Optimal. Leaf size=47 \[ \frac{9}{8} a^2 b x^{8/3}+\frac{a^3 x^3}{3}+\frac{9}{7} a b^2 x^{7/3}+\frac{b^3 x^2}{2} \]
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Rubi [A] time = 0.0300391, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {263, 266, 43} \[ \frac{9}{8} a^2 b x^{8/3}+\frac{a^3 x^3}{3}+\frac{9}{7} a b^2 x^{7/3}+\frac{b^3 x^2}{2} \]
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 )^3 x^2 \, dx &=\int \left (b+a \sqrt [3]{x}\right )^3 x \, dx\\ &=3 \operatorname{Subst}\left (\int x^5 (b+a x)^3 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (b^3 x^5+3 a b^2 x^6+3 a^2 b x^7+a^3 x^8\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{b^3 x^2}{2}+\frac{9}{7} a b^2 x^{7/3}+\frac{9}{8} a^2 b x^{8/3}+\frac{a^3 x^3}{3}\\ \end{align*}
Mathematica [A] time = 0.0184443, size = 47, normalized size = 1. \[ \frac{9}{8} a^2 b x^{8/3}+\frac{a^3 x^3}{3}+\frac{9}{7} a b^2 x^{7/3}+\frac{b^3 x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 36, normalized size = 0.8 \begin{align*}{\frac{{b}^{3}{x}^{2}}{2}}+{\frac{9\,{b}^{2}a}{7}{x}^{{\frac{7}{3}}}}+{\frac{9\,b{a}^{2}}{8}{x}^{{\frac{8}{3}}}}+{\frac{{a}^{3}{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956926, size = 50, normalized size = 1.06 \begin{align*} \frac{1}{168} \,{\left (56 \, a^{3} + \frac{189 \, a^{2} b}{x^{\frac{1}{3}}} + \frac{216 \, a b^{2}}{x^{\frac{2}{3}}} + \frac{84 \, b^{3}}{x}\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42199, size = 90, normalized size = 1.91 \begin{align*} \frac{1}{3} \, a^{3} x^{3} + \frac{9}{8} \, a^{2} b x^{\frac{8}{3}} + \frac{9}{7} \, a b^{2} x^{\frac{7}{3}} + \frac{1}{2} \, b^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.42408, size = 42, normalized size = 0.89 \begin{align*} \frac{a^{3} x^{3}}{3} + \frac{9 a^{2} b x^{\frac{8}{3}}}{8} + \frac{9 a b^{2} x^{\frac{7}{3}}}{7} + \frac{b^{3} x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15235, size = 47, normalized size = 1. \begin{align*} \frac{1}{3} \, a^{3} x^{3} + \frac{9}{8} \, a^{2} b x^{\frac{8}{3}} + \frac{9}{7} \, a b^{2} x^{\frac{7}{3}} + \frac{1}{2} \, b^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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