3.2718 \(\int x^{-1-9 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=24 \[ -\frac{x^{-9 n} \left (a+b x^n\right )^9}{9 a n} \]

[Out]

-(a + b*x^n)^9/(9*a*n*x^(9*n))

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Rubi [A]  time = 0.0216856, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{x^{-9 n} \left (a+b x^n\right )^9}{9 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 9*n)*(a + b*x^n)^8,x]

[Out]

-(a + b*x^n)^9/(9*a*n*x^(9*n))

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Rubi in Sympy [A]  time = 3.10034, size = 19, normalized size = 0.79 \[ - \frac{x^{- 9 n} \left (a + b x^{n}\right )^{9}}{9 a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-9*n)*(a+b*x**n)**8,x)

[Out]

-x**(-9*n)*(a + b*x**n)**9/(9*a*n)

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Mathematica [B]  time = 0.0528804, size = 111, normalized size = 4.62 \[ -\frac{x^{-9 n} \left (a^8+9 a^7 b x^n+36 a^6 b^2 x^{2 n}+84 a^5 b^3 x^{3 n}+126 a^4 b^4 x^{4 n}+126 a^3 b^5 x^{5 n}+84 a^2 b^6 x^{6 n}+36 a b^7 x^{7 n}+9 b^8 x^{8 n}\right )}{9 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 9*n)*(a + b*x^n)^8,x]

[Out]

-(a^8 + 9*a^7*b*x^n + 36*a^6*b^2*x^(2*n) + 84*a^5*b^3*x^(3*n) + 126*a^4*b^4*x^(4
*n) + 126*a^3*b^5*x^(5*n) + 84*a^2*b^6*x^(6*n) + 36*a*b^7*x^(7*n) + 9*b^8*x^(8*n
))/(9*n*x^(9*n))

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Maple [B]  time = 0., size = 136, normalized size = 5.7 \[ -{\frac{{b}^{8}}{n{x}^{n}}}-4\,{\frac{a{b}^{7}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{28\,{a}^{2}{b}^{6}}{3\,n \left ({x}^{n} \right ) ^{3}}}-14\,{\frac{{a}^{3}{b}^{5}}{n \left ({x}^{n} \right ) ^{4}}}-14\,{\frac{{a}^{4}{b}^{4}}{n \left ({x}^{n} \right ) ^{5}}}-{\frac{28\,{a}^{5}{b}^{3}}{3\,n \left ({x}^{n} \right ) ^{6}}}-4\,{\frac{{a}^{6}{b}^{2}}{n \left ({x}^{n} \right ) ^{7}}}-{\frac{b{a}^{7}}{n \left ({x}^{n} \right ) ^{8}}}-{\frac{{a}^{8}}{9\,n \left ({x}^{n} \right ) ^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-9*n)*(a+b*x^n)^8,x)

[Out]

-b^8/n/(x^n)-4*a*b^7/n/(x^n)^2-28/3*a^2*b^6/n/(x^n)^3-14*a^3*b^5/n/(x^n)^4-14*a^
4*b^4/n/(x^n)^5-28/3*a^5*b^3/n/(x^n)^6-4*a^6*b^2/n/(x^n)^7-a^7*b/n/(x^n)^8-1/9*a
^8/n/(x^n)^9

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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^(-9*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239559, size = 150, normalized size = 6.25 \[ -\frac{9 \, b^{8} x^{8 \, n} + 36 \, a b^{7} x^{7 \, n} + 84 \, a^{2} b^{6} x^{6 \, n} + 126 \, a^{3} b^{5} x^{5 \, n} + 126 \, a^{4} b^{4} x^{4 \, n} + 84 \, a^{5} b^{3} x^{3 \, n} + 36 \, a^{6} b^{2} x^{2 \, n} + 9 \, a^{7} b x^{n} + a^{8}}{9 \, n x^{9 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-9*n - 1),x, algorithm="fricas")

[Out]

-1/9*(9*b^8*x^(8*n) + 36*a*b^7*x^(7*n) + 84*a^2*b^6*x^(6*n) + 126*a^3*b^5*x^(5*n
) + 126*a^4*b^4*x^(4*n) + 84*a^5*b^3*x^(3*n) + 36*a^6*b^2*x^(2*n) + 9*a^7*b*x^n
+ a^8)/(n*x^(9*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**(-1-9*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.232475, size = 161, normalized size = 6.71 \[ -\frac{{\left (9 \, b^{8} e^{\left (8 \, n{\rm ln}\left (x\right )\right )} + 36 \, a b^{7} e^{\left (7 \, n{\rm ln}\left (x\right )\right )} + 84 \, a^{2} b^{6} e^{\left (6 \, n{\rm ln}\left (x\right )\right )} + 126 \, a^{3} b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 126 \, a^{4} b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 84 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 36 \, a^{6} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 9 \, a^{7} b e^{\left (n{\rm ln}\left (x\right )\right )} + a^{8}\right )} e^{\left (-9 \, n{\rm ln}\left (x\right )\right )}}{9 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-9*n - 1),x, algorithm="giac")

[Out]

-1/9*(9*b^8*e^(8*n*ln(x)) + 36*a*b^7*e^(7*n*ln(x)) + 84*a^2*b^6*e^(6*n*ln(x)) +
126*a^3*b^5*e^(5*n*ln(x)) + 126*a^4*b^4*e^(4*n*ln(x)) + 84*a^5*b^3*e^(3*n*ln(x))
+ 36*a^6*b^2*e^(2*n*ln(x)) + 9*a^7*b*e^(n*ln(x)) + a^8)*e^(-9*n*ln(x))/n