3.2575 \(\int x^{-1-7 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=134 \[ -\frac{a^8 x^{-7 n}}{7 n}-\frac{4 a^7 b x^{-6 n}}{3 n}-\frac{28 a^6 b^2 x^{-5 n}}{5 n}-\frac{14 a^5 b^3 x^{-4 n}}{n}-\frac{70 a^4 b^4 x^{-3 n}}{3 n}-\frac{28 a^3 b^5 x^{-2 n}}{n}-\frac{28 a^2 b^6 x^{-n}}{n}+8 a b^7 \log (x)+\frac{b^8 x^n}{n} \]

[Out]

-a^8/(7*n*x^(7*n)) - (4*a^7*b)/(3*n*x^(6*n)) - (28*a^6*b^2)/(5*n*x^(5*n)) - (14*
a^5*b^3)/(n*x^(4*n)) - (70*a^4*b^4)/(3*n*x^(3*n)) - (28*a^3*b^5)/(n*x^(2*n)) - (
28*a^2*b^6)/(n*x^n) + (b^8*x^n)/n + 8*a*b^7*Log[x]

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Rubi [A]  time = 0.157496, antiderivative size = 134, 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 \[ -\frac{a^8 x^{-7 n}}{7 n}-\frac{4 a^7 b x^{-6 n}}{3 n}-\frac{28 a^6 b^2 x^{-5 n}}{5 n}-\frac{14 a^5 b^3 x^{-4 n}}{n}-\frac{70 a^4 b^4 x^{-3 n}}{3 n}-\frac{28 a^3 b^5 x^{-2 n}}{n}-\frac{28 a^2 b^6 x^{-n}}{n}+8 a b^7 \log (x)+\frac{b^8 x^n}{n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 7*n)*(a + b*x^n)^8,x]

[Out]

-a^8/(7*n*x^(7*n)) - (4*a^7*b)/(3*n*x^(6*n)) - (28*a^6*b^2)/(5*n*x^(5*n)) - (14*
a^5*b^3)/(n*x^(4*n)) - (70*a^4*b^4)/(3*n*x^(3*n)) - (28*a^3*b^5)/(n*x^(2*n)) - (
28*a^2*b^6)/(n*x^n) + (b^8*x^n)/n + 8*a*b^7*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-7*n)*(a+b*x**n)**8,x)

[Out]

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

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Mathematica [A]  time = 0.190376, size = 111, normalized size = 0.83 \[ 8 a b^7 \log (x)-\frac{x^{-7 n} \left (15 a^8+140 a^7 b x^n+588 a^6 b^2 x^{2 n}+1470 a^5 b^3 x^{3 n}+2450 a^4 b^4 x^{4 n}+2940 a^3 b^5 x^{5 n}+2940 a^2 b^6 x^{6 n}-105 b^8 x^{8 n}\right )}{105 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 7*n)*(a + b*x^n)^8,x]

[Out]

-(15*a^8 + 140*a^7*b*x^n + 588*a^6*b^2*x^(2*n) + 1470*a^5*b^3*x^(3*n) + 2450*a^4
*b^4*x^(4*n) + 2940*a^3*b^5*x^(5*n) + 2940*a^2*b^6*x^(6*n) - 105*b^8*x^(8*n))/(1
05*n*x^(7*n)) + 8*a*b^7*Log[x]

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Maple [A]  time = 0.043, size = 127, normalized size = 1. \[ 8\,a{b}^{7}\ln \left ( x \right ) +{\frac{{b}^{8}{x}^{n}}{n}}-28\,{\frac{{a}^{2}{b}^{6}}{n{x}^{n}}}-28\,{\frac{{a}^{3}{b}^{5}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{70\,{a}^{4}{b}^{4}}{3\,n \left ({x}^{n} \right ) ^{3}}}-14\,{\frac{{a}^{5}{b}^{3}}{n \left ({x}^{n} \right ) ^{4}}}-{\frac{28\,{a}^{6}{b}^{2}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{4\,b{a}^{7}}{3\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{{a}^{8}}{7\,n \left ({x}^{n} \right ) ^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-7*n)*(a+b*x^n)^8,x)

[Out]

8*a*b^7*ln(x)+b^8*x^n/n-28*a^2*b^6/n/(x^n)-28*a^3*b^5/n/(x^n)^2-70/3*a^4*b^4/n/(
x^n)^3-14*a^5*b^3/n/(x^n)^4-28/5*a^6*b^2/n/(x^n)^5-4/3*a^7*b/n/(x^n)^6-1/7*a^8/n
/(x^n)^7

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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^(-7*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228666, size = 157, normalized size = 1.17 \[ \frac{840 \, a b^{7} n x^{7 \, n} \log \left (x\right ) + 105 \, b^{8} x^{8 \, n} - 2940 \, a^{2} b^{6} x^{6 \, n} - 2940 \, a^{3} b^{5} x^{5 \, n} - 2450 \, a^{4} b^{4} x^{4 \, n} - 1470 \, a^{5} b^{3} x^{3 \, n} - 588 \, a^{6} b^{2} x^{2 \, n} - 140 \, a^{7} b x^{n} - 15 \, a^{8}}{105 \, n x^{7 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-7*n - 1),x, algorithm="fricas")

[Out]

1/105*(840*a*b^7*n*x^(7*n)*log(x) + 105*b^8*x^(8*n) - 2940*a^2*b^6*x^(6*n) - 294
0*a^3*b^5*x^(5*n) - 2450*a^4*b^4*x^(4*n) - 1470*a^5*b^3*x^(3*n) - 588*a^6*b^2*x^
(2*n) - 140*a^7*b*x^n - 15*a^8)/(n*x^(7*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**(-1-7*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.236093, size = 167, normalized size = 1.25 \[ \frac{{\left (840 \, a b^{7} n e^{\left (7 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + 105 \, b^{8} e^{\left (8 \, n{\rm ln}\left (x\right )\right )} - 2940 \, a^{2} b^{6} e^{\left (6 \, n{\rm ln}\left (x\right )\right )} - 2940 \, a^{3} b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} - 2450 \, a^{4} b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} - 1470 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} - 588 \, a^{6} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 140 \, a^{7} b e^{\left (n{\rm ln}\left (x\right )\right )} - 15 \, a^{8}\right )} e^{\left (-7 \, n{\rm ln}\left (x\right )\right )}}{105 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-7*n - 1),x, algorithm="giac")

[Out]

1/105*(840*a*b^7*n*e^(7*n*ln(x))*ln(x) + 105*b^8*e^(8*n*ln(x)) - 2940*a^2*b^6*e^
(6*n*ln(x)) - 2940*a^3*b^5*e^(5*n*ln(x)) - 2450*a^4*b^4*e^(4*n*ln(x)) - 1470*a^5
*b^3*e^(3*n*ln(x)) - 588*a^6*b^2*e^(2*n*ln(x)) - 140*a^7*b*e^(n*ln(x)) - 15*a^8)
*e^(-7*n*ln(x))/n