3.2574 \(\int x^{-1-6 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=135 \[ -\frac{a^8 x^{-6 n}}{6 n}-\frac{8 a^7 b x^{-5 n}}{5 n}-\frac{7 a^6 b^2 x^{-4 n}}{n}-\frac{56 a^5 b^3 x^{-3 n}}{3 n}-\frac{35 a^4 b^4 x^{-2 n}}{n}-\frac{56 a^3 b^5 x^{-n}}{n}+28 a^2 b^6 \log (x)+\frac{8 a b^7 x^n}{n}+\frac{b^8 x^{2 n}}{2 n} \]

[Out]

-a^8/(6*n*x^(6*n)) - (8*a^7*b)/(5*n*x^(5*n)) - (7*a^6*b^2)/(n*x^(4*n)) - (56*a^5
*b^3)/(3*n*x^(3*n)) - (35*a^4*b^4)/(n*x^(2*n)) - (56*a^3*b^5)/(n*x^n) + (8*a*b^7
*x^n)/n + (b^8*x^(2*n))/(2*n) + 28*a^2*b^6*Log[x]

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Rubi [A]  time = 0.158556, 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 \[ -\frac{a^8 x^{-6 n}}{6 n}-\frac{8 a^7 b x^{-5 n}}{5 n}-\frac{7 a^6 b^2 x^{-4 n}}{n}-\frac{56 a^5 b^3 x^{-3 n}}{3 n}-\frac{35 a^4 b^4 x^{-2 n}}{n}-\frac{56 a^3 b^5 x^{-n}}{n}+28 a^2 b^6 \log (x)+\frac{8 a b^7 x^n}{n}+\frac{b^8 x^{2 n}}{2 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 6*n)*(a + b*x^n)^8,x]

[Out]

-a^8/(6*n*x^(6*n)) - (8*a^7*b)/(5*n*x^(5*n)) - (7*a^6*b^2)/(n*x^(4*n)) - (56*a^5
*b^3)/(3*n*x^(3*n)) - (35*a^4*b^4)/(n*x^(2*n)) - (56*a^3*b^5)/(n*x^n) + (8*a*b^7
*x^n)/n + (b^8*x^(2*n))/(2*n) + 28*a^2*b^6*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{8} x^{- 6 n}}{6 n} - \frac{8 a^{7} b x^{- 5 n}}{5 n} - \frac{7 a^{6} b^{2} x^{- 4 n}}{n} - \frac{56 a^{5} b^{3} x^{- 3 n}}{3 n} - \frac{35 a^{4} b^{4} x^{- 2 n}}{n} - \frac{56 a^{3} b^{5} x^{- n}}{n} + \frac{28 a^{2} b^{6} \log{\left (x^{n} \right )}}{n} + \frac{8 a b^{7} x^{n}}{n} + \frac{b^{8} \int ^{x^{n}} x\, dx}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-6*n)*(a+b*x**n)**8,x)

[Out]

-a**8*x**(-6*n)/(6*n) - 8*a**7*b*x**(-5*n)/(5*n) - 7*a**6*b**2*x**(-4*n)/n - 56*
a**5*b**3*x**(-3*n)/(3*n) - 35*a**4*b**4*x**(-2*n)/n - 56*a**3*b**5*x**(-n)/n +
28*a**2*b**6*log(x**n)/n + 8*a*b**7*x**n/n + b**8*Integral(x, (x, x**n))/n

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Mathematica [A]  time = 0.155288, size = 111, normalized size = 0.82 \[ 28 a^2 b^6 \log (x)-\frac{x^{-6 n} \left (5 a^8+48 a^7 b x^n+210 a^6 b^2 x^{2 n}+560 a^5 b^3 x^{3 n}+1050 a^4 b^4 x^{4 n}+1680 a^3 b^5 x^{5 n}-240 a b^7 x^{7 n}-15 b^8 x^{8 n}\right )}{30 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 6*n)*(a + b*x^n)^8,x]

[Out]

-(5*a^8 + 48*a^7*b*x^n + 210*a^6*b^2*x^(2*n) + 560*a^5*b^3*x^(3*n) + 1050*a^4*b^
4*x^(4*n) + 1680*a^3*b^5*x^(5*n) - 240*a*b^7*x^(7*n) - 15*b^8*x^(8*n))/(30*n*x^(
6*n)) + 28*a^2*b^6*Log[x]

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Maple [A]  time = 0.043, size = 128, normalized size = 1. \[ 28\,{a}^{2}{b}^{6}\ln \left ( x \right ) +{\frac{{b}^{8} \left ({x}^{n} \right ) ^{2}}{2\,n}}+8\,{\frac{a{b}^{7}{x}^{n}}{n}}-56\,{\frac{{a}^{3}{b}^{5}}{n{x}^{n}}}-35\,{\frac{{a}^{4}{b}^{4}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{56\,{a}^{5}{b}^{3}}{3\,n \left ({x}^{n} \right ) ^{3}}}-7\,{\frac{{a}^{6}{b}^{2}}{n \left ({x}^{n} \right ) ^{4}}}-{\frac{8\,b{a}^{7}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{{a}^{8}}{6\,n \left ({x}^{n} \right ) ^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-6*n)*(a+b*x^n)^8,x)

[Out]

28*a^2*b^6*ln(x)+1/2*b^8/n*(x^n)^2+8*a*b^7*x^n/n-56*a^3*b^5/n/(x^n)-35*a^4*b^4/n
/(x^n)^2-56/3*a^5*b^3/n/(x^n)^3-7*a^6*b^2/n/(x^n)^4-8/5*a^7*b/n/(x^n)^5-1/6*a^8/
n/(x^n)^6

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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^(-6*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228656, size = 157, normalized size = 1.16 \[ \frac{840 \, a^{2} b^{6} n x^{6 \, n} \log \left (x\right ) + 15 \, b^{8} x^{8 \, n} + 240 \, a b^{7} x^{7 \, n} - 1680 \, a^{3} b^{5} x^{5 \, n} - 1050 \, a^{4} b^{4} x^{4 \, n} - 560 \, a^{5} b^{3} x^{3 \, n} - 210 \, a^{6} b^{2} x^{2 \, n} - 48 \, a^{7} b x^{n} - 5 \, a^{8}}{30 \, n x^{6 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-6*n - 1),x, algorithm="fricas")

[Out]

1/30*(840*a^2*b^6*n*x^(6*n)*log(x) + 15*b^8*x^(8*n) + 240*a*b^7*x^(7*n) - 1680*a
^3*b^5*x^(5*n) - 1050*a^4*b^4*x^(4*n) - 560*a^5*b^3*x^(3*n) - 210*a^6*b^2*x^(2*n
) - 48*a^7*b*x^n - 5*a^8)/(n*x^(6*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**(-1-6*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.236683, size = 167, normalized size = 1.24 \[ \frac{{\left (840 \, a^{2} b^{6} n e^{\left (6 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + 15 \, b^{8} e^{\left (8 \, n{\rm ln}\left (x\right )\right )} + 240 \, a b^{7} e^{\left (7 \, n{\rm ln}\left (x\right )\right )} - 1680 \, a^{3} b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} - 1050 \, a^{4} b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} - 560 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} - 210 \, a^{6} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 48 \, a^{7} b e^{\left (n{\rm ln}\left (x\right )\right )} - 5 \, a^{8}\right )} e^{\left (-6 \, n{\rm ln}\left (x\right )\right )}}{30 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-6*n - 1),x, algorithm="giac")

[Out]

1/30*(840*a^2*b^6*n*e^(6*n*ln(x))*ln(x) + 15*b^8*e^(8*n*ln(x)) + 240*a*b^7*e^(7*
n*ln(x)) - 1680*a^3*b^5*e^(5*n*ln(x)) - 1050*a^4*b^4*e^(4*n*ln(x)) - 560*a^5*b^3
*e^(3*n*ln(x)) - 210*a^6*b^2*e^(2*n*ln(x)) - 48*a^7*b*e^(n*ln(x)) - 5*a^8)*e^(-6
*n*ln(x))/n