Optimal. Leaf size=48 \[ -\frac{3 a^2 b x^{-n}}{n}-\frac{a^3 x^{-2 n}}{2 n}+3 a b^2 \log (x)+\frac{b^3 x^n}{n} \]
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Rubi [A] time = 0.0214762, 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, Rules used = {266, 43} \[ -\frac{3 a^2 b x^{-n}}{n}-\frac{a^3 x^{-2 n}}{2 n}+3 a b^2 \log (x)+\frac{b^3 x^n}{n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-2 n} \left (a+b x^n\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^3} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (b^3+\frac{a^3}{x^3}+\frac{3 a^2 b}{x^2}+\frac{3 a b^2}{x}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^3 x^{-2 n}}{2 n}-\frac{3 a^2 b x^{-n}}{n}+\frac{b^3 x^n}{n}+3 a b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0258998, size = 44, normalized size = 0.92 \[ \frac{-3 a^2 b x^{-n}-\frac{1}{2} a^3 x^{-2 n}+3 a b^2 n \log (x)+b^3 x^n}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 61, normalized size = 1.3 \begin{align*}{\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\,{b}^{2}a\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}-{\frac{{a}^{3}}{2\,n}}-3\,{\frac{b{a}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{n}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33808, size = 108, normalized size = 2.25 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2356, size = 69, normalized size = 1.44 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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