3.2570 \(\int x^{-1-2 n} \left (a+b x^n\right )^8 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.151508, 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 \[ -\frac{a^8 x^{-2 n}}{2 n}-\frac{8 a^7 b x^{-n}}{n}+28 a^6 b^2 \log (x)+\frac{56 a^5 b^3 x^n}{n}+\frac{35 a^4 b^4 x^{2 n}}{n}+\frac{56 a^3 b^5 x^{3 n}}{3 n}+\frac{7 a^2 b^6 x^{4 n}}{n}+\frac{8 a b^7 x^{5 n}}{5 n}+\frac{b^8 x^{6 n}}{6 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 2*n)*(a + b*x^n)^8,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-2*n)*(a+b*x**n)**8,x)

[Out]

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

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Mathematica [A]  time = 0.0519688, size = 116, normalized size = 0.86 \[ \frac{x^{-2 n} \left (-15 a^8-240 a^7 b x^n+840 a^6 b^2 n x^{2 n} \log (x)+1680 a^5 b^3 x^{3 n}+1050 a^4 b^4 x^{4 n}+560 a^3 b^5 x^{5 n}+210 a^2 b^6 x^{6 n}+48 a b^7 x^{7 n}+5 b^8 x^{8 n}\right )}{30 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 2*n)*(a + b*x^n)^8,x]

[Out]

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

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Maple [A]  time = 0.045, size = 128, normalized size = 1. \[ 28\,{a}^{6}{b}^{2}\ln \left ( x \right ) +{\frac{{b}^{8} \left ({x}^{n} \right ) ^{6}}{6\,n}}+{\frac{8\,a{b}^{7} \left ({x}^{n} \right ) ^{5}}{5\,n}}+7\,{\frac{{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{4}}{n}}+{\frac{56\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{3}}{3\,n}}+35\,{\frac{{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{2}}{n}}+56\,{\frac{{a}^{5}{b}^{3}{x}^{n}}{n}}-8\,{\frac{b{a}^{7}}{n{x}^{n}}}-{\frac{{a}^{8}}{2\,n \left ({x}^{n} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-2*n)*(a+b*x^n)^8,x)

[Out]

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

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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^(-2*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228129, size = 157, normalized size = 1.16 \[ \frac{840 \, a^{6} b^{2} n x^{2 \, n} \log \left (x\right ) + 5 \, b^{8} x^{8 \, n} + 48 \, a b^{7} x^{7 \, n} + 210 \, a^{2} b^{6} x^{6 \, n} + 560 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} + 1680 \, a^{5} b^{3} x^{3 \, n} - 240 \, a^{7} b x^{n} - 15 \, a^{8}}{30 \, n x^{2 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-2*n - 1),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233199, size = 167, normalized size = 1.24 \[ \frac{{\left (840 \, a^{6} b^{2} n e^{\left (2 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + 5 \, b^{8} e^{\left (8 \, n{\rm ln}\left (x\right )\right )} + 48 \, a b^{7} e^{\left (7 \, n{\rm ln}\left (x\right )\right )} + 210 \, a^{2} b^{6} e^{\left (6 \, n{\rm ln}\left (x\right )\right )} + 560 \, a^{3} b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 1050 \, a^{4} b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 1680 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} - 240 \, a^{7} b e^{\left (n{\rm ln}\left (x\right )\right )} - 15 \, a^{8}\right )} e^{\left (-2 \, n{\rm ln}\left (x\right )\right )}}{30 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-2*n - 1),x, algorithm="giac")

[Out]

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