Optimal. Leaf size=34 \[ \frac{a^2 x^4}{4}+\frac{6}{11} a b x^{11/3}+\frac{3}{10} b^2 x^{10/3} \]
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Rubi [A] time = 0.027699, antiderivative size = 34, 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} \[ \frac{a^2 x^4}{4}+\frac{6}{11} a b x^{11/3}+\frac{3}{10} b^2 x^{10/3} \]
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^3 \, dx &=\int \left (b+a \sqrt [3]{x}\right )^2 x^{7/3} \, dx\\ &=3 \operatorname{Subst}\left (\int x^9 (b+a x)^2 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (b^2 x^9+2 a b x^{10}+a^2 x^{11}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3}{10} b^2 x^{10/3}+\frac{6}{11} a b x^{11/3}+\frac{a^2 x^4}{4}\\ \end{align*}
Mathematica [A] time = 0.0102911, size = 34, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{6}{11} a b x^{11/3}+\frac{3}{10} b^2 x^{10/3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 25, normalized size = 0.7 \begin{align*}{\frac{3\,{b}^{2}}{10}{x}^{{\frac{10}{3}}}}+{\frac{6\,ab}{11}{x}^{{\frac{11}{3}}}}+{\frac{{a}^{2}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.999271, size = 35, normalized size = 1.03 \begin{align*} \frac{1}{220} \,{\left (55 \, a^{2} + \frac{120 \, a b}{x^{\frac{1}{3}}} + \frac{66 \, b^{2}}{x^{\frac{2}{3}}}\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43084, size = 72, normalized size = 2.12 \begin{align*} \frac{1}{4} \, a^{2} x^{4} + \frac{6}{11} \, a b x^{\frac{11}{3}} + \frac{3}{10} \, b^{2} x^{\frac{10}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.49883, size = 31, normalized size = 0.91 \begin{align*} \frac{a^{2} x^{4}}{4} + \frac{6 a b x^{\frac{11}{3}}}{11} + \frac{3 b^{2} x^{\frac{10}{3}}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19087, size = 32, normalized size = 0.94 \begin{align*} \frac{1}{4} \, a^{2} x^{4} + \frac{6}{11} \, a b x^{\frac{11}{3}} + \frac{3}{10} \, b^{2} x^{\frac{10}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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