3.2539 \(\int x^{-1-3 n} \left (a+b x^n\right )^3 \, dx\)

Optimal. Leaf size=52 \[ -\frac{a^3 x^{-3 n}}{3 n}-\frac{3 a^2 b x^{-2 n}}{2 n}-\frac{3 a b^2 x^{-n}}{n}+b^3 \log (x) \]

[Out]

-a^3/(3*n*x^(3*n)) - (3*a^2*b)/(2*n*x^(2*n)) - (3*a*b^2)/(n*x^n) + b^3*Log[x]

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Rubi [A]  time = 0.0628968, antiderivative size = 52, 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 \[ -\frac{a^3 x^{-3 n}}{3 n}-\frac{3 a^2 b x^{-2 n}}{2 n}-\frac{3 a b^2 x^{-n}}{n}+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 3*n)*(a + b*x^n)^3,x]

[Out]

-a^3/(3*n*x^(3*n)) - (3*a^2*b)/(2*n*x^(2*n)) - (3*a*b^2)/(n*x^n) + b^3*Log[x]

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Rubi in Sympy [A]  time = 10.6021, size = 48, normalized size = 0.92 \[ - \frac{a^{3} x^{- 3 n}}{3 n} - \frac{3 a^{2} b x^{- 2 n}}{2 n} - \frac{3 a b^{2} x^{- n}}{n} + \frac{b^{3} \log{\left (x^{n} \right )}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-3*n)*(a+b*x**n)**3,x)

[Out]

-a**3*x**(-3*n)/(3*n) - 3*a**2*b*x**(-2*n)/(2*n) - 3*a*b**2*x**(-n)/n + b**3*log
(x**n)/n

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Mathematica [A]  time = 0.0474944, size = 43, normalized size = 0.83 \[ b^3 \log (x)-\frac{a x^{-3 n} \left (2 a^2+9 a b x^n+18 b^2 x^{2 n}\right )}{6 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 3*n)*(a + b*x^n)^3,x]

[Out]

-(a*(2*a^2 + 9*a*b*x^n + 18*b^2*x^(2*n)))/(6*n*x^(3*n)) + b^3*Log[x]

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Maple [A]  time = 0.024, size = 61, normalized size = 1.2 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}} \left ({b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}-{\frac{{a}^{3}}{3\,n}}-3\,{\frac{a{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}-{\frac{3\,{a}^{2}b{{\rm e}^{n\ln \left ( x \right ) }}}{2\,n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-3*n)*(a+b*x^n)^3,x)

[Out]

(b^3*ln(x)*exp(n*ln(x))^3-1/3*a^3/n-3*a*b^2/n*exp(n*ln(x))^2-3/2*a^2*b/n*exp(n*l
n(x)))/exp(n*ln(x))^3

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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)^3*x^(-3*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225226, size = 69, normalized size = 1.33 \[ \frac{6 \, b^{3} n x^{3 \, n} \log \left (x\right ) - 18 \, a b^{2} x^{2 \, n} - 9 \, a^{2} b x^{n} - 2 \, a^{3}}{6 \, n x^{3 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(-3*n - 1),x, algorithm="fricas")

[Out]

1/6*(6*b^3*n*x^(3*n)*log(x) - 18*a*b^2*x^(2*n) - 9*a^2*b*x^n - 2*a^3)/(n*x^(3*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**(-1-3*n)*(a+b*x**n)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220839, size = 73, normalized size = 1.4 \[ \frac{{\left (6 \, b^{3} n e^{\left (3 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) - 18 \, a b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 9 \, a^{2} b e^{\left (n{\rm ln}\left (x\right )\right )} - 2 \, a^{3}\right )} e^{\left (-3 \, n{\rm ln}\left (x\right )\right )}}{6 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(-3*n - 1),x, algorithm="giac")

[Out]

1/6*(6*b^3*n*e^(3*n*ln(x))*ln(x) - 18*a*b^2*e^(2*n*ln(x)) - 9*a^2*b*e^(n*ln(x))
- 2*a^3)*e^(-3*n*ln(x))/n