3.2557 \(\int x^{-1-9 n} \left (a+b x^n\right )^5 \, dx\)

Optimal. Leaf size=97 \[ -\frac{a^5 x^{-9 n}}{9 n}-\frac{5 a^4 b x^{-8 n}}{8 n}-\frac{10 a^3 b^2 x^{-7 n}}{7 n}-\frac{5 a^2 b^3 x^{-6 n}}{3 n}-\frac{a b^4 x^{-5 n}}{n}-\frac{b^5 x^{-4 n}}{4 n} \]

[Out]

-a^5/(9*n*x^(9*n)) - (5*a^4*b)/(8*n*x^(8*n)) - (10*a^3*b^2)/(7*n*x^(7*n)) - (5*a
^2*b^3)/(3*n*x^(6*n)) - (a*b^4)/(n*x^(5*n)) - b^5/(4*n*x^(4*n))

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Rubi [A]  time = 0.103649, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{a^5 x^{-9 n}}{9 n}-\frac{5 a^4 b x^{-8 n}}{8 n}-\frac{10 a^3 b^2 x^{-7 n}}{7 n}-\frac{5 a^2 b^3 x^{-6 n}}{3 n}-\frac{a b^4 x^{-5 n}}{n}-\frac{b^5 x^{-4 n}}{4 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 9*n)*(a + b*x^n)^5,x]

[Out]

-a^5/(9*n*x^(9*n)) - (5*a^4*b)/(8*n*x^(8*n)) - (10*a^3*b^2)/(7*n*x^(7*n)) - (5*a
^2*b^3)/(3*n*x^(6*n)) - (a*b^4)/(n*x^(5*n)) - b^5/(4*n*x^(4*n))

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Rubi in Sympy [A]  time = 16.9831, size = 85, normalized size = 0.88 \[ - \frac{a^{5} x^{- 9 n}}{9 n} - \frac{5 a^{4} b x^{- 8 n}}{8 n} - \frac{10 a^{3} b^{2} x^{- 7 n}}{7 n} - \frac{5 a^{2} b^{3} x^{- 6 n}}{3 n} - \frac{a b^{4} x^{- 5 n}}{n} - \frac{b^{5} x^{- 4 n}}{4 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-9*n)*(a+b*x**n)**5,x)

[Out]

-a**5*x**(-9*n)/(9*n) - 5*a**4*b*x**(-8*n)/(8*n) - 10*a**3*b**2*x**(-7*n)/(7*n)
- 5*a**2*b**3*x**(-6*n)/(3*n) - a*b**4*x**(-5*n)/n - b**5*x**(-4*n)/(4*n)

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Mathematica [A]  time = 0.0367907, size = 74, normalized size = 0.76 \[ -\frac{x^{-9 n} \left (56 a^5+315 a^4 b x^n+720 a^3 b^2 x^{2 n}+840 a^2 b^3 x^{3 n}+504 a b^4 x^{4 n}+126 b^5 x^{5 n}\right )}{504 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 9*n)*(a + b*x^n)^5,x]

[Out]

-(56*a^5 + 315*a^4*b*x^n + 720*a^3*b^2*x^(2*n) + 840*a^2*b^3*x^(3*n) + 504*a*b^4
*x^(4*n) + 126*b^5*x^(5*n))/(504*n*x^(9*n))

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Maple [A]  time = 0.035, size = 88, normalized size = 0.9 \[ -{\frac{{b}^{5}}{4\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{a{b}^{4}}{n \left ({x}^{n} \right ) ^{5}}}-{\frac{5\,{a}^{2}{b}^{3}}{3\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{10\,{a}^{3}{b}^{2}}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{5\,{a}^{4}b}{8\,n \left ({x}^{n} \right ) ^{8}}}-{\frac{{a}^{5}}{9\,n \left ({x}^{n} \right ) ^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-9*n)*(a+b*x^n)^5,x)

[Out]

-1/4*b^5/n/(x^n)^4-a*b^4/n/(x^n)^5-5/3*a^2*b^3/n/(x^n)^6-10/7*a^3*b^2/n/(x^n)^7-
5/8*a^4*b/n/(x^n)^8-1/9*a^5/n/(x^n)^9

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-9*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22556, size = 100, normalized size = 1.03 \[ -\frac{126 \, b^{5} x^{5 \, n} + 504 \, a b^{4} x^{4 \, n} + 840 \, a^{2} b^{3} x^{3 \, n} + 720 \, a^{3} b^{2} x^{2 \, n} + 315 \, a^{4} b x^{n} + 56 \, a^{5}}{504 \, n x^{9 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-9*n - 1),x, algorithm="fricas")

[Out]

-1/504*(126*b^5*x^(5*n) + 504*a*b^4*x^(4*n) + 840*a^2*b^3*x^(3*n) + 720*a^3*b^2*
x^(2*n) + 315*a^4*b*x^n + 56*a^5)/(n*x^(9*n))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1-9*n)*(a+b*x**n)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227272, size = 107, normalized size = 1.1 \[ -\frac{{\left (126 \, b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 504 \, a b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 840 \, a^{2} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 720 \, a^{3} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 315 \, a^{4} b e^{\left (n{\rm ln}\left (x\right )\right )} + 56 \, a^{5}\right )} e^{\left (-9 \, n{\rm ln}\left (x\right )\right )}}{504 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-9*n - 1),x, algorithm="giac")

[Out]

-1/504*(126*b^5*e^(5*n*ln(x)) + 504*a*b^4*e^(4*n*ln(x)) + 840*a^2*b^3*e^(3*n*ln(
x)) + 720*a^3*b^2*e^(2*n*ln(x)) + 315*a^4*b*e^(n*ln(x)) + 56*a^5)*e^(-9*n*ln(x))
/n