3.2573 \(\int x^{-1-5 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=133 \[ -\frac{a^8 x^{-5 n}}{5 n}-\frac{2 a^7 b x^{-4 n}}{n}-\frac{28 a^6 b^2 x^{-3 n}}{3 n}-\frac{28 a^5 b^3 x^{-2 n}}{n}-\frac{70 a^4 b^4 x^{-n}}{n}+56 a^3 b^5 \log (x)+\frac{28 a^2 b^6 x^n}{n}+\frac{4 a b^7 x^{2 n}}{n}+\frac{b^8 x^{3 n}}{3 n} \]

[Out]

-a^8/(5*n*x^(5*n)) - (2*a^7*b)/(n*x^(4*n)) - (28*a^6*b^2)/(3*n*x^(3*n)) - (28*a^
5*b^3)/(n*x^(2*n)) - (70*a^4*b^4)/(n*x^n) + (28*a^2*b^6*x^n)/n + (4*a*b^7*x^(2*n
))/n + (b^8*x^(3*n))/(3*n) + 56*a^3*b^5*Log[x]

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Rubi [A]  time = 0.155108, antiderivative size = 133, 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 \[ -\frac{a^8 x^{-5 n}}{5 n}-\frac{2 a^7 b x^{-4 n}}{n}-\frac{28 a^6 b^2 x^{-3 n}}{3 n}-\frac{28 a^5 b^3 x^{-2 n}}{n}-\frac{70 a^4 b^4 x^{-n}}{n}+56 a^3 b^5 \log (x)+\frac{28 a^2 b^6 x^n}{n}+\frac{4 a b^7 x^{2 n}}{n}+\frac{b^8 x^{3 n}}{3 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 5*n)*(a + b*x^n)^8,x]

[Out]

-a^8/(5*n*x^(5*n)) - (2*a^7*b)/(n*x^(4*n)) - (28*a^6*b^2)/(3*n*x^(3*n)) - (28*a^
5*b^3)/(n*x^(2*n)) - (70*a^4*b^4)/(n*x^n) + (28*a^2*b^6*x^n)/n + (4*a*b^7*x^(2*n
))/n + (b^8*x^(3*n))/(3*n) + 56*a^3*b^5*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{8} x^{- 5 n}}{5 n} - \frac{2 a^{7} b x^{- 4 n}}{n} - \frac{28 a^{6} b^{2} x^{- 3 n}}{3 n} - \frac{28 a^{5} b^{3} x^{- 2 n}}{n} - \frac{70 a^{4} b^{4} x^{- n}}{n} + \frac{56 a^{3} b^{5} \log{\left (x^{n} \right )}}{n} + \frac{28 a^{2} b^{6} x^{n}}{n} + \frac{8 a b^{7} \int ^{x^{n}} x\, dx}{n} + \frac{b^{8} x^{3 n}}{3 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-5*n)*(a+b*x**n)**8,x)

[Out]

-a**8*x**(-5*n)/(5*n) - 2*a**7*b*x**(-4*n)/n - 28*a**6*b**2*x**(-3*n)/(3*n) - 28
*a**5*b**3*x**(-2*n)/n - 70*a**4*b**4*x**(-n)/n + 56*a**3*b**5*log(x**n)/n + 28*
a**2*b**6*x**n/n + 8*a*b**7*Integral(x, (x, x**n))/n + b**8*x**(3*n)/(3*n)

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Mathematica [A]  time = 0.152126, size = 111, normalized size = 0.83 \[ 56 a^3 b^5 \log (x)-\frac{x^{-5 n} \left (3 a^8+30 a^7 b x^n+140 a^6 b^2 x^{2 n}+420 a^5 b^3 x^{3 n}+1050 a^4 b^4 x^{4 n}-420 a^2 b^6 x^{6 n}-60 a b^7 x^{7 n}-5 b^8 x^{8 n}\right )}{15 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 5*n)*(a + b*x^n)^8,x]

[Out]

-(3*a^8 + 30*a^7*b*x^n + 140*a^6*b^2*x^(2*n) + 420*a^5*b^3*x^(3*n) + 1050*a^4*b^
4*x^(4*n) - 420*a^2*b^6*x^(6*n) - 60*a*b^7*x^(7*n) - 5*b^8*x^(8*n))/(15*n*x^(5*n
)) + 56*a^3*b^5*Log[x]

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Maple [A]  time = 0.045, size = 128, normalized size = 1. \[ 56\,{a}^{3}{b}^{5}\ln \left ( x \right ) +{\frac{{b}^{8} \left ({x}^{n} \right ) ^{3}}{3\,n}}+4\,{\frac{a{b}^{7} \left ({x}^{n} \right ) ^{2}}{n}}+28\,{\frac{{a}^{2}{b}^{6}{x}^{n}}{n}}-70\,{\frac{{a}^{4}{b}^{4}}{n{x}^{n}}}-28\,{\frac{{a}^{5}{b}^{3}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{28\,{a}^{6}{b}^{2}}{3\,n \left ({x}^{n} \right ) ^{3}}}-2\,{\frac{b{a}^{7}}{n \left ({x}^{n} \right ) ^{4}}}-{\frac{{a}^{8}}{5\,n \left ({x}^{n} \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-5*n)*(a+b*x^n)^8,x)

[Out]

56*a^3*b^5*ln(x)+1/3*b^8/n*(x^n)^3+4*a*b^7/n*(x^n)^2+28*a^2*b^6*x^n/n-70*a^4*b^4
/n/(x^n)-28*a^5*b^3/n/(x^n)^2-28/3*a^6*b^2/n/(x^n)^3-2*a^7*b/n/(x^n)^4-1/5*a^8/n
/(x^n)^5

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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)^8*x^(-5*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228435, size = 157, normalized size = 1.18 \[ \frac{840 \, a^{3} b^{5} n x^{5 \, n} \log \left (x\right ) + 5 \, b^{8} x^{8 \, n} + 60 \, a b^{7} x^{7 \, n} + 420 \, a^{2} b^{6} x^{6 \, n} - 1050 \, a^{4} b^{4} x^{4 \, n} - 420 \, a^{5} b^{3} x^{3 \, n} - 140 \, a^{6} b^{2} x^{2 \, n} - 30 \, a^{7} b x^{n} - 3 \, a^{8}}{15 \, n x^{5 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-5*n - 1),x, algorithm="fricas")

[Out]

1/15*(840*a^3*b^5*n*x^(5*n)*log(x) + 5*b^8*x^(8*n) + 60*a*b^7*x^(7*n) + 420*a^2*
b^6*x^(6*n) - 1050*a^4*b^4*x^(4*n) - 420*a^5*b^3*x^(3*n) - 140*a^6*b^2*x^(2*n) -
30*a^7*b*x^n - 3*a^8)/(n*x^(5*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**(-1-5*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.236424, size = 167, normalized size = 1.26 \[ \frac{{\left (840 \, a^{3} b^{5} n e^{\left (5 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + 5 \, b^{8} e^{\left (8 \, n{\rm ln}\left (x\right )\right )} + 60 \, a b^{7} e^{\left (7 \, n{\rm ln}\left (x\right )\right )} + 420 \, a^{2} b^{6} e^{\left (6 \, n{\rm ln}\left (x\right )\right )} - 1050 \, a^{4} b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} - 420 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} - 140 \, a^{6} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 30 \, a^{7} b e^{\left (n{\rm ln}\left (x\right )\right )} - 3 \, a^{8}\right )} e^{\left (-5 \, n{\rm ln}\left (x\right )\right )}}{15 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-5*n - 1),x, algorithm="giac")

[Out]

1/15*(840*a^3*b^5*n*e^(5*n*ln(x))*ln(x) + 5*b^8*e^(8*n*ln(x)) + 60*a*b^7*e^(7*n*
ln(x)) + 420*a^2*b^6*e^(6*n*ln(x)) - 1050*a^4*b^4*e^(4*n*ln(x)) - 420*a^5*b^3*e^
(3*n*ln(x)) - 140*a^6*b^2*e^(2*n*ln(x)) - 30*a^7*b*e^(n*ln(x)) - 3*a^8)*e^(-5*n*
ln(x))/n