Optimal. Leaf size=36 \[ \frac{\left (a+b x^3\right )^{4/3}}{4 b^2}-\frac{a \sqrt [3]{a+b x^3}}{b^2} \]
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Rubi [A] time = 0.0219188, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{\left (a+b x^3\right )^{4/3}}{4 b^2}-\frac{a \sqrt [3]{a+b x^3}}{b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^3\right )^{2/3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{2/3}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{2/3}}+\frac{\sqrt [3]{a+b x}}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac{a \sqrt [3]{a+b x^3}}{b^2}+\frac{\left (a+b x^3\right )^{4/3}}{4 b^2}\\ \end{align*}
Mathematica [A] time = 0.0124883, size = 27, normalized size = 0.75 \[ \frac{\left (b x^3-3 a\right ) \sqrt [3]{a+b x^3}}{4 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-b{x}^{3}+3\,a}{4\,{b}^{2}}\sqrt [3]{b{x}^{3}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02863, size = 41, normalized size = 1.14 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}}}{4 \, b^{2}} - \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}} a}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42485, size = 55, normalized size = 1.53 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b x^{3} - 3 \, a\right )}}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.906806, size = 44, normalized size = 1.22 \begin{align*} \begin{cases} - \frac{3 a \sqrt [3]{a + b x^{3}}}{4 b^{2}} + \frac{x^{3} \sqrt [3]{a + b x^{3}}}{4 b} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 a^{\frac{2}{3}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07938, size = 36, normalized size = 1. \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}} - 4 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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