Optimal. Leaf size=40 \[ 3 a^2 b \log (x)+\frac{a^3 x^2}{2}-\frac{3 a b^2}{2 x^2}-\frac{b^3}{4 x^4} \]
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Rubi [A] time = 0.0196545, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {263, 266, 43} \[ 3 a^2 b \log (x)+\frac{a^3 x^2}{2}-\frac{3 a b^2}{2 x^2}-\frac{b^3}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x^2}\right )^3 x \, dx &=\int \frac{\left (b+a x^2\right )^3}{x^5} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(b+a x)^3}{x^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^3+\frac{b^3}{x^3}+\frac{3 a b^2}{x^2}+\frac{3 a^2 b}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{b^3}{4 x^4}-\frac{3 a b^2}{2 x^2}+\frac{a^3 x^2}{2}+3 a^2 b \log (x)\\ \end{align*}
Mathematica [A] time = 0.0048519, size = 40, normalized size = 1. \[ 3 a^2 b \log (x)+\frac{a^3 x^2}{2}-\frac{3 a b^2}{2 x^2}-\frac{b^3}{4 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 35, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}}{4\,{x}^{4}}}-{\frac{3\,{b}^{2}a}{2\,{x}^{2}}}+{\frac{{x}^{2}{a}^{3}}{2}}+3\,{a}^{2}b\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03739, size = 50, normalized size = 1.25 \begin{align*} \frac{1}{2} \, a^{3} x^{2} + \frac{3}{2} \, a^{2} b \log \left (x^{2}\right ) - \frac{6 \, a b^{2} x^{2} + b^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46929, size = 85, normalized size = 2.12 \begin{align*} \frac{2 \, a^{3} x^{6} + 12 \, a^{2} b x^{4} \log \left (x\right ) - 6 \, a b^{2} x^{2} - b^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.332937, size = 36, normalized size = 0.9 \begin{align*} \frac{a^{3} x^{2}}{2} + 3 a^{2} b \log{\left (x \right )} - \frac{6 a b^{2} x^{2} + b^{3}}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1878, size = 49, normalized size = 1.22 \begin{align*} \frac{1}{2} \, a^{3} x^{2} + 3 \, a^{2} b \log \left ({\left | x \right |}\right ) - \frac{6 \, a b^{2} x^{2} + b^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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