3.2537 \(\int x^{-1-n} \left (a+b x^n\right )^3 \, dx\)

Optimal. Leaf size=49 \[ -\frac{a^3 x^{-n}}{n}+3 a^2 b \log (x)+\frac{3 a b^2 x^n}{n}+\frac{b^3 x^{2 n}}{2 n} \]

[Out]

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

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Rubi [A]  time = 0.0639627, antiderivative size = 49, 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 \[ -\frac{a^3 x^{-n}}{n}+3 a^2 b \log (x)+\frac{3 a b^2 x^n}{n}+\frac{b^3 x^{2 n}}{2 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - n)*(a + b*x^n)^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0500665, size = 46, normalized size = 0.94 \[ -\frac{2 a^3 x^{-n}-6 a^2 b n \log (x)-6 a b^2 x^n-b^3 x^{2 n}}{2 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.023, size = 62, normalized size = 1.3 \[{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ( 3\,{a}^{2}b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}-{\frac{{a}^{3}}{n}}+{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{2\,n}}+3\,{\frac{a{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-n)*(a+b*x^n)^3,x)

[Out]

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

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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^(-n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225691, size = 65, normalized size = 1.33 \[ \frac{6 \, a^{2} b n x^{n} \log \left (x\right ) + b^{3} x^{3 \, n} + 6 \, a b^{2} x^{2 \, n} - 2 \, a^{3}}{2 \, n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222891, size = 72, normalized size = 1.47 \[ \frac{{\left (6 \, a^{2} b n e^{\left (n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 6 \, a b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 2 \, a^{3}\right )} e^{\left (-n{\rm ln}\left (x\right )\right )}}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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