def With36(Pm, Qm, a, b, c, n, n2, p, x):
if isinstance(x, (int, Integer, float, Float)):
return False
m = Expon(Pm, x)
if And(Equal(Expon(Qm, x), m + S(-1)), ZeroQ(-Qm*m*Coeff(Pm, x, m) + Coeff(Qm, x, m + S(-1))*D(Pm, x))):
return True
return False
def replacement34(p, Pm, b, n2, c, a, n, Qm, x):
m = Expon(Pm, x)
rubi.append(34)
return Dist(
Coeff(Qm, x, m + S(-1)) / (m * Coeff(Pm, x, m)),
Subst(Int((a + b * x**n + c * x**(S(2) * n))**p, x), x, Pm), x)
def replacement33(p, Pm, b, a, n, Qm, x):
m = Expon(Pm, x)
rubi.append(33)
return Dist(
Coeff(Qm, x, m + S(-1)) / (m * Coeff(Pm, x, m)),
Subst(Int((a + b * x**n)**p, x), x, Pm), x)
def replacement36(Pm, Qm, a, b, c, n, n2, p, x):
m = Expon(Pm, x)
return Dist(
Coeff(Qm, x, m + S(-1)) / (m * Coeff(Pm, x, m)),
Subst(Int((a + b * x**n + c * x**(S(2) * n))**p, x), x, Pm),
x,
)
def replacement35(Pm, Qm, a, b, n, p, x):
m = Expon(Pm, x)
return Dist(
Coeff(Qm, x, m + S(-1)) / (m * Coeff(Pm, x, m)),
Subst(Int((a + b * x**n)**p, x), x, Pm),
x,
)
def With35(a, n, Pm, c, Qm, n2, x, b, p):
m = Expon(Pm, x)
return Coeff(Qm, x, m + S(-1)) * Subst(
Int((a + b * x**n + c * x**(S(2) * n))**p, x), x,
Pm) / (m * Coeff(Pm, x, m))
def With34(a, Pm, n, Qm, x, b, p):
m = Expon(Pm, x)
return Coeff(Qm, x, m + S(-1)) * Subst(Int(
(a + b * x**n)**p, x), x, Pm) / (m * Coeff(Pm, x, m))
def integrand_simplification(rubi):
pattern1 = Pattern(
Integral((a_ + x_**WC('n', S(1)) * WC('b', S(1)))**WC('p', S(1)) *
WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda a: ZeroQ(a)))
rule1 = ReplacementRule(pattern1,
lambda n, a, u, x, b, p: Int(u * (b * x**n)**p, x))
rubi.add(rule1)
pattern2 = Pattern(
Integral((x_**WC('n', S(1)) * WC('b', S(1)) + WC('a', S(0)))**WC(
'p', S(1)) * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda b: ZeroQ(b)))
rule2 = ReplacementRule(pattern2,
lambda a, n, u, x, b, p: Int(a**p * u, x))
rubi.add(rule2)
pattern3 = Pattern(
Integral(
(a_ + x_**WC('j', S(1)) * WC('c', S(1)) +
x_**WC('n', S(1)) * WC('b', S(1)))**WC('p', S(1)) * WC('u', S(1)),
x_), CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda n, j: ZeroQ(j - S(2) * n)),
CustomConstraint(lambda a: ZeroQ(a)))
rule3 = ReplacementRule(
pattern3, lambda n, a, u, c, j, x, b, p: Int(
u * (b * x**n + c * x**(S(2) * n))**p, x))
rubi.add(rule3)
pattern4 = Pattern(
Integral((x_**WC('j', S(1)) * WC('c', S(1)) +
x_**WC('n', S(1)) * WC('b', S(1)) + WC('a', S(0)))**WC(
'p', S(1)) * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda n, j: ZeroQ(j - S(2) * n)),
CustomConstraint(lambda b: ZeroQ(b)))
rule4 = ReplacementRule(
pattern4,
lambda a, n, u, c, j, x, b, p: Int(u * (a + c * x**(S(2) * n))**p, x))
rubi.add(rule4)
pattern5 = Pattern(
Integral((x_**WC('j', S(1)) * WC('c', S(1)) +
x_**WC('n', S(1)) * WC('b', S(1)) + WC('a', S(0)))**WC(
'p', S(1)) * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda n, j: ZeroQ(j - S(2) * n)),
CustomConstraint(lambda c: ZeroQ(c)))
rule5 = ReplacementRule(
pattern5, lambda a, n, u, c, j, x, b, p: Int(u * (a + b * x**n)**p, x))
rubi.add(rule5)
pattern6 = Pattern(
Integral((v_ * WC('a', S(1)) + v_ * WC('b', S(1)) + WC('w', S(0)))**WC(
'p', S(1)) * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda v, x: NFreeQ(v, x)))
rule6 = ReplacementRule(
pattern6, lambda a, u, w, v, x, b, p: Int(u * (v * (a + b) + w)**p, x))
rubi.add(rule6)
pattern7 = Pattern(Integral(Pm_**p_ * WC('u', S(1)), x_),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda Pm, x: PolyQ(Pm, x)),
CustomConstraint(lambda p: Not(RationalQ(p))),
CustomConstraint(lambda p: RationalQ(p)))
rule7 = ReplacementRule(pattern7, lambda Pm, u, x, p: Int(Pm**p * u, x))
rubi.add(rule7)
pattern8 = Pattern(Integral(a_, x_),
CustomConstraint(lambda a, x: FreeQ(a, x)))
rule8 = ReplacementRule(pattern8, lambda a, x: a * x)
rubi.add(rule8)
pattern9 = Pattern(Integral(a_ * (b_ + x_ * WC('c', S(1))), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)))
rule9 = ReplacementRule(
pattern9, lambda a, x, c, b: a * (b + c * x)**S(2) / (S(2) * c))
rubi.add(rule9)
pattern10 = Pattern(Integral(-u_, x_))
rule10 = ReplacementRule(pattern10, lambda x, u: Int(u, x) * I)
rubi.add(rule10)
pattern11 = Pattern(Integral(u_ * Complex(S(0), a_), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda a: EqQ(a**S(2), S(1))))
rule11 = ReplacementRule(pattern11,
lambda a, x, u: Int(u, x) * Complex(I, a))
rubi.add(rule11)
pattern12 = Pattern(Integral(a_ * u_, x_),
CustomConstraint(lambda a, x: FreeQ(a, x)))
rule12 = ReplacementRule(pattern12, lambda a, x, u: a * Int(u, x))
rubi.add(rule12)
pattern13 = Pattern(Integral(u_, x_), CustomConstraint(lambda u: SumQ(u)))
rule13 = ReplacementRule(pattern13, lambda x, u: IntSum(u, x))
rubi.add(rule13)
pattern14 = Pattern(
Integral(v_**WC('m', S(1)) * (b_ * v_)**n_ * WC('u', S(1)), x_),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda m: IntegerQ(m)))
rule14 = ReplacementRule(
pattern14,
lambda n, u, m, v, x, b: b**(-m) * Int(u * (b * v)**(m + n), x))
rubi.add(rule14)
pattern15 = Pattern(
Integral((v_ * WC('a', S(1)))**m_ *
(v_ * WC('b', S(1)))**n_ * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda m: Not(IntegerQ(m))),
CustomConstraint(lambda n: PositiveIntegerQ(n + S(1) / 2)),
CustomConstraint(lambda n, m: IntegerQ(m + n)))
rule15 = ReplacementRule(
pattern15, lambda a, n, u, m, v, x, b: a**(m + S(1) / 2) * b**
(n + S(-1) / 2) * sqrt(b * v) * Int(u * v**(m + n), x) / sqrt(a * v))
rubi.add(rule15)
pattern16 = Pattern(
Integral((v_ * WC('a', S(1)))**m_ *
(v_ * WC('b', S(1)))**n_ * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda m: Not(IntegerQ(m))),
CustomConstraint(lambda n: NegativeIntegerQ(n + S(-1) / 2)),
CustomConstraint(lambda n, m: IntegerQ(m + n)))
rule16 = ReplacementRule(
pattern16, lambda a, n, u, m, v, x, b: a**(m + S(-1) / 2) * b**
(n + S(1) / 2) * sqrt(a * v) * Int(u * v**(m + n), x) / sqrt(b * v))
rubi.add(rule16)
pattern17 = Pattern(
Integral((v_ * WC('a', S(1)))**m_ *
(v_ * WC('b', S(1)))**n_ * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda m: Not(IntegerQ(m))),
CustomConstraint(lambda n: Not(IntegerQ(n))),
CustomConstraint(lambda n, m: IntegerQ(m + n)))
rule17 = ReplacementRule(
pattern17, lambda a, n, u, m, v, x, b: a**(m + n) * (a * v)**(-n) *
(b * v)**n * Int(u * v**(m + n), x))
rubi.add(rule17)
pattern18 = Pattern(
Integral((v_ * WC('a', S(1)))**m_ *
(v_ * WC('b', S(1)))**n_ * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda m: Not(IntegerQ(m))),
CustomConstraint(lambda n: Not(IntegerQ(n))),
CustomConstraint(lambda n, m: Not(IntegerQ(m + n))))
rule18 = ReplacementRule(
pattern18,
lambda a, n, u, m, v, x, b: a**(-IntPart(n)) * b**IntPart(n) *
(a * v)**(-FracPart(n)) *
(b * v)**FracPart(n) * Int(u * (a * v)**(m + n), x))
rubi.add(rule18)
pattern19 = Pattern(
Integral((a_ + v_ * WC('b', S(1)))**WC('m', S(1)) *
(c_ + v_ * WC('d', S(1)))**WC('n', S(1)) * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda d, x: FreeQ(d, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda a, d, c, b: ZeroQ(-a * d + b * c)),
CustomConstraint(lambda m: IntegerQ(m)),
CustomConstraint(lambda n, a, c, x, d, b: Not(IntegerQ(n)) | SimplerQ(
c + d * x, a + b * x)))
rule19 = ReplacementRule(
pattern19, lambda n, a, u, m, c, x, v, d, b:
(b / d)**m * Int(u * (c + d * v)**(m + n), x))
rubi.add(rule19)
pattern20 = Pattern(
Integral((a_ + v_ * WC('b', S(1)))**m_ *
(c_ + v_ * WC('d', S(1)))**n_ * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda d, x: FreeQ(d, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda a, d, c, b: ZeroQ(-a * d + b * c)),
CustomConstraint(lambda d, b: PositiveQ(b / d)),
CustomConstraint(lambda n, m: Not(IntegerQ(m) | IntegerQ(n))))
rule20 = ReplacementRule(
pattern20, lambda a, n, u, m, c, x, v, d, b:
(b / d)**m * Int(u * (c + d * v)**(m + n), x))
rubi.add(rule20)
pattern21 = Pattern(
Integral((a_ + v_ * WC('b', S(1)))**m_ *
(c_ + v_ * WC('d', S(1)))**n_ * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda d, x: FreeQ(d, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda a, d, c, b: ZeroQ(-a * d + b * c)),
CustomConstraint(lambda n, d, b, m: Not(
IntegerQ(m) | IntegerQ(n) | PositiveQ(b / d))))
rule21 = ReplacementRule(
pattern21, lambda a, n, u, m, c, x, v, d, b: (a + b * v)**m *
(c + d * v)**(-m) * Int(u * (c + d * v)**(m + n), x))
rubi.add(rule21)
pattern22 = Pattern(
Integral(
(v_ * WC('a', S(1)))**m_ *
(v_**S(2) * WC('c', S(1)) + v_ * WC('b', S(1))) * WC('u', S(1)),
x_), CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda m: RationalQ(m)),
CustomConstraint(lambda m: LessEqual(m, S(-1))))
rule22 = ReplacementRule(
pattern22,
lambda a, u, m, c, v, x, b: Int(u * (a * v)**(m + S(1)) *
(b + c * v), x) / a)
rubi.add(rule22)
pattern23 = Pattern(
Integral(
(a_ + v_ * WC('b', S(1)))**m_ *
(v_**S(2) * WC('C', S(1)) + v_ * WC('B', S(1)) + WC('A', S(0))) *
WC('u', S(1)), x_), CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda A, x: FreeQ(A, x)),
CustomConstraint(lambda B, x: FreeQ(B, x)),
CustomConstraint(lambda C, x: FreeQ(C, x)),
CustomConstraint(lambda a, C, A, B, b: ZeroQ(A * b**S(2) - B * a * b +
C * a**S(2))),
CustomConstraint(lambda m: RationalQ(m)),
CustomConstraint(lambda m: LessEqual(m, S(-1))))
rule23 = ReplacementRule(
pattern23, lambda a, u, C, m, A, B, v, x, b: Int(
u * (a + b * v)**
(m + S(1)) * Simp(B * b - C * a + C * b * v, x), x) / b**S(2))
rubi.add(rule23)
pattern24 = Pattern(
Integral((a_ + x_**WC('n', S(1)) * WC('b', S(1)))**WC('m', S(1)) *
(c_ + x_**WC('q', S(1)) * WC('d', S(1)))**WC('p', S(1)) *
WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda d, x: FreeQ(d, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda n, q: ZeroQ(n + q)),
CustomConstraint(lambda p: IntegerQ(p)),
CustomConstraint(lambda a, d, c, b: ZeroQ(a * c - b * d)),
CustomConstraint(lambda n, m: Not(IntegerQ(m) & NegQ(n))))
rule24 = ReplacementRule(
pattern24, lambda n, a, u, m, c, x, q, d, b, p:
(d / a)**p * Int(u * x**(-n * p) * (a + b * x**n)**(m + p), x))
rubi.add(rule24)
pattern25 = Pattern(
Integral((a_ + x_**WC('n', S(1)) * WC('b', S(1)))**WC('m', S(1)) *
(c_ + x_**j_ * WC('d', S(1)))**WC('p', S(1)) * WC('u', S(1)),
x_), CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda d, x: FreeQ(d, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda n, j: ZeroQ(j - S(2) * n)),
CustomConstraint(lambda m, p: ZeroQ(m + p)),
CustomConstraint(lambda a, d, c, b: ZeroQ(a**S(2) * d + b**S(2) * c)),
CustomConstraint(lambda a: PositiveQ(a)),
CustomConstraint(lambda d: NegativeQ(d)))
rule25 = ReplacementRule(
pattern25, lambda n, a, u, m, c, j, x, d, b, p:
(-b**S(2) / d)**m * Int(u * (a - b * x**n)**(-m), x))
rubi.add(rule25)
pattern26 = Pattern(
Integral((a_ + x_**S(2) * WC('c', S(1)) + x_ * WC('b', S(1)))**WC(
'p', S(1)) * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda a, c, b: ZeroQ(-S(4) * a * c + b**S(2))),
CustomConstraint(lambda p: IntegerQ(p)))
rule26 = ReplacementRule(
pattern26, lambda a, u, c, x, b, p: Int(
S(2)**(-S(2) * p) * c**(-p) * u *
(b + S(2) * c * x)**(S(2) * p), x))
rubi.add(rule26)
pattern27 = Pattern(
Integral(
(a_ + x_**n_ * WC('b', S(1)) + x_**WC('n2', S(1)) * WC('c', S(1)))
**WC('p', S(1)) * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda n, n2: ZeroQ(-S(2) * n + n2)),
CustomConstraint(lambda a, c, b: ZeroQ(-S(4) * a * c + b**S(2))),
CustomConstraint(lambda p: IntegerQ(p)))
rule27 = ReplacementRule(
pattern27, lambda a, n, u, c, b, x, n2, p: c**
(-p) * Int(u * (b / S(2) + c * x**n)**(S(2) * p), x))
rubi.add(rule27)
pattern28 = Pattern(
Integral(
(d_ + x_ * WC('e', S(1))) *
(x_**S(2) * WC('c', S(1)) + x_ * WC('b', S(1)) + WC('a', S(0)))**
WC('p', S(1)), x_), CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda d, x: FreeQ(d, x)),
CustomConstraint(lambda e, x: FreeQ(e, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda d, c, b, e: ZeroQ(-b * e + S(2) * c * d)))
rule28 = ReplacementRule(
pattern28, lambda a, e, c, x, d, b, p: d * Subst(
Int(x**p, x), x, a + b * x + c * x**S(2)) / b)
rubi.add(rule28)
pattern29 = Pattern(
Integral(
(x_**WC('p', S(1)) * WC('a', S(1)) +
x_**WC('q', S(1)) * WC('b', S(1)))**WC('m', S(1)) * WC('u', S(1)),
x_), CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda q, x: FreeQ(q, x)),
CustomConstraint(lambda m: IntegerQ(m)),
CustomConstraint(lambda q, p: PosQ(-p + q)))
rule29 = ReplacementRule(
pattern29, lambda a, u, m, q, x, b, p: Int(
u * x**(m * p) * (a + b * x**(-p + q))**m, x))
rubi.add(rule29)
pattern30 = Pattern(
Integral(
(x_**WC('p', S(1)) * WC('a', S(1)) +
x_**WC('q', S(1)) * WC('b', S(1)) +
x_**WC('r', S(1)) * WC('c', S(1)))**WC('m', S(1)) * WC('u', S(1)),
x_), CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda q, x: FreeQ(q, x)),
CustomConstraint(lambda r, x: FreeQ(r, x)),
CustomConstraint(lambda m: IntegerQ(m)),
CustomConstraint(lambda q, p: PosQ(-p + q)),
CustomConstraint(lambda r, p: PosQ(-p + r)))
rule30 = ReplacementRule(
pattern30, lambda a, u, m, c, r, q, x, b, p: Int(
u * x**(m * p) * (a + b * x**(-p + q) + c * x**(-p + r))**m, x))
rubi.add(rule30)
pattern31 = Pattern(
Integral(x_**WC('m', S(1)) / (a_ + x_**n_ * WC('b', S(1))), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda n, m: ZeroQ(m - n + S(1))))
rule31 = ReplacementRule(
pattern31, lambda a, n, m, x, b: log(RemoveContent(a + b * x**n, x)) /
(b * n))
rubi.add(rule31)
pattern32 = Pattern(
Integral(x_**WC('m', S(1)) * (a_ + x_**n_ * WC('b', S(1)))**p_, x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda n, m: ZeroQ(m - n + S(1))),
CustomConstraint(lambda p: NonzeroQ(p + S(1))))
rule32 = ReplacementRule(
pattern32, lambda a, n, m, x, b, p: (a + b * x**n)**(p + S(1)) /
(b * n * (p + S(1))))
rubi.add(rule32)
pattern33 = Pattern(
Integral(
x_**WC('m', S(1)) *
(a1_ + x_**WC('n', S(1)) * WC('b1', S(1)))**p_ *
(a2_ + x_**WC('n', S(1)) * WC('b2', S(1)))**p_, x_),
CustomConstraint(lambda a1, x: FreeQ(a1, x)),
CustomConstraint(lambda b1, x: FreeQ(b1, x)),
CustomConstraint(lambda a2, x: FreeQ(a2, x)),
CustomConstraint(lambda b2, x: FreeQ(b2, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda a1, a2, b2, b1: ZeroQ(a1 * b2 + a2 * b1)),
CustomConstraint(lambda n, m: ZeroQ(m - S(2) * n + S(1))),
CustomConstraint(lambda p: NonzeroQ(p + S(1))))
rule33 = ReplacementRule(
pattern33, lambda n, b2, m, b1, a2, a1, x, p:
(a1 + b1 * x**n)**(p + S(1)) * (a2 + b2 * x**n)**(p + S(1)) /
(S(2) * b1 * b2 * n * (p + S(1))))
rubi.add(rule33)
pattern34 = Pattern(
Integral(
Qm_ * (Pm_**WC('n', S(1)) * WC('b', S(1)) + WC('a', S(0)))**WC(
'p', S(1)), x_), CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda Pm, x: PolyQ(Pm, x)),
CustomConstraint(lambda x, Qm: PolyQ(Qm, x)),
CustomConstraint(lambda Qm, p, a, x, m, b, Pm, n: Equal(
Expon(Qm, x), m + S(-1)) & ZeroQ(-Qm * m * Coeff(Pm, x, m) + Coeff(
Qm, x, m + S(-1)) * D(Pm, x))))
def With34(a, Pm, n, Qm, x, b, p):
m = Expon(Pm, x)
return Coeff(Qm, x, m + S(-1)) * Subst(Int(
(a + b * x**n)**p, x), x, Pm) / (m * Coeff(Pm, x, m))
rule34 = ReplacementRule(
pattern34, lambda a, Pm, n, Qm, x, b, p: With34(a, Pm, n, Qm, x, b, p))
rubi.add(rule34)
pattern35 = Pattern(
Integral(
Qm_ * (Pm_**WC('n', S(1)) * WC('b', S(1)) + Pm_**WC('n2', S(1)) *
WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda n, n2: ZeroQ(-S(2) * n + n2)),
CustomConstraint(lambda Pm, x: PolyQ(Pm, x)),
CustomConstraint(lambda x, Qm: PolyQ(Qm, x)),
CustomConstraint(lambda Qm, c, p, a, x, m, b, Pm, n: Equal(
Expon(Qm, x), m + S(-1)) & ZeroQ(-Qm * m * Coeff(Pm, x, m) + Coeff(
Qm, x, m + S(-1)) * D(Pm, x))))
def With35(a, n, Pm, c, Qm, n2, x, b, p):
m = Expon(Pm, x)
return Coeff(Qm, x, m + S(-1)) * Subst(
Int((a + b * x**n + c * x**(S(2) * n))**p, x), x,
Pm) / (m * Coeff(Pm, x, m))
rule35 = ReplacementRule(
pattern35, lambda a, n, Pm, c, Qm, n2, x, b, p: With35(
a, n, Pm, c, Qm, n2, x, b, p))
rubi.add(rule35)
pattern36 = Pattern(
Integral(Pq_**m_ * Qr_**p_ * WC('u', S(1)), x_),
CustomConstraint(lambda m: PositiveIntegerQ(m)),
CustomConstraint(lambda p: NegativeIntegerQ(p)),
CustomConstraint(lambda x, Pq: PolyQ(Pq, x)),
CustomConstraint(lambda x, Qr: PolyQ(Qr, x)),
CustomConstraint(
lambda m, p, u, gcd, Pq, Qr, x: NonzeroQ(gcd + S(-1))))
def With36(u, m, Pq, x, Qr, p):
gcd = PolyGCD(Pq, Qr, x)
return Int(
gcd**(m + p) * u * PolynomialQuotient(Pq, gcd, x)**m *
PolynomialQuotient(Qr, gcd, x)**p, x)
rule36 = ReplacementRule(
pattern36, lambda u, m, Pq, x, Qr, p: With36(u, m, Pq, x, Qr, p))
rubi.add(rule36)
pattern37 = Pattern(
Integral(Pq_ * Qr_**p_ * WC('u', S(1)), x_),
CustomConstraint(lambda p: NegativeIntegerQ(p)),
CustomConstraint(lambda x, Pq: PolyQ(Pq, x)),
CustomConstraint(lambda x, Qr: PolyQ(Qr, x)),
CustomConstraint(lambda p, u, gcd, Pq, Qr, x: NonzeroQ(gcd + S(-1))))
def With37(u, Pq, x, Qr, p):
gcd = PolyGCD(Pq, Qr, x)
return Int(
gcd**(p + S(1)) * u * PolynomialQuotient(Pq, gcd, x) *
PolynomialQuotient(Qr, gcd, x)**p, x)
rule37 = ReplacementRule(pattern37,
lambda u, Pq, x, Qr, p: With37(u, Pq, x, Qr, p))
rubi.add(rule37)
return rubi