3.2572 \(\int x^{-1-4 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=135 \[ -\frac{a^8 x^{-4 n}}{4 n}-\frac{8 a^7 b x^{-3 n}}{3 n}-\frac{14 a^6 b^2 x^{-2 n}}{n}-\frac{56 a^5 b^3 x^{-n}}{n}+70 a^4 b^4 \log (x)+\frac{56 a^3 b^5 x^n}{n}+\frac{14 a^2 b^6 x^{2 n}}{n}+\frac{8 a b^7 x^{3 n}}{3 n}+\frac{b^8 x^{4 n}}{4 n} \]

[Out]

-a^8/(4*n*x^(4*n)) - (8*a^7*b)/(3*n*x^(3*n)) - (14*a^6*b^2)/(n*x^(2*n)) - (56*a^
5*b^3)/(n*x^n) + (56*a^3*b^5*x^n)/n + (14*a^2*b^6*x^(2*n))/n + (8*a*b^7*x^(3*n))
/(3*n) + (b^8*x^(4*n))/(4*n) + 70*a^4*b^4*Log[x]

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Rubi [A]  time = 0.157235, antiderivative size = 135, 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 \[ -\frac{a^8 x^{-4 n}}{4 n}-\frac{8 a^7 b x^{-3 n}}{3 n}-\frac{14 a^6 b^2 x^{-2 n}}{n}-\frac{56 a^5 b^3 x^{-n}}{n}+70 a^4 b^4 \log (x)+\frac{56 a^3 b^5 x^n}{n}+\frac{14 a^2 b^6 x^{2 n}}{n}+\frac{8 a b^7 x^{3 n}}{3 n}+\frac{b^8 x^{4 n}}{4 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 4*n)*(a + b*x^n)^8,x]

[Out]

-a^8/(4*n*x^(4*n)) - (8*a^7*b)/(3*n*x^(3*n)) - (14*a^6*b^2)/(n*x^(2*n)) - (56*a^
5*b^3)/(n*x^n) + (56*a^3*b^5*x^n)/n + (14*a^2*b^6*x^(2*n))/n + (8*a*b^7*x^(3*n))
/(3*n) + (b^8*x^(4*n))/(4*n) + 70*a^4*b^4*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{8} x^{- 4 n}}{4 n} - \frac{8 a^{7} b x^{- 3 n}}{3 n} - \frac{14 a^{6} b^{2} x^{- 2 n}}{n} - \frac{56 a^{5} b^{3} x^{- n}}{n} + \frac{70 a^{4} b^{4} \log{\left (x^{n} \right )}}{n} + \frac{56 a^{3} b^{5} x^{n}}{n} + \frac{28 a^{2} b^{6} \int ^{x^{n}} x\, dx}{n} + \frac{8 a b^{7} x^{3 n}}{3 n} + \frac{b^{8} x^{4 n}}{4 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-4*n)*(a+b*x**n)**8,x)

[Out]

-a**8*x**(-4*n)/(4*n) - 8*a**7*b*x**(-3*n)/(3*n) - 14*a**6*b**2*x**(-2*n)/n - 56
*a**5*b**3*x**(-n)/n + 70*a**4*b**4*log(x**n)/n + 56*a**3*b**5*x**n/n + 28*a**2*
b**6*Integral(x, (x, x**n))/n + 8*a*b**7*x**(3*n)/(3*n) + b**8*x**(4*n)/(4*n)

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Mathematica [A]  time = 0.0562005, size = 111, normalized size = 0.82 \[ -\frac{3 a^8 x^{-4 n}+32 a^7 b x^{-3 n}+168 a^6 b^2 x^{-2 n}+672 a^5 b^3 x^{-n}-840 a^4 b^4 n \log (x)-672 a^3 b^5 x^n-168 a^2 b^6 x^{2 n}-32 a b^7 x^{3 n}-3 b^8 x^{4 n}}{12 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 4*n)*(a + b*x^n)^8,x]

[Out]

-((3*a^8)/x^(4*n) + (32*a^7*b)/x^(3*n) + (168*a^6*b^2)/x^(2*n) + (672*a^5*b^3)/x
^n - 672*a^3*b^5*x^n - 168*a^2*b^6*x^(2*n) - 32*a*b^7*x^(3*n) - 3*b^8*x^(4*n) -
840*a^4*b^4*n*Log[x])/(12*n)

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Maple [A]  time = 0.044, size = 128, normalized size = 1. \[ 70\,{a}^{4}{b}^{4}\ln \left ( x \right ) +{\frac{{b}^{8} \left ({x}^{n} \right ) ^{4}}{4\,n}}+{\frac{8\,a{b}^{7} \left ({x}^{n} \right ) ^{3}}{3\,n}}+14\,{\frac{{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{2}}{n}}+56\,{\frac{{a}^{3}{b}^{5}{x}^{n}}{n}}-56\,{\frac{{a}^{5}{b}^{3}}{n{x}^{n}}}-14\,{\frac{{a}^{6}{b}^{2}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{8\,b{a}^{7}}{3\,n \left ({x}^{n} \right ) ^{3}}}-{\frac{{a}^{8}}{4\,n \left ({x}^{n} \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-4*n)*(a+b*x^n)^8,x)

[Out]

70*a^4*b^4*ln(x)+1/4*b^8/n*(x^n)^4+8/3*a*b^7/n*(x^n)^3+14*a^2*b^6/n*(x^n)^2+56*a
^3*b^5*x^n/n-56*a^5*b^3/n/(x^n)-14*a^6*b^2/n/(x^n)^2-8/3*a^7*b/n/(x^n)^3-1/4*a^8
/n/(x^n)^4

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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^(-4*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226898, size = 157, normalized size = 1.16 \[ \frac{840 \, a^{4} b^{4} n x^{4 \, n} \log \left (x\right ) + 3 \, b^{8} x^{8 \, n} + 32 \, a b^{7} x^{7 \, n} + 168 \, a^{2} b^{6} x^{6 \, n} + 672 \, a^{3} b^{5} x^{5 \, n} - 672 \, a^{5} b^{3} x^{3 \, n} - 168 \, a^{6} b^{2} x^{2 \, n} - 32 \, a^{7} b x^{n} - 3 \, a^{8}}{12 \, n x^{4 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-4*n - 1),x, algorithm="fricas")

[Out]

1/12*(840*a^4*b^4*n*x^(4*n)*log(x) + 3*b^8*x^(8*n) + 32*a*b^7*x^(7*n) + 168*a^2*
b^6*x^(6*n) + 672*a^3*b^5*x^(5*n) - 672*a^5*b^3*x^(3*n) - 168*a^6*b^2*x^(2*n) -
32*a^7*b*x^n - 3*a^8)/(n*x^(4*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**(-1-4*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23332, size = 167, normalized size = 1.24 \[ \frac{{\left (840 \, a^{4} b^{4} n e^{\left (4 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + 3 \, b^{8} e^{\left (8 \, n{\rm ln}\left (x\right )\right )} + 32 \, a b^{7} e^{\left (7 \, n{\rm ln}\left (x\right )\right )} + 168 \, a^{2} b^{6} e^{\left (6 \, n{\rm ln}\left (x\right )\right )} + 672 \, a^{3} b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} - 672 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} - 168 \, a^{6} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 32 \, a^{7} b e^{\left (n{\rm ln}\left (x\right )\right )} - 3 \, a^{8}\right )} e^{\left (-4 \, n{\rm ln}\left (x\right )\right )}}{12 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-4*n - 1),x, algorithm="giac")

[Out]

1/12*(840*a^4*b^4*n*e^(4*n*ln(x))*ln(x) + 3*b^8*e^(8*n*ln(x)) + 32*a*b^7*e^(7*n*
ln(x)) + 168*a^2*b^6*e^(6*n*ln(x)) + 672*a^3*b^5*e^(5*n*ln(x)) - 672*a^5*b^3*e^(
3*n*ln(x)) - 168*a^6*b^2*e^(2*n*ln(x)) - 32*a^7*b*e^(n*ln(x)) - 3*a^8)*e^(-4*n*l
n(x))/n