Optimal. Leaf size=40 \[ \frac{3}{2} a^2 b x^2+\frac{a^3 x^4}{4}+3 a b^2 \log (x)-\frac{b^3}{2 x^2} \]
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Rubi [A] time = 0.0224021, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ \frac{3}{2} a^2 b x^2+\frac{a^3 x^4}{4}+3 a b^2 \log (x)-\frac{b^3}{2 x^2} \]
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^3 \, dx &=\int \frac{\left (b+a x^2\right )^3}{x^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(b+a x)^3}{x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (3 a^2 b+\frac{b^3}{x^2}+\frac{3 a b^2}{x}+a^3 x\right ) \, dx,x,x^2\right )\\ &=-\frac{b^3}{2 x^2}+\frac{3}{2} a^2 b x^2+\frac{a^3 x^4}{4}+3 a b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0068502, size = 40, normalized size = 1. \[ \frac{3}{2} a^2 b x^2+\frac{a^3 x^4}{4}+3 a b^2 \log (x)-\frac{b^3}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 35, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}}{2\,{x}^{2}}}+{\frac{3\,{a}^{2}b{x}^{2}}{2}}+{\frac{{a}^{3}{x}^{4}}{4}}+3\,a{b}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967636, size = 49, normalized size = 1.22 \begin{align*} \frac{1}{4} \, a^{3} x^{4} + \frac{3}{2} \, a^{2} b x^{2} + \frac{3}{2} \, a b^{2} \log \left (x^{2}\right ) - \frac{b^{3}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44169, size = 85, normalized size = 2.12 \begin{align*} \frac{a^{3} x^{6} + 6 \, a^{2} b x^{4} + 12 \, a b^{2} x^{2} \log \left (x\right ) - 2 \, b^{3}}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.289655, size = 37, normalized size = 0.92 \begin{align*} \frac{a^{3} x^{4}}{4} + \frac{3 a^{2} b x^{2}}{2} + 3 a b^{2} \log{\left (x \right )} - \frac{b^{3}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19244, size = 47, normalized size = 1.18 \begin{align*} \frac{1}{4} \, a^{3} x^{4} + \frac{3}{2} \, a^{2} b x^{2} + 3 \, a b^{2} \log \left ({\left | x \right |}\right ) - \frac{b^{3}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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