Optimal. Leaf size=34 \[ \frac{a^2 x^2}{2}+\frac{6}{5} a b x^{5/3}+\frac{3}{4} b^2 x^{4/3} \]
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Rubi [A] time = 0.0195002, antiderivative size = 34, 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} \[ \frac{a^2 x^2}{2}+\frac{6}{5} a b x^{5/3}+\frac{3}{4} b^2 x^{4/3} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a+\frac{b}{\sqrt [3]{x}}\right )^2 x \, dx &=\int \left (b+a \sqrt [3]{x}\right )^2 \sqrt [3]{x} \, dx\\ &=3 \operatorname{Subst}\left (\int x^3 (b+a x)^2 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (b^2 x^3+2 a b x^4+a^2 x^5\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3}{4} b^2 x^{4/3}+\frac{6}{5} a b x^{5/3}+\frac{a^2 x^2}{2}\\ \end{align*}
Mathematica [A] time = 0.0098686, size = 34, normalized size = 1. \[ \frac{a^2 x^2}{2}+\frac{6}{5} a b x^{5/3}+\frac{3}{4} b^2 x^{4/3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 25, normalized size = 0.7 \begin{align*}{\frac{3\,{b}^{2}}{4}{x}^{{\frac{4}{3}}}}+{\frac{6\,ab}{5}{x}^{{\frac{5}{3}}}}+{\frac{{a}^{2}{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01205, size = 35, normalized size = 1.03 \begin{align*} \frac{1}{20} \,{\left (10 \, a^{2} + \frac{24 \, a b}{x^{\frac{1}{3}}} + \frac{15 \, b^{2}}{x^{\frac{2}{3}}}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48472, size = 66, normalized size = 1.94 \begin{align*} \frac{1}{2} \, a^{2} x^{2} + \frac{6}{5} \, a b x^{\frac{5}{3}} + \frac{3}{4} \, b^{2} x^{\frac{4}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.326758, size = 31, normalized size = 0.91 \begin{align*} \frac{a^{2} x^{2}}{2} + \frac{6 a b x^{\frac{5}{3}}}{5} + \frac{3 b^{2} x^{\frac{4}{3}}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16425, size = 32, normalized size = 0.94 \begin{align*} \frac{1}{2} \, a^{2} x^{2} + \frac{6}{5} \, a b x^{\frac{5}{3}} + \frac{3}{4} \, b^{2} x^{\frac{4}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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