Optimal. Leaf size=39 \[ \frac{3}{2} a^2 b x^2+a^3 \log (x)+\frac{3}{4} a b^2 x^4+\frac{b^3 x^6}{6} \]
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Rubi [A] time = 0.0184897, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{3}{2} a^2 b x^2+a^3 \log (x)+\frac{3}{4} a b^2 x^4+\frac{b^3 x^6}{6} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^3}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (3 a^2 b+\frac{a^3}{x}+3 a b^2 x+b^3 x^2\right ) \, dx,x,x^2\right )\\ &=\frac{3}{2} a^2 b x^2+\frac{3}{4} a b^2 x^4+\frac{b^3 x^6}{6}+a^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0034781, size = 39, normalized size = 1. \[ \frac{3}{2} a^2 b x^2+a^3 \log (x)+\frac{3}{4} a b^2 x^4+\frac{b^3 x^6}{6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 34, normalized size = 0.9 \begin{align*}{\frac{3\,{a}^{2}b{x}^{2}}{2}}+{\frac{3\,a{b}^{2}{x}^{4}}{4}}+{\frac{{b}^{3}{x}^{6}}{6}}+{a}^{3}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.0592, size = 49, normalized size = 1.26 \begin{align*} \frac{1}{6} \, b^{3} x^{6} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{2} \, a^{2} b x^{2} + \frac{1}{2} \, a^{3} \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98528, size = 78, normalized size = 2. \begin{align*} \frac{1}{6} \, b^{3} x^{6} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.251546, size = 37, normalized size = 0.95 \begin{align*} a^{3} \log{\left (x \right )} + \frac{3 a^{2} b x^{2}}{2} + \frac{3 a b^{2} x^{4}}{4} + \frac{b^{3} x^{6}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.90448, size = 49, normalized size = 1.26 \begin{align*} \frac{1}{6} \, b^{3} x^{6} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{2} \, a^{2} b x^{2} + \frac{1}{2} \, a^{3} \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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