Optimal. Leaf size=40 \[ 3 a^2 b \log (x)-\frac{a^3}{2 x^2}+\frac{3}{2} a b^2 x^2+\frac{b^3 x^4}{4} \]
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Rubi [A] time = 0.0225053, 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 = {1593, 266, 43} \[ 3 a^2 b \log (x)-\frac{a^3}{2 x^2}+\frac{3}{2} a b^2 x^2+\frac{b^3 x^4}{4} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (\frac{a}{x}+b x\right )^3 \, dx &=\int \frac{\left (a+b x^2\right )^3}{x^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (3 a b^2+\frac{a^3}{x^2}+\frac{3 a^2 b}{x}+b^3 x\right ) \, dx,x,x^2\right )\\ &=-\frac{a^3}{2 x^2}+\frac{3}{2} a b^2 x^2+\frac{b^3 x^4}{4}+3 a^2 b \log (x)\\ \end{align*}
Mathematica [A] time = 0.0068147, size = 40, normalized size = 1. \[ 3 a^2 b \log (x)-\frac{a^3}{2 x^2}+\frac{3}{2} a b^2 x^2+\frac{b^3 x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 35, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{2\,{x}^{2}}}+{\frac{3\,a{b}^{2}{x}^{2}}{2}}+{\frac{{b}^{3}{x}^{4}}{4}}+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 = 1.15406, size = 46, normalized size = 1.15 \begin{align*} \frac{1}{4} \, b^{3} x^{4} + \frac{3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b \log \left (x\right ) - \frac{a^{3}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.770119, size = 85, normalized size = 2.12 \begin{align*} \frac{b^{3} x^{6} + 6 \, a b^{2} x^{4} + 12 \, a^{2} b x^{2} \log \left (x\right ) - 2 \, a^{3}}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.419012, size = 37, normalized size = 0.92 \begin{align*} - \frac{a^{3}}{2 x^{2}} + 3 a^{2} b \log{\left (x \right )} + \frac{3 a b^{2} x^{2}}{2} + \frac{b^{3} x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11189, size = 62, normalized size = 1.55 \begin{align*} \frac{1}{4} \, b^{3} x^{4} + \frac{3}{2} \, a b^{2} x^{2} + \frac{3}{2} \, a^{2} b \log \left (x^{2}\right ) - \frac{3 \, a^{2} b x^{2} + a^{3}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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