3.2538 \(\int x^{-1-2 n} \left (a+b x^n\right )^3 \, dx\)

Optimal. Leaf size=48 \[ -\frac{a^3 x^{-2 n}}{2 n}-\frac{3 a^2 b x^{-n}}{n}+3 a b^2 \log (x)+\frac{b^3 x^n}{n} \]

[Out]

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

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Rubi [A]  time = 0.061919, antiderivative size = 48, 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 \[ -\frac{a^3 x^{-2 n}}{2 n}-\frac{3 a^2 b x^{-n}}{n}+3 a b^2 \log (x)+\frac{b^3 x^n}{n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 2*n)*(a + b*x^n)^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-2*n)*(a+b*x**n)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0696971, size = 45, normalized size = 0.94 \[ -\frac{a^3 x^{-2 n}+6 a^2 b x^{-n}-6 a b^2 n \log (x)-2 b^3 x^n}{2 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 2*n)*(a + b*x^n)^3,x]

[Out]

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

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Maple [A]  time = 0.023, size = 61, normalized size = 1.3 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}+3\,a{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}-{\frac{{a}^{3}}{2\,n}}-3\,{\frac{{a}^{2}b{{\rm e}^{n\ln \left ( x \right ) }}}{n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-2*n)*(a+b*x^n)^3,x)

[Out]

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

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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^(-2*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225111, size = 69, normalized size = 1.44 \[ \frac{6 \, a b^{2} n x^{2 \, n} \log \left (x\right ) + 2 \, b^{3} x^{3 \, n} - 6 \, a^{2} b x^{n} - a^{3}}{2 \, n x^{2 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(-2*n - 1),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220434, size = 73, normalized size = 1.52 \[ \frac{{\left (6 \, a b^{2} n e^{\left (2 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + 2 \, b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} - 6 \, a^{2} b e^{\left (n{\rm ln}\left (x\right )\right )} - a^{3}\right )} e^{\left (-2 \, n{\rm ln}\left (x\right )\right )}}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(-2*n - 1),x, algorithm="giac")

[Out]

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