3.2582 \(\int x^{-1-14 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=151 \[ -\frac{a^8 x^{-14 n}}{14 n}-\frac{8 a^7 b x^{-13 n}}{13 n}-\frac{7 a^6 b^2 x^{-12 n}}{3 n}-\frac{56 a^5 b^3 x^{-11 n}}{11 n}-\frac{7 a^4 b^4 x^{-10 n}}{n}-\frac{56 a^3 b^5 x^{-9 n}}{9 n}-\frac{7 a^2 b^6 x^{-8 n}}{2 n}-\frac{8 a b^7 x^{-7 n}}{7 n}-\frac{b^8 x^{-6 n}}{6 n} \]

[Out]

-a^8/(14*n*x^(14*n)) - (8*a^7*b)/(13*n*x^(13*n)) - (7*a^6*b^2)/(3*n*x^(12*n)) -
(56*a^5*b^3)/(11*n*x^(11*n)) - (7*a^4*b^4)/(n*x^(10*n)) - (56*a^3*b^5)/(9*n*x^(9
*n)) - (7*a^2*b^6)/(2*n*x^(8*n)) - (8*a*b^7)/(7*n*x^(7*n)) - b^8/(6*n*x^(6*n))

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Rubi [A]  time = 0.171064, antiderivative size = 151, 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 \[ -\frac{a^8 x^{-14 n}}{14 n}-\frac{8 a^7 b x^{-13 n}}{13 n}-\frac{7 a^6 b^2 x^{-12 n}}{3 n}-\frac{56 a^5 b^3 x^{-11 n}}{11 n}-\frac{7 a^4 b^4 x^{-10 n}}{n}-\frac{56 a^3 b^5 x^{-9 n}}{9 n}-\frac{7 a^2 b^6 x^{-8 n}}{2 n}-\frac{8 a b^7 x^{-7 n}}{7 n}-\frac{b^8 x^{-6 n}}{6 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 14*n)*(a + b*x^n)^8,x]

[Out]

-a^8/(14*n*x^(14*n)) - (8*a^7*b)/(13*n*x^(13*n)) - (7*a^6*b^2)/(3*n*x^(12*n)) -
(56*a^5*b^3)/(11*n*x^(11*n)) - (7*a^4*b^4)/(n*x^(10*n)) - (56*a^3*b^5)/(9*n*x^(9
*n)) - (7*a^2*b^6)/(2*n*x^(8*n)) - (8*a*b^7)/(7*n*x^(7*n)) - b^8/(6*n*x^(6*n))

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Rubi in Sympy [A]  time = 28.08, size = 138, normalized size = 0.91 \[ - \frac{a^{8} x^{- 14 n}}{14 n} - \frac{8 a^{7} b x^{- 13 n}}{13 n} - \frac{7 a^{6} b^{2} x^{- 12 n}}{3 n} - \frac{56 a^{5} b^{3} x^{- 11 n}}{11 n} - \frac{7 a^{4} b^{4} x^{- 10 n}}{n} - \frac{56 a^{3} b^{5} x^{- 9 n}}{9 n} - \frac{7 a^{2} b^{6} x^{- 8 n}}{2 n} - \frac{8 a b^{7} x^{- 7 n}}{7 n} - \frac{b^{8} x^{- 6 n}}{6 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-14*n)*(a+b*x**n)**8,x)

[Out]

-a**8*x**(-14*n)/(14*n) - 8*a**7*b*x**(-13*n)/(13*n) - 7*a**6*b**2*x**(-12*n)/(3
*n) - 56*a**5*b**3*x**(-11*n)/(11*n) - 7*a**4*b**4*x**(-10*n)/n - 56*a**3*b**5*x
**(-9*n)/(9*n) - 7*a**2*b**6*x**(-8*n)/(2*n) - 8*a*b**7*x**(-7*n)/(7*n) - b**8*x
**(-6*n)/(6*n)

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Mathematica [A]  time = 0.0525172, size = 113, normalized size = 0.75 \[ -\frac{x^{-14 n} \left (1287 a^8+11088 a^7 b x^n+42042 a^6 b^2 x^{2 n}+91728 a^5 b^3 x^{3 n}+126126 a^4 b^4 x^{4 n}+112112 a^3 b^5 x^{5 n}+63063 a^2 b^6 x^{6 n}+20592 a b^7 x^{7 n}+3003 b^8 x^{8 n}\right )}{18018 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 14*n)*(a + b*x^n)^8,x]

[Out]

-(1287*a^8 + 11088*a^7*b*x^n + 42042*a^6*b^2*x^(2*n) + 91728*a^5*b^3*x^(3*n) + 1
26126*a^4*b^4*x^(4*n) + 112112*a^3*b^5*x^(5*n) + 63063*a^2*b^6*x^(6*n) + 20592*a
*b^7*x^(7*n) + 3003*b^8*x^(8*n))/(18018*n*x^(14*n))

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Maple [A]  time = 0.042, size = 136, normalized size = 0.9 \[ -{\frac{{b}^{8}}{6\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{8\,a{b}^{7}}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{7\,{a}^{2}{b}^{6}}{2\,n \left ({x}^{n} \right ) ^{8}}}-{\frac{56\,{a}^{3}{b}^{5}}{9\,n \left ({x}^{n} \right ) ^{9}}}-7\,{\frac{{a}^{4}{b}^{4}}{n \left ({x}^{n} \right ) ^{10}}}-{\frac{56\,{a}^{5}{b}^{3}}{11\,n \left ({x}^{n} \right ) ^{11}}}-{\frac{7\,{a}^{6}{b}^{2}}{3\,n \left ({x}^{n} \right ) ^{12}}}-{\frac{8\,b{a}^{7}}{13\,n \left ({x}^{n} \right ) ^{13}}}-{\frac{{a}^{8}}{14\,n \left ({x}^{n} \right ) ^{14}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-14*n)*(a+b*x^n)^8,x)

[Out]

-1/6*b^8/n/(x^n)^6-8/7*a*b^7/n/(x^n)^7-7/2*a^2*b^6/n/(x^n)^8-56/9*a^3*b^5/n/(x^n
)^9-7*a^4*b^4/n/(x^n)^10-56/11*a^5*b^3/n/(x^n)^11-7/3*a^6*b^2/n/(x^n)^12-8/13*a^
7*b/n/(x^n)^13-1/14*a^8/n/(x^n)^14

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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^(-14*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22651, size = 153, normalized size = 1.01 \[ -\frac{3003 \, b^{8} x^{8 \, n} + 20592 \, a b^{7} x^{7 \, n} + 63063 \, a^{2} b^{6} x^{6 \, n} + 112112 \, a^{3} b^{5} x^{5 \, n} + 126126 \, a^{4} b^{4} x^{4 \, n} + 91728 \, a^{5} b^{3} x^{3 \, n} + 42042 \, a^{6} b^{2} x^{2 \, n} + 11088 \, a^{7} b x^{n} + 1287 \, a^{8}}{18018 \, n x^{14 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-14*n - 1),x, algorithm="fricas")

[Out]

-1/18018*(3003*b^8*x^(8*n) + 20592*a*b^7*x^(7*n) + 63063*a^2*b^6*x^(6*n) + 11211
2*a^3*b^5*x^(5*n) + 126126*a^4*b^4*x^(4*n) + 91728*a^5*b^3*x^(3*n) + 42042*a^6*b
^2*x^(2*n) + 11088*a^7*b*x^n + 1287*a^8)/(n*x^(14*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**(-1-14*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.235877, size = 163, normalized size = 1.08 \[ -\frac{{\left (3003 \, b^{8} e^{\left (8 \, n{\rm ln}\left (x\right )\right )} + 20592 \, a b^{7} e^{\left (7 \, n{\rm ln}\left (x\right )\right )} + 63063 \, a^{2} b^{6} e^{\left (6 \, n{\rm ln}\left (x\right )\right )} + 112112 \, a^{3} b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 126126 \, a^{4} b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 91728 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 42042 \, a^{6} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 11088 \, a^{7} b e^{\left (n{\rm ln}\left (x\right )\right )} + 1287 \, a^{8}\right )} e^{\left (-14 \, n{\rm ln}\left (x\right )\right )}}{18018 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-14*n - 1),x, algorithm="giac")

[Out]

-1/18018*(3003*b^8*e^(8*n*ln(x)) + 20592*a*b^7*e^(7*n*ln(x)) + 63063*a^2*b^6*e^(
6*n*ln(x)) + 112112*a^3*b^5*e^(5*n*ln(x)) + 126126*a^4*b^4*e^(4*n*ln(x)) + 91728
*a^5*b^3*e^(3*n*ln(x)) + 42042*a^6*b^2*e^(2*n*ln(x)) + 11088*a^7*b*e^(n*ln(x)) +
1287*a^8)*e^(-14*n*ln(x))/n