Optimal. Leaf size=52 \[ -\frac{3 a^2 b x^{-2 n}}{2 n}-\frac{a^3 x^{-3 n}}{3 n}-\frac{3 a b^2 x^{-n}}{n}+b^3 \log (x) \]
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Rubi [A] time = 0.0217421, 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, Rules used = {266, 43} \[ -\frac{3 a^2 b x^{-2 n}}{2 n}-\frac{a^3 x^{-3 n}}{3 n}-\frac{3 a b^2 x^{-n}}{n}+b^3 \log (x) \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-3 n} \left (a+b x^n\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^4} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^3}{x^4}+\frac{3 a^2 b}{x^3}+\frac{3 a b^2}{x^2}+\frac{b^3}{x}\right ) \, dx,x,x^n\right )}{n}\\ &=-\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)\\ \end{align*}
Mathematica [A] time = 0.0398875, 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.
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Maple [A] time = 0.013, size = 61, normalized size = 1.2 \begin{align*}{\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}}-{\frac{3\,b{a}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{2\,n}}-3\,{\frac{{b}^{2}a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{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 = 1.48915, size = 112, normalized size = 2.15 \begin{align*} \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}} \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 = 1.21462, size = 69, normalized size = 1.33 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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