Optimal. Leaf size=28 \[ a^2 \log (x)+6 a b \sqrt [3]{x}+\frac{3}{2} b^2 x^{2/3} \]
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Rubi [A] time = 0.0112806, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ a^2 \log (x)+6 a b \sqrt [3]{x}+\frac{3}{2} b^2 x^{2/3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt [3]{x}\right )^2}{x} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (2 a b+\frac{a^2}{x}+b^2 x\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=6 a b \sqrt [3]{x}+\frac{3}{2} b^2 x^{2/3}+a^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0102526, size = 28, normalized size = 1. \[ a^2 \log (x)+6 a b \sqrt [3]{x}+\frac{3}{2} b^2 x^{2/3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 23, normalized size = 0.8 \begin{align*} 6\,ab\sqrt [3]{x}+{\frac{3\,{b}^{2}}{2}{x}^{{\frac{2}{3}}}}+{a}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962925, size = 30, normalized size = 1.07 \begin{align*} a^{2} \log \left (x\right ) + \frac{3}{2} \, b^{2} x^{\frac{2}{3}} + 6 \, a b x^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47879, size = 73, normalized size = 2.61 \begin{align*} 3 \, a^{2} \log \left (x^{\frac{1}{3}}\right ) + \frac{3}{2} \, b^{2} x^{\frac{2}{3}} + 6 \, a b x^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.275664, size = 27, normalized size = 0.96 \begin{align*} a^{2} \log{\left (x \right )} + 6 a b \sqrt [3]{x} + \frac{3 b^{2} x^{\frac{2}{3}}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10493, size = 31, normalized size = 1.11 \begin{align*} a^{2} \log \left ({\left | x \right |}\right ) + \frac{3}{2} \, b^{2} x^{\frac{2}{3}} + 6 \, a b x^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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