3.2581 \(\int x^{-1-13 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=131 \[ -\frac{b^4 x^{-9 n} \left (a+b x^n\right )^9}{6435 a^5 n}+\frac{b^3 x^{-10 n} \left (a+b x^n\right )^9}{715 a^4 n}-\frac{b^2 x^{-11 n} \left (a+b x^n\right )^9}{143 a^3 n}+\frac{b x^{-12 n} \left (a+b x^n\right )^9}{39 a^2 n}-\frac{x^{-13 n} \left (a+b x^n\right )^9}{13 a n} \]

[Out]

-(a + b*x^n)^9/(13*a*n*x^(13*n)) + (b*(a + b*x^n)^9)/(39*a^2*n*x^(12*n)) - (b^2*
(a + b*x^n)^9)/(143*a^3*n*x^(11*n)) + (b^3*(a + b*x^n)^9)/(715*a^4*n*x^(10*n)) -
(b^4*(a + b*x^n)^9)/(6435*a^5*n*x^(9*n))

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Rubi [A]  time = 0.150569, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{b^4 x^{-9 n} \left (a+b x^n\right )^9}{6435 a^5 n}+\frac{b^3 x^{-10 n} \left (a+b x^n\right )^9}{715 a^4 n}-\frac{b^2 x^{-11 n} \left (a+b x^n\right )^9}{143 a^3 n}+\frac{b x^{-12 n} \left (a+b x^n\right )^9}{39 a^2 n}-\frac{x^{-13 n} \left (a+b x^n\right )^9}{13 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 13*n)*(a + b*x^n)^8,x]

[Out]

-(a + b*x^n)^9/(13*a*n*x^(13*n)) + (b*(a + b*x^n)^9)/(39*a^2*n*x^(12*n)) - (b^2*
(a + b*x^n)^9)/(143*a^3*n*x^(11*n)) + (b^3*(a + b*x^n)^9)/(715*a^4*n*x^(10*n)) -
(b^4*(a + b*x^n)^9)/(6435*a^5*n*x^(9*n))

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Rubi in Sympy [A]  time = 27.9394, size = 136, normalized size = 1.04 \[ - \frac{a^{8} x^{- 13 n}}{13 n} - \frac{2 a^{7} b x^{- 12 n}}{3 n} - \frac{28 a^{6} b^{2} x^{- 11 n}}{11 n} - \frac{28 a^{5} b^{3} x^{- 10 n}}{5 n} - \frac{70 a^{4} b^{4} x^{- 9 n}}{9 n} - \frac{7 a^{3} b^{5} x^{- 8 n}}{n} - \frac{4 a^{2} b^{6} x^{- 7 n}}{n} - \frac{4 a b^{7} x^{- 6 n}}{3 n} - \frac{b^{8} x^{- 5 n}}{5 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-13*n)*(a+b*x**n)**8,x)

[Out]

-a**8*x**(-13*n)/(13*n) - 2*a**7*b*x**(-12*n)/(3*n) - 28*a**6*b**2*x**(-11*n)/(1
1*n) - 28*a**5*b**3*x**(-10*n)/(5*n) - 70*a**4*b**4*x**(-9*n)/(9*n) - 7*a**3*b**
5*x**(-8*n)/n - 4*a**2*b**6*x**(-7*n)/n - 4*a*b**7*x**(-6*n)/(3*n) - b**8*x**(-5
*n)/(5*n)

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Mathematica [A]  time = 0.0516625, size = 113, normalized size = 0.86 \[ -\frac{x^{-13 n} \left (495 a^8+4290 a^7 b x^n+16380 a^6 b^2 x^{2 n}+36036 a^5 b^3 x^{3 n}+50050 a^4 b^4 x^{4 n}+45045 a^3 b^5 x^{5 n}+25740 a^2 b^6 x^{6 n}+8580 a b^7 x^{7 n}+1287 b^8 x^{8 n}\right )}{6435 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 13*n)*(a + b*x^n)^8,x]

[Out]

-(495*a^8 + 4290*a^7*b*x^n + 16380*a^6*b^2*x^(2*n) + 36036*a^5*b^3*x^(3*n) + 500
50*a^4*b^4*x^(4*n) + 45045*a^3*b^5*x^(5*n) + 25740*a^2*b^6*x^(6*n) + 8580*a*b^7*
x^(7*n) + 1287*b^8*x^(8*n))/(6435*n*x^(13*n))

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Maple [A]  time = 0.04, size = 136, normalized size = 1. \[ -{\frac{{b}^{8}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{4\,a{b}^{7}}{3\,n \left ({x}^{n} \right ) ^{6}}}-4\,{\frac{{a}^{2}{b}^{6}}{n \left ({x}^{n} \right ) ^{7}}}-7\,{\frac{{a}^{3}{b}^{5}}{n \left ({x}^{n} \right ) ^{8}}}-{\frac{70\,{a}^{4}{b}^{4}}{9\,n \left ({x}^{n} \right ) ^{9}}}-{\frac{28\,{a}^{5}{b}^{3}}{5\,n \left ({x}^{n} \right ) ^{10}}}-{\frac{28\,{a}^{6}{b}^{2}}{11\,n \left ({x}^{n} \right ) ^{11}}}-{\frac{2\,b{a}^{7}}{3\,n \left ({x}^{n} \right ) ^{12}}}-{\frac{{a}^{8}}{13\,n \left ({x}^{n} \right ) ^{13}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-13*n)*(a+b*x^n)^8,x)

[Out]

-1/5*b^8/n/(x^n)^5-4/3*a*b^7/n/(x^n)^6-4*a^2*b^6/n/(x^n)^7-7*a^3*b^5/n/(x^n)^8-7
0/9*a^4*b^4/n/(x^n)^9-28/5*a^5*b^3/n/(x^n)^10-28/11*a^6*b^2/n/(x^n)^11-2/3*a^7*b
/n/(x^n)^12-1/13*a^8/n/(x^n)^13

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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^(-13*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227498, size = 153, normalized size = 1.17 \[ -\frac{1287 \, b^{8} x^{8 \, n} + 8580 \, a b^{7} x^{7 \, n} + 25740 \, a^{2} b^{6} x^{6 \, n} + 45045 \, a^{3} b^{5} x^{5 \, n} + 50050 \, a^{4} b^{4} x^{4 \, n} + 36036 \, a^{5} b^{3} x^{3 \, n} + 16380 \, a^{6} b^{2} x^{2 \, n} + 4290 \, a^{7} b x^{n} + 495 \, a^{8}}{6435 \, n x^{13 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-13*n - 1),x, algorithm="fricas")

[Out]

-1/6435*(1287*b^8*x^(8*n) + 8580*a*b^7*x^(7*n) + 25740*a^2*b^6*x^(6*n) + 45045*a
^3*b^5*x^(5*n) + 50050*a^4*b^4*x^(4*n) + 36036*a^5*b^3*x^(3*n) + 16380*a^6*b^2*x
^(2*n) + 4290*a^7*b*x^n + 495*a^8)/(n*x^(13*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**(-1-13*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233937, size = 163, normalized size = 1.24 \[ -\frac{{\left (1287 \, b^{8} e^{\left (8 \, n{\rm ln}\left (x\right )\right )} + 8580 \, a b^{7} e^{\left (7 \, n{\rm ln}\left (x\right )\right )} + 25740 \, a^{2} b^{6} e^{\left (6 \, n{\rm ln}\left (x\right )\right )} + 45045 \, a^{3} b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 50050 \, a^{4} b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 36036 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 16380 \, a^{6} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 4290 \, a^{7} b e^{\left (n{\rm ln}\left (x\right )\right )} + 495 \, a^{8}\right )} e^{\left (-13 \, n{\rm ln}\left (x\right )\right )}}{6435 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-13*n - 1),x, algorithm="giac")

[Out]

-1/6435*(1287*b^8*e^(8*n*ln(x)) + 8580*a*b^7*e^(7*n*ln(x)) + 25740*a^2*b^6*e^(6*
n*ln(x)) + 45045*a^3*b^5*e^(5*n*ln(x)) + 50050*a^4*b^4*e^(4*n*ln(x)) + 36036*a^5
*b^3*e^(3*n*ln(x)) + 16380*a^6*b^2*e^(2*n*ln(x)) + 4290*a^7*b*e^(n*ln(x)) + 495*
a^8)*e^(-13*n*ln(x))/n