3.2558 \(\int x^{-1-10 n} \left (a+b x^n\right )^5 \, dx\)

Optimal. Leaf size=99 \[ -\frac{a^5 x^{-10 n}}{10 n}-\frac{5 a^4 b x^{-9 n}}{9 n}-\frac{5 a^3 b^2 x^{-8 n}}{4 n}-\frac{10 a^2 b^3 x^{-7 n}}{7 n}-\frac{5 a b^4 x^{-6 n}}{6 n}-\frac{b^5 x^{-5 n}}{5 n} \]

[Out]

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

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Rubi [A]  time = 0.103916, antiderivative size = 99, 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 \[ -\frac{a^5 x^{-10 n}}{10 n}-\frac{5 a^4 b x^{-9 n}}{9 n}-\frac{5 a^3 b^2 x^{-8 n}}{4 n}-\frac{10 a^2 b^3 x^{-7 n}}{7 n}-\frac{5 a b^4 x^{-6 n}}{6 n}-\frac{b^5 x^{-5 n}}{5 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 10*n)*(a + b*x^n)^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 16.9153, size = 88, normalized size = 0.89 \[ - \frac{a^{5} x^{- 10 n}}{10 n} - \frac{5 a^{4} b x^{- 9 n}}{9 n} - \frac{5 a^{3} b^{2} x^{- 8 n}}{4 n} - \frac{10 a^{2} b^{3} x^{- 7 n}}{7 n} - \frac{5 a b^{4} x^{- 6 n}}{6 n} - \frac{b^{5} x^{- 5 n}}{5 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-10*n)*(a+b*x**n)**5,x)

[Out]

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

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Mathematica [A]  time = 0.0384869, size = 74, normalized size = 0.75 \[ -\frac{x^{-10 n} \left (126 a^5+700 a^4 b x^n+1575 a^3 b^2 x^{2 n}+1800 a^2 b^3 x^{3 n}+1050 a b^4 x^{4 n}+252 b^5 x^{5 n}\right )}{1260 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 10*n)*(a + b*x^n)^5,x]

[Out]

-(126*a^5 + 700*a^4*b*x^n + 1575*a^3*b^2*x^(2*n) + 1800*a^2*b^3*x^(3*n) + 1050*a
*b^4*x^(4*n) + 252*b^5*x^(5*n))/(1260*n*x^(10*n))

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Maple [A]  time = 0.037, size = 88, normalized size = 0.9 \[ -{\frac{{b}^{5}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{5\,a{b}^{4}}{6\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{10\,{a}^{2}{b}^{3}}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{5\,{a}^{3}{b}^{2}}{4\,n \left ({x}^{n} \right ) ^{8}}}-{\frac{5\,{a}^{4}b}{9\,n \left ({x}^{n} \right ) ^{9}}}-{\frac{{a}^{5}}{10\,n \left ({x}^{n} \right ) ^{10}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-10*n)*(a+b*x^n)^5,x)

[Out]

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

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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^(-10*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227062, size = 100, normalized size = 1.01 \[ -\frac{252 \, b^{5} x^{5 \, n} + 1050 \, a b^{4} x^{4 \, n} + 1800 \, a^{2} b^{3} x^{3 \, n} + 1575 \, a^{3} b^{2} x^{2 \, n} + 700 \, a^{4} b x^{n} + 126 \, a^{5}}{1260 \, n x^{10 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-10*n - 1),x, algorithm="fricas")

[Out]

-1/1260*(252*b^5*x^(5*n) + 1050*a*b^4*x^(4*n) + 1800*a^2*b^3*x^(3*n) + 1575*a^3*
b^2*x^(2*n) + 700*a^4*b*x^n + 126*a^5)/(n*x^(10*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**(-1-10*n)*(a+b*x**n)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228078, size = 107, normalized size = 1.08 \[ -\frac{{\left (252 \, b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 1050 \, a b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 1800 \, a^{2} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 1575 \, a^{3} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 700 \, a^{4} b e^{\left (n{\rm ln}\left (x\right )\right )} + 126 \, a^{5}\right )} e^{\left (-10 \, n{\rm ln}\left (x\right )\right )}}{1260 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-10*n - 1),x, algorithm="giac")

[Out]

-1/1260*(252*b^5*e^(5*n*ln(x)) + 1050*a*b^4*e^(4*n*ln(x)) + 1800*a^2*b^3*e^(3*n*
ln(x)) + 1575*a^3*b^2*e^(2*n*ln(x)) + 700*a^4*b*e^(n*ln(x)) + 126*a^5)*e^(-10*n*
ln(x))/n