), Int(sqrt(e + f*x**S(2))/(c + d*x**S(2))**(S(3)/2), x), x)
def replacement1014(a, b, c, d, e, f, x):
return Dist((-a*f + b*e)/(-a*d + b*c), Int(sqrt(e + f*x**S(2))/((a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x) - Dist((-c*f + d*e)/(-a*d + b*c), Int(sqrt(e + f*x**S(2))/(c + d*x**S(2))**(S(3)/2), x), x)
def replacement1015(a, b, c, d, e, f, x):
return Dist(d/b**S(2), Int(sqrt(e + f*x**S(2))*(-a*d + S(2)*b*c + b*d*x**S(2))/sqrt(c + d*x**S(2)), x), x) + Dist((-a*d + b*c)**S(2)/b**S(2), Int(sqrt(e + f*x**S(2))/((a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x)
def replacement1016(a, b, c, d, e, f, q, r, x):
return Dist(b*(-a*f + b*e)/(-a*d + b*c)**S(2), Int((c + d*x**S(2))**(q + S(2))*(e + f*x**S(2))**(r + S(-1))/(a + b*x**S(2)), x), x) - Dist((-a*d + b*c)**(S(-2)), Int((c + d*x**S(2))**q*(e + f*x**S(2))**(r + S(-1))*(-a*d**S(2)*e - b*c**S(2)*f + S(2)*b*c*d*e + d**S(2)*x**S(2)*(-a*f + b*e)), x), x)
def replacement1017(a, b, c, d, e, f, q, r, x):
return Dist(d/b, Int((c + d*x**S(2))**(q + S(-1))*(e + f*x**S(2))**r, x), x) + Dist((-a*d + b*c)/b, Int((c + d*x**S(2))**(q + S(-1))*(e + f*x**S(2))**r/(a + b*x**S(2)), x), x)
def replacement1018(a, b, c, d, e, f, q, r, x):
return Dist(b**S(2)/(-a*d + b*c)**S(2), Int((c + d*x**S(2))**(q + S(2))*(e + f*x**S(2))**r/(a + b*x**S(2)), x), x) - Dist(d/(-a*d + b*c)**S(2), Int((c + d*x**S(2))**q*(e + f*x**S(2))**r*(-a*d + S(2)*b*c + b*d*x**S(2)), x), x)
def replacement1019(a, b, c, d, e, f, q, r, x):
return Dist(b/(-a*d + b*c), Int((c + d*x**S(2))**(q + S(1))*(e + f*x**S(2))**r/(a + b*x**S(2)), x), x) - Dist(d/(-a*d + b*c), Int((c + d*x**S(2))**q*(e + f*x**S(2))**r, x), x)
def replacement1020(a, b, c, d, e, f, x):
return Dist((-a**S(2)*d*f + b**S(2)*c*e)/(S(2)*a*b**S(2)), Int(S(1)/((a + b*x**S(2))*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) + Dist(d*f/(S(2)*a*b**S(2)), Int((a - b*x**S(2))/(sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) + Simp(x*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))/(S(2)*a*(a + b*x**S(2))), x)
def replacement1021(a, b, c, d, e, f, x):
return Dist((S(3)*a**S(2)*d*f - S(2)*a*b*(c*f + d*e) + b**S(2)*c*e)/(S(2)*a*(-a*d + b*c)*(-a*f + b*e)), Int(S(1)/((a + b*x**S(2))*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) - Dist(d*f/(S(2)*a*(-a*d + b*c)*(-a*f + b*e)), Int((a + b*x**S(2))/(sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) + Simp(b**S(2)*x*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))/(S(2)*a*(a + b*x**S(2))*(-a*d + b*c)*(-a*f + b*e)), x)
def replacement1022(a, b, c, d, e, f, n, p, q, r, x):
return Dist(d/b, Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*(e + f*x**n)**r, x), x) + Dist((-a*d + b*c)/b, Int((a + b*x**n)**p*(c + d*x**n)**(q + S(-1))*(e + f*x**n)**r, x), x)
def replacement1023(a, b, c, d, e, f, n, p, q, r, x):
return Dist(b/(-a*d + b*c), Int((a + b*x**n)**p*(c + d*x**n)**(q + S(1))*(e + f*x**n)**r, x), x) - Dist(d/(-a*d + b*c), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
def replacement1024(a, b, c, d, e, f, x):
return Dist(sqrt(a*(e + f*x**S(2))/(e*(a + b*x**S(2))))*sqrt(c + d*x**S(2))/(c*sqrt(a*(c + d*x**S(2))/(c*(a + b*x**S(2))))*sqrt(e + f*x**S(2))), Subst(Int(S(1)/(sqrt(S(1) - x**S(2)*(-a*d + b*c)/c)*sqrt(S(1) - x**S(2)*(-a*f + b*e)/e)), x), x, x/sqrt(a + b*x**S(2))), x)
def replacement1025(a, b, c, d, e, f, x):
return Dist(a*sqrt(a*(e + f*x**S(2))/(e*(a + b*x**S(2))))*sqrt(c + d*x**S(2))/(c*sqrt(a*(c + d*x**S(2))/(c*(a + b*x**S(2))))*sqrt(e + f*x**S(2))), Subst(Int(S(1)/(sqrt(S(1) - x**S(2)*(-a*d + b*c)/c)*sqrt(S(1) - x**S(2)*(-a*f + b*e)/e)*(-b*x**S(2) + S(1))), x), x, x/sqrt(a + b*x**S(2))), x)
def replacement1026(a, b, c, d, e, f, x):
return Dist(sqrt(a*(e + f*x**S(2))/(e*(a + b*x**S(2))))*sqrt(c + d*x**S(2))/(a*sqrt(a*(c + d*x**S(2))/(c*(a + b*x**S(2))))*sqrt(e + f*x**S(2))), Subst(Int(sqrt(S(1) - x**S(2)*(-a*d + b*c)/c)/sqrt(S(1) - x**S(2)*(-a*f + b*e)/e), x), x, x/sqrt(a + b*x**S(2))), x)
def replacement1027(a, b, c, d, e, f, x):
return -Dist(c*(-c*f + d*e)/(S(2)*f), Int(sqrt(a + b*x**S(2))/((c + d*x**S(2))**(S(3)/2)*sqrt(e + f*x**S(2))), x), x) - Dist((-a*d*f - b*c*f + b*d*e)/(S(2)*d*f), Int(sqrt(c + d*x**S(2))/(sqrt(a + b*x**S(2))*sqrt(e + f*x**S(2))), x), x) + Dist(b*c*(-c*f + d*e)/(S(2)*d*f), Int(S(1)/(sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) + Simp(d*x*sqrt(a + b*x**S(2))*sqrt(e + f*x**S(2))/(S(2)*f*sqrt(c + d*x**S(2))), x)
def replacement1028(a, b, c, d, e, f, x):
return -Dist((-a*d*f - b*c*f + b*d*e)/(S(2)*f**S(2)), Int(sqrt(e + f*x**S(2))/(sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x) + Dist(e*(-a*f + b*e)/(S(2)*f), Int(sqrt(c + d*x**S(2))/(sqrt(a + b*x**S(2))*(e + f*x**S(2))**(S(3)/2)), x), x) + Dist((-a*f + b*e)*(-S(2)*c*f + d*e)/(S(2)*f**S(2)), Int(S(1)/(sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) + Simp(x*sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))/(S(2)*sqrt(e + f*x**S(2))), x)
def replacement1029(a, b, c, d, e, f, x):
return Dist(b/f, Int(sqrt(c + d*x**S(2))/(sqrt(a + b*x**S(2))*sqrt(e + f*x**S(2))), x), x) - Dist((-a*f + b*e)/f, Int(sqrt(c + d*x**S(2))/(sqrt(a + b*x**S(2))*(e + f*x**S(2))**(S(3)/2)), x), x)
def With1030(a, b, c, d, e, f, n, p, q, r, x):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand((a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x)
if SumQ(u):
return True
return False
def replacement1030(a, b, c, d, e, f, n, p, q, r, x):
u = ExpandIntegrand((a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x)
return Int(u, x)
def replacement1031(a, b, c, d, e, f, n, p, q, r, x):
return -Subst(Int((a + b*x**(-n))**p*(c + d*x**(-n))**q*(e + f*x**(-n))**r/x**S(2), x), x, S(1)/x)
def replacement1032(a, b, c, d, e, f, n, p, q, r, x):
return Int((a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x)
def replacement1033(a, b, c, d, e, f, n, p, q, r, u, v, w, x):
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x, u), x)
def replacement1034(a, b, c, d, e, f, mn, n, p, q, r, x):
return Int(x**(-n*q)*(a + b*x**n)**p*(e + f*x**n)**r*(c*x**n + d)**q, x)
def replacement1035(a, b, c, d, e, f, mn, n, p, q, r, x):
return Int(x**(n*(p + r))*(c + d*x**(-n))**q*(a*x**(-n) + b)**p*(e*x**(-n) + f)**r, x)
def replacement1036(a, b, c, d, e, f, mn, n, p, q, r, x):
return Dist(x**(n*FracPart(q))*(c + d*x**(-n))**FracPart(q)*(c*x**n + d)**(-FracPart(q)), Int(x**(-n*q)*(a + b*x**n)**p*(e + f*x**n)**r*(c*x**n + d)**q, x), x)
def replacement1037(a, b, c, d, e1, e2, f1, f2, n, n2, p, q, r, x):
return Int((a + b*x**n)**p*(c + d*x**n)**q*(e1*e2 + f1*f2*x**n)**r, x)
def replacement1038(a, b, c, d, e1, e2, f1, f2, n, n2, p, q, r, x):
return Dist((e1 + f1*x**(n/S(2)))**FracPart(r)*(e2 + f2*x**(n/S(2)))**FracPart(r)*(e1*e2 + f1*f2*x**n)**(-FracPart(r)), Int((a + b*x**n)**p*(c + d*x**n)**q*(e1*e2 + f1*f2*x**n)**r, x), x)
def replacement1039(b, c, d, e, f, g, m, n, p, q, r, x):
return Dist(b**(S(1) - (m + S(1))/n)*g**m/n, Subst(Int((b*x)**(p + S(-1) + (m + S(1))/n)*(c + d*x)**q*(e + f*x)**r, x), x, x**n), x)
def replacement1040(b, c, d, e, f, g, m, n, p, q, r, x):
return Dist(b**IntPart(p)*g**m*x**(-n*FracPart(p))*(b*x**n)**FracPart(p), Int(x**(m + n*p)*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
def replacement1041(b, c, d, e, f, g, m, n, p, q, r, x):
return Dist(g**IntPart(m)*x**(-FracPart(m))*(g*x)**FracPart(m), Int(x**m*(b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
def replacement1042(a, b, c, d, e, f, g, m, n, p, q, r, x):
return Int(ExpandIntegrand((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
def replacement1043(a, b, c, d, e, f, m, n, p, q, r, x):
return Dist(S(1)/n, Subst(Int((a + b*x)**p*(c + d*x)**q*(e + f*x)**r, x), x, x**n), x)
def replacement1044(a, b, c, d, e, f, m, n, p, q, r, x):
return Int(x**(m + n*(p + q + r))*(a*x**(-n) + b)**p*(c*x**(-n) + d)**q*(e*x**(-n) + f)**r, x)
def replacement1045(a, b, c, d, e, f, m, n, p, q, r, x):
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b*x)**p*(c + d*x)**q*(e + f*x)**r, x), x, x**n), x)
def replacement1046(a, b, c, d, e, f, g, m, n, p, q, r, x):
return Dist(g**IntPart(m)*x**(-FracPart(m))*(g*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
def With1047(a, b, c, d, e, f, m, n, p, q, r, x):
if isinstance(x, (int, Integer, float, Float)):
return False
k = GCD(m + S(1), n)
if Unequal(k, S(1)):
return True
return False
def replacement1047(a, b, c, d, e, f, m, n, p, q, r, x):
k = GCD(m + S(1), n)
return Dist(S(1)/k, Subst(Int(x**(S(-1) + (m + S(1))/k)*(a + b*x**(n/k))**p*(c + d*x**(n/k))**q*(e + f*x**(n/k))**r, x), x, x**k), x)
def With1048(a, b, c, d, e, f, g, m, n, p, q, r, x):
k = Denominator(m)
return Dist(k/g, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*g**(-n)*x**(k*n))**p*(c + d*g**(-n)*x**(k*n))**q*(e + f*g**(-n)*x**(k*n))**r, x), x, (g*x)**(S(1)/k)), x)
def replacement1049(a, b, c, d, e, f, g, m, n, p, q, x):
return Dist(S(1)/(a*b*n*(p + S(1))), Int((g*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*Simp(c*(b*e*n*(p + S(1)) + (m + S(1))*(-a*f + b*e)) + d*x**n*(b*e*n*(p + S(1)) + (-a*f + b*e)*(m + n*q + S(1))), x), x), x) - Simp((g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*(-a*f + b*e)/(a*b*g*n*(p + S(1))), x)
def replacement1050(a, b, c, d, e, f, g, m, n, p, q, x):
return -Dist(g**n/(b*n*(p + S(1))*(-a*d + b*c)), Int((g*x)**(m - n)*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(c*(-a*f + b*e)*(m - n + S(1)) + x**n*(-b*n*(p + S(1))*(c*f - d*e) + d*(-a*f + b*e)*(m + n*q + S(1))), x), x), x) + Simp(g**(n + S(-1))*(g*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))*(-a*f + b*e)/(b*n*(p + S(1))*(-a*d + b*c)), x)
def replacement1051(a, b, c, d, e, f, g, m, n, p, q, x):
return Dist(S(1)/(a*n*(p + S(1))*(-a*d + b*c)), Int((g*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(c*(m + S(1))*(-a*f + b*e) + d*x**n*(-a*f + b*e)*(m + n*(p + q + S(2)) + S(1)) + e*n*(p + S(1))*(-a*d + b*c), x), x), x) - Simp((g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))*(-a*f + b*e)/(a*g*n*(p + S(1))*(-a*d + b*c)), x)
def replacement1052(a, b, c, d, e, f, g, m, n, p, q, x):
return -Dist(g**(-n)/(a*(m + S(1))), Int((g*x)**(m + n)*(a + b*x**n)**p*(c + d*x**n)**(q + S(-1))*Simp(c*(m + S(1))*(-a*f + b*e) + d*x**n*(b*e*n*(p + q + S(1)) + (m + S(1))*(-a*f + b*e)) + e*n*(a*d*q + b*c*(p + S(1))), x), x), x) + Simp(e*(g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(a*g*(m + S(1))), x)
def replacement1053(a, b, c, d, e, f, g, m, n, p, q, x):
return Dist(S(1)/(b*(m + n*(p + q + S(1)) + S(1))), Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**(q + S(-1))*Simp(c*(b*e*n*(p + q + S(1)) + (m + S(1))*(-a*f + b*e)) + x**n*(b*d*e*n*(p + q + S(1)) + d*(m + S(1))*(-a*f + b*e) + f*n*q*(-a*d + b*c)), x), x), x) + Simp(f*(g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(b*g*(m + n*(p + q + S(1)) + S(1))), x)
def replacement1054(a, b, c, d, e, f, g, m, n, p, q, x):
return -Dist(g**n/(b*d*(m + n*(p + q + S(1)) + S(1))), Int((g*x)**(m - n)*(a + b*x**n)**p*(c + d*x**n)**q*Simp(a*c*f*(m - n + S(1)) + x**n*(a*d*f*(m + n*q + S(1)) + b*(c*f*(m + n*p + S(1)) - d*e*(m + n*(p + q + S(1)) + S(1)))), x), x), x) + Simp(f*g**(n + S(-1))*(g*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(b*d*(m + n*(p + q + S(1)) + S(1))), x)
def replacement1055(a, b, c, d, e, f, g, m, n, p, q, x):
return Dist(g**(-n)/(a*c*(m + S(1))), Int((g*x)**(m + n)*(a + b*x**n)**p*(c + d*x**n)**q*Simp(a*c*f*(m + S(1)) - b*d*e*x**n*(m + n*(p + q + S(2)) + S(1)) - e*n*(a*d*q + b*c*p) - e*(a*d + b*c)*(m + n + S(1)), x), x), x) + Simp(e*(g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(a*c*g*(m + S(1))), x)
def replacement1056(a, b, c, d, e, f, g, m, n, p, x):
return Int(ExpandIntegrand((g*x)**m*(a + b*x**n)**p*(e + f*x**n)/(c + d*x**n), x), x)
def replacement1057(a, b, c, d, e, f, g, m, n, p, q, x):
return Dist(e, Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x) + Dist(e**(-n)*f, Int((g*x)**(m + n)*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
def replacement1058(a, b, c, d, e, f, g, m, n, p, q, r, x):
return Dist(e, Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**(r + S(-1)), x), x) + Dist(e**(-n)*f, Int((g*x)**(m + n)*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**(r + S(-1)), x), x)
def replacement1059(a, b, c, d, e, f, m, n, p, q, r, x):
return -Subst(Int(x**(-m + S(-2))*(a + b*x**(-n))**p*(c + d*x**(-n))**q*(e + f*x**(-n))**r, x), x, S(1)/x)
def With1060(a, b, c, d, e, f, g, m, n, p, q, r, x):
k = Denominator(m)
return -Dist(k/g, Subst(Int(x**(-k*(m + S(1)) + S(-1))*(a + b*g**(-n)*x**(-k*n))**p*(c + d*g**(-n)*x**(-k*n))**q*(e + f*g**(-n)*x**(-k*n))**r, x), x, (g*x)**(-S(1)/k)), x)
def replacement1061(a, b, c, d, e, f, g, m, n, p, q, r, x):
return -Dist((g*x)**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(a + b*x**(-n))**p*(c + d*x**(-n))**q*(e + f*x**(-n))**r, x), x, S(1)/x), x)
def With1062(a, b, c, d, e, f, m, n, p, q, r, x):
k = Denominator(n)
return Dist(k, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*x**(k*n))**p*(c + d*x**(k*n))**q*(e + f*x**(k*n))**r, x), x, x**(S(1)/k)), x)
def replacement1063(a, b, c, d, e, f, g, m, n, p, q, r, x):
return Dist(g**IntPart(m)*x**(-FracPart(m))*(g*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
def replacement1064(a, b, c, d, e, f, m, n, p, q, r, x):
return Dist(S(1)/(m + S(1)), Subst(Int((a + b*x**(n/(m + S(1))))**p*(c + d*x**(n/(m + S(1))))**q*(e + f*x**(n/(m + S(1))))**r, x), x, x**(m + S(1))), x)
def replacement1065(a, b, c, d, e, f, g, m, n, p, q, r, x):
return Dist(g**IntPart(m)*x**(-FracPart(m))*(g*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
def replacement1066(a, b, c, d, e, f, g, m, n, p, q, x):
return Dist(S(1)/(a*b*n*(p + S(1))), Int((g*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*Simp(c*(b*e*n*(p + S(1)) + (m + S(1))*(-a*f + b*e)) + d*x**n*(b*e*n*(p + S(1)) + (-a*f + b*e)*(m + n*q + S(1))), x), x), x) - Simp((g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*(-a*f + b*e)/(a*b*g*n*(p + S(1))), x)
def replacement1067(a, b, c, d, e, f, g, m, n, p, q, x):
return Dist(S(1)/(a*n*(p + S(1))*(-a*d + b*c)), Int((g*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(c*(m + S(1))*(-a*f + b*e) + d*x**n*(-a*f + b*e)*(m + n*(p + q + S(2)) + S(1)) + e*n*(p + S(1))*(-a*d + b*c), x), x), x) - Simp((g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))*(-a*f + b*e)/(a*g*n*(p + S(1))*(-a*d + b*c)), x)
def replacement1068(a, b, c, d, e, f, g, m, n, p, q, x):
return Dist(S(1)/(b*(m + n*(p + q + S(1)) + S(1))), Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**(q + S(-1))*Simp(c*(b*e*n*(p + q + S(1)) + (m + S(1))*(-a*f + b*e)) + x**n*(b*d*e*n*(p + q + S(1)) + d*(m + S(1))*(-a*f + b*e) + f*n*q*(-a*d + b*c)), x), x), x) + Simp(f*(g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(b*g*(m + n*(p + q + S(1)) + S(1))), x)
def replacement1069(a, b, c, d, e, f, g, m, n, p, x):
return Int(ExpandIntegrand((g*x)**m*(a + b*x**n)**p*(e + f*x**n)/(c + d*x**n), x), x)
def replacement1070(a, b, c, d, e, f, g, m, n, p, q, x):
return Dist(e, Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x) + Dist(f*x**(-m)*(g*x)**m, Int(x**(m + n)*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
def replacement1071(a, b, c, d, e, f, m, mn, n, p, q, r, x):
return Int(x**(m - n*q)*(a + b*x**n)**p*(e + f*x**n)**r*(c*x**n + d)**q, x)
def replacement1072(a, b, c, d, e, f, m, mn, n, p, q, r, x):
return Int(x**(m + n*(p + r))*(c + d*x**(-n))**q*(a*x**(-n) + b)**p*(e*x**(-n) + f)**r, x)
def replacement1073(a, b, c, d, e, f, m, mn, n, p, q, r, x):
return Dist(x**(n*FracPart(q))*(c + d*x**(-n))**FracPart(q)*(c*x**n + d)**(-FracPart(q)), Int(x**(m - n*q)*(a + b*x**n)**p*(e + f*x**n)**r*(c*x**n + d)**q, x), x)
def replacement1074(a, b, c, d, e, f, g, m, mn, n, p, q, r, x):
return Dist(g**IntPart(m)*x**(-FracPart(m))*(g*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**(-n))**q*(e + f*x**n)**r, x), x)
def replacement1075(a, b, c, d, e, f, g, m, n, p, q, r, x):
return Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x)
def replacement1076(a, b, c, d, e, f, m, n, p, q, r, u, v, x):
return Dist(u**m*v**(-m)/Coefficient(v, x, S(1)), Subst(Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x, v), x)
def replacement1077(a, b, c, d, e1, e2, f1, f2, g, m, n, n2, p, q, r, x):
return Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**q*(e1*e2 + f1*f2*x**n)**r, x)
def replacement1078(a, b, c, d, e1, e2, f1, f2, g, m, n, n2, p, q, r, x):
return Dist((e1 + f1*x**(n/S(2)))**FracPart(r)*(e2 + f2*x**(n/S(2)))**FracPart(r)*(e1*e2 + f1*f2*x**n)**(-FracPart(r)), Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**q*(e1*e2 + f1*f2*x**n)**r, x), x)
Server : Apache/2.4.41 (Ubuntu)