3.2576 \(\int x^{-1-8 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=140 \[ -\frac{a^8 x^{-8 n}}{8 n}-\frac{8 a^7 b x^{-7 n}}{7 n}-\frac{14 a^6 b^2 x^{-6 n}}{3 n}-\frac{56 a^5 b^3 x^{-5 n}}{5 n}-\frac{35 a^4 b^4 x^{-4 n}}{2 n}-\frac{56 a^3 b^5 x^{-3 n}}{3 n}-\frac{14 a^2 b^6 x^{-2 n}}{n}-\frac{8 a b^7 x^{-n}}{n}+b^8 \log (x) \]

[Out]

-a^8/(8*n*x^(8*n)) - (8*a^7*b)/(7*n*x^(7*n)) - (14*a^6*b^2)/(3*n*x^(6*n)) - (56*
a^5*b^3)/(5*n*x^(5*n)) - (35*a^4*b^4)/(2*n*x^(4*n)) - (56*a^3*b^5)/(3*n*x^(3*n))
- (14*a^2*b^6)/(n*x^(2*n)) - (8*a*b^7)/(n*x^n) + b^8*Log[x]

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Rubi [A]  time = 0.159078, antiderivative size = 140, 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 \[ -\frac{a^8 x^{-8 n}}{8 n}-\frac{8 a^7 b x^{-7 n}}{7 n}-\frac{14 a^6 b^2 x^{-6 n}}{3 n}-\frac{56 a^5 b^3 x^{-5 n}}{5 n}-\frac{35 a^4 b^4 x^{-4 n}}{2 n}-\frac{56 a^3 b^5 x^{-3 n}}{3 n}-\frac{14 a^2 b^6 x^{-2 n}}{n}-\frac{8 a b^7 x^{-n}}{n}+b^8 \log (x) \]

Antiderivative was successfully verified.

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

[Out]

-a^8/(8*n*x^(8*n)) - (8*a^7*b)/(7*n*x^(7*n)) - (14*a^6*b^2)/(3*n*x^(6*n)) - (56*
a^5*b^3)/(5*n*x^(5*n)) - (35*a^4*b^4)/(2*n*x^(4*n)) - (56*a^3*b^5)/(3*n*x^(3*n))
- (14*a^2*b^6)/(n*x^(2*n)) - (8*a*b^7)/(n*x^n) + b^8*Log[x]

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Rubi in Sympy [A]  time = 27.3092, size = 131, normalized size = 0.94 \[ - \frac{a^{8} x^{- 8 n}}{8 n} - \frac{8 a^{7} b x^{- 7 n}}{7 n} - \frac{14 a^{6} b^{2} x^{- 6 n}}{3 n} - \frac{56 a^{5} b^{3} x^{- 5 n}}{5 n} - \frac{35 a^{4} b^{4} x^{- 4 n}}{2 n} - \frac{56 a^{3} b^{5} x^{- 3 n}}{3 n} - \frac{14 a^{2} b^{6} x^{- 2 n}}{n} - \frac{8 a b^{7} x^{- n}}{n} + \frac{b^{8} \log{\left (x^{n} \right )}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**8*x**(-8*n)/(8*n) - 8*a**7*b*x**(-7*n)/(7*n) - 14*a**6*b**2*x**(-6*n)/(3*n)
- 56*a**5*b**3*x**(-5*n)/(5*n) - 35*a**4*b**4*x**(-4*n)/(2*n) - 56*a**3*b**5*x**
(-3*n)/(3*n) - 14*a**2*b**6*x**(-2*n)/n - 8*a*b**7*x**(-n)/n + b**8*log(x**n)/n

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Mathematica [A]  time = 0.0975139, size = 108, normalized size = 0.77 \[ b^8 \log (x)-\frac{a x^{-8 n} \left (105 a^7+960 a^6 b x^n+3920 a^5 b^2 x^{2 n}+9408 a^4 b^3 x^{3 n}+14700 a^3 b^4 x^{4 n}+15680 a^2 b^5 x^{5 n}+11760 a b^6 x^{6 n}+6720 b^7 x^{7 n}\right )}{840 n} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(105*a^7 + 960*a^6*b*x^n + 3920*a^5*b^2*x^(2*n) + 9408*a^4*b^3*x^(3*n) + 147
00*a^3*b^4*x^(4*n) + 15680*a^2*b^5*x^(5*n) + 11760*a*b^6*x^(6*n) + 6720*b^7*x^(7
*n)))/(840*n*x^(8*n)) + b^8*Log[x]

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Maple [A]  time = 0.043, size = 129, normalized size = 0.9 \[{b}^{8}\ln \left ( x \right ) -8\,{\frac{a{b}^{7}}{n{x}^{n}}}-14\,{\frac{{a}^{2}{b}^{6}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{56\,{a}^{3}{b}^{5}}{3\,n \left ({x}^{n} \right ) ^{3}}}-{\frac{35\,{a}^{4}{b}^{4}}{2\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{56\,{a}^{5}{b}^{3}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{14\,{a}^{6}{b}^{2}}{3\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{8\,b{a}^{7}}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{{a}^{8}}{8\,n \left ({x}^{n} \right ) ^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-8*n)*(a+b*x^n)^8,x)

[Out]

b^8*ln(x)-8*a*b^7/n/(x^n)-14*a^2*b^6/n/(x^n)^2-56/3*a^3*b^5/n/(x^n)^3-35/2*a^4*b
^4/n/(x^n)^4-56/5*a^5*b^3/n/(x^n)^5-14/3*a^6*b^2/n/(x^n)^6-8/7*a^7*b/n/(x^n)^7-1
/8*a^8/n/(x^n)^8

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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^(-8*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230016, size = 157, normalized size = 1.12 \[ \frac{840 \, b^{8} n x^{8 \, n} \log \left (x\right ) - 6720 \, a b^{7} x^{7 \, n} - 11760 \, a^{2} b^{6} x^{6 \, n} - 15680 \, a^{3} b^{5} x^{5 \, n} - 14700 \, a^{4} b^{4} x^{4 \, n} - 9408 \, a^{5} b^{3} x^{3 \, n} - 3920 \, a^{6} b^{2} x^{2 \, n} - 960 \, a^{7} b x^{n} - 105 \, a^{8}}{840 \, n x^{8 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/840*(840*b^8*n*x^(8*n)*log(x) - 6720*a*b^7*x^(7*n) - 11760*a^2*b^6*x^(6*n) - 1
5680*a^3*b^5*x^(5*n) - 14700*a^4*b^4*x^(4*n) - 9408*a^5*b^3*x^(3*n) - 3920*a^6*b
^2*x^(2*n) - 960*a^7*b*x^n - 105*a^8)/(n*x^(8*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**(-1-8*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23567, size = 167, normalized size = 1.19 \[ \frac{{\left (840 \, b^{8} n e^{\left (8 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) - 6720 \, a b^{7} e^{\left (7 \, n{\rm ln}\left (x\right )\right )} - 11760 \, a^{2} b^{6} e^{\left (6 \, n{\rm ln}\left (x\right )\right )} - 15680 \, a^{3} b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} - 14700 \, a^{4} b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} - 9408 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} - 3920 \, a^{6} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 960 \, a^{7} b e^{\left (n{\rm ln}\left (x\right )\right )} - 105 \, a^{8}\right )} e^{\left (-8 \, n{\rm ln}\left (x\right )\right )}}{840 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/840*(840*b^8*n*e^(8*n*ln(x))*ln(x) - 6720*a*b^7*e^(7*n*ln(x)) - 11760*a^2*b^6*
e^(6*n*ln(x)) - 15680*a^3*b^5*e^(5*n*ln(x)) - 14700*a^4*b^4*e^(4*n*ln(x)) - 9408
*a^5*b^3*e^(3*n*ln(x)) - 3920*a^6*b^2*e^(2*n*ln(x)) - 960*a^7*b*e^(n*ln(x)) - 10
5*a^8)*e^(-8*n*ln(x))/n