Optimal. Leaf size=34 \[ \frac{a^2 x^5}{5}+\frac{3}{7} a b x^{14/3}+\frac{3}{13} b^2 x^{13/3} \]
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Rubi [A] time = 0.0307636, 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} \[ \frac{a^2 x^5}{5}+\frac{3}{7} a b x^{14/3}+\frac{3}{13} b^2 x^{13/3} \]
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^4 \, dx &=\int \left (b+a \sqrt [3]{x}\right )^2 x^{10/3} \, dx\\ &=3 \operatorname{Subst}\left (\int x^{12} (b+a x)^2 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (b^2 x^{12}+2 a b x^{13}+a^2 x^{14}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3}{13} b^2 x^{13/3}+\frac{3}{7} a b x^{14/3}+\frac{a^2 x^5}{5}\\ \end{align*}
Mathematica [A] time = 0.012112, size = 34, normalized size = 1. \[ \frac{a^2 x^5}{5}+\frac{3}{7} a b x^{14/3}+\frac{3}{13} b^2 x^{13/3} \]
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}}{13}{x}^{{\frac{13}{3}}}}+{\frac{3\,ab}{7}{x}^{{\frac{14}{3}}}}+{\frac{{x}^{5}{a}^{2}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0044, size = 35, normalized size = 1.03 \begin{align*} \frac{1}{455} \,{\left (91 \, a^{2} + \frac{195 \, a b}{x^{\frac{1}{3}}} + \frac{105 \, b^{2}}{x^{\frac{2}{3}}}\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46761, size = 70, normalized size = 2.06 \begin{align*} \frac{1}{5} \, a^{2} x^{5} + \frac{3}{7} \, a b x^{\frac{14}{3}} + \frac{3}{13} \, b^{2} x^{\frac{13}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.16411, size = 31, normalized size = 0.91 \begin{align*} \frac{a^{2} x^{5}}{5} + \frac{3 a b x^{\frac{14}{3}}}{7} + \frac{3 b^{2} x^{\frac{13}{3}}}{13} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20793, size = 32, normalized size = 0.94 \begin{align*} \frac{1}{5} \, a^{2} x^{5} + \frac{3}{7} \, a b x^{\frac{14}{3}} + \frac{3}{13} \, b^{2} x^{\frac{13}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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