Optimal. Leaf size=42 \[ \frac{9}{5} a^2 b x^{5/3}+\frac{a^3 x^2}{2}+\frac{9}{4} a b^2 x^{4/3}+b^3 x \]
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Rubi [A] time = 0.0235856, antiderivative size = 42, 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, 190, 43} \[ \frac{9}{5} a^2 b x^{5/3}+\frac{a^3 x^2}{2}+\frac{9}{4} a b^2 x^{4/3}+b^3 x \]
Antiderivative was successfully verified.
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Rule 263
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \left (a+\frac{b}{\sqrt [3]{x}}\right )^3 x \, dx &=\int \left (b+a \sqrt [3]{x}\right )^3 \, dx\\ &=3 \operatorname{Subst}\left (\int x^2 (b+a x)^3 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (b^3 x^2+3 a b^2 x^3+3 a^2 b x^4+a^3 x^5\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=b^3 x+\frac{9}{4} a b^2 x^{4/3}+\frac{9}{5} a^2 b x^{5/3}+\frac{a^3 x^2}{2}\\ \end{align*}
Mathematica [A] time = 0.0158789, size = 42, normalized size = 1. \[ \frac{9}{5} a^2 b x^{5/3}+\frac{a^3 x^2}{2}+\frac{9}{4} a b^2 x^{4/3}+b^3 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 33, normalized size = 0.8 \begin{align*}{b}^{3}x+{\frac{9\,{b}^{2}a}{4}{x}^{{\frac{4}{3}}}}+{\frac{9\,b{a}^{2}}{5}{x}^{{\frac{5}{3}}}}+{\frac{{x}^{2}{a}^{3}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966784, size = 50, normalized size = 1.19 \begin{align*} \frac{1}{20} \,{\left (10 \, a^{3} + \frac{36 \, a^{2} b}{x^{\frac{1}{3}}} + \frac{45 \, a b^{2}}{x^{\frac{2}{3}}} + \frac{20 \, b^{3}}{x}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41798, size = 82, normalized size = 1.95 \begin{align*} \frac{1}{2} \, a^{3} x^{2} + \frac{9}{5} \, a^{2} b x^{\frac{5}{3}} + \frac{9}{4} \, a b^{2} x^{\frac{4}{3}} + b^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.644069, size = 39, normalized size = 0.93 \begin{align*} \frac{a^{3} x^{2}}{2} + \frac{9 a^{2} b x^{\frac{5}{3}}}{5} + \frac{9 a b^{2} x^{\frac{4}{3}}}{4} + b^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19377, size = 43, normalized size = 1.02 \begin{align*} \frac{1}{2} \, a^{3} x^{2} + \frac{9}{5} \, a^{2} b x^{\frac{5}{3}} + \frac{9}{4} \, a b^{2} x^{\frac{4}{3}} + b^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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