Optimal. Leaf size=34 \[ \frac{a^2 x^2}{2}+\frac{6}{7} a b x^{7/3}+\frac{3}{8} b^2 x^{8/3} \]
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Rubi [A] time = 0.0202066, antiderivative size = 34, 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} \[ \frac{a^2 x^2}{2}+\frac{6}{7} a b x^{7/3}+\frac{3}{8} b^2 x^{8/3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt [3]{x}\right )^2 x \, dx &=3 \operatorname{Subst}\left (\int x^5 (a+b x)^2 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (a^2 x^5+2 a b x^6+b^2 x^7\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{a^2 x^2}{2}+\frac{6}{7} a b x^{7/3}+\frac{3}{8} b^2 x^{8/3}\\ \end{align*}
Mathematica [A] time = 0.0150255, size = 34, normalized size = 1. \[ \frac{a^2 x^2}{2}+\frac{6}{7} a b x^{7/3}+\frac{3}{8} b^2 x^{8/3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 25, normalized size = 0.7 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{6\,ab}{7}{x}^{{\frac{7}{3}}}}+{\frac{3\,{b}^{2}}{8}{x}^{{\frac{8}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.970035, size = 132, normalized size = 3.88 \begin{align*} \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8}}{8 \, b^{6}} - \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a}{7 \, b^{6}} + \frac{5 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{2}}{b^{6}} - \frac{6 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{3}}{b^{6}} + \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{4}}{4 \, b^{6}} - \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{5}}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45875, size = 66, normalized size = 1.94 \begin{align*} \frac{3}{8} \, b^{2} x^{\frac{8}{3}} + \frac{6}{7} \, a b x^{\frac{7}{3}} + \frac{1}{2} \, a^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.61413, size = 31, normalized size = 0.91 \begin{align*} \frac{a^{2} x^{2}}{2} + \frac{6 a b x^{\frac{7}{3}}}{7} + \frac{3 b^{2} x^{\frac{8}{3}}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13096, size = 32, normalized size = 0.94 \begin{align*} \frac{3}{8} \, b^{2} x^{\frac{8}{3}} + \frac{6}{7} \, a b x^{\frac{7}{3}} + \frac{1}{2} \, a^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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