Optimal. Leaf size=80 \[ \frac{3 a^2 \left (a+b x^3\right )^{8/3}}{8 b^4}-\frac{a^3 \left (a+b x^3\right )^{5/3}}{5 b^4}+\frac{\left (a+b x^3\right )^{14/3}}{14 b^4}-\frac{3 a \left (a+b x^3\right )^{11/3}}{11 b^4} \]
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Rubi [A] time = 0.0489658, antiderivative size = 80, 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} \[ \frac{3 a^2 \left (a+b x^3\right )^{8/3}}{8 b^4}-\frac{a^3 \left (a+b x^3\right )^{5/3}}{5 b^4}+\frac{\left (a+b x^3\right )^{14/3}}{14 b^4}-\frac{3 a \left (a+b x^3\right )^{11/3}}{11 b^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{11} \left (a+b x^3\right )^{2/3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^3 (a+b x)^{2/3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^{2/3}}{b^3}+\frac{3 a^2 (a+b x)^{5/3}}{b^3}-\frac{3 a (a+b x)^{8/3}}{b^3}+\frac{(a+b x)^{11/3}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^3 \left (a+b x^3\right )^{5/3}}{5 b^4}+\frac{3 a^2 \left (a+b x^3\right )^{8/3}}{8 b^4}-\frac{3 a \left (a+b x^3\right )^{11/3}}{11 b^4}+\frac{\left (a+b x^3\right )^{14/3}}{14 b^4}\\ \end{align*}
Mathematica [A] time = 0.0254731, size = 50, normalized size = 0.62 \[ \frac{\left (a+b x^3\right )^{5/3} \left (135 a^2 b x^3-81 a^3-180 a b^2 x^6+220 b^3 x^9\right )}{3080 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-220\,{b}^{3}{x}^{9}+180\,a{b}^{2}{x}^{6}-135\,{a}^{2}b{x}^{3}+81\,{a}^{3}}{3080\,{b}^{4}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06835, size = 86, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{14}{3}}}{14 \, b^{4}} - \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{11}{3}} a}{11 \, b^{4}} + \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} a^{2}}{8 \, b^{4}} - \frac{{\left (b x^{3} + a\right )}^{\frac{5}{3}} a^{3}}{5 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72476, size = 136, normalized size = 1.7 \begin{align*} \frac{{\left (220 \, b^{4} x^{12} + 40 \, a b^{3} x^{9} - 45 \, a^{2} b^{2} x^{6} + 54 \, a^{3} b x^{3} - 81 \, a^{4}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{3080 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.96267, size = 110, normalized size = 1.38 \begin{align*} \begin{cases} - \frac{81 a^{4} \left (a + b x^{3}\right )^{\frac{2}{3}}}{3080 b^{4}} + \frac{27 a^{3} x^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}{1540 b^{3}} - \frac{9 a^{2} x^{6} \left (a + b x^{3}\right )^{\frac{2}{3}}}{616 b^{2}} + \frac{a x^{9} \left (a + b x^{3}\right )^{\frac{2}{3}}}{77 b} + \frac{x^{12} \left (a + b x^{3}\right )^{\frac{2}{3}}}{14} & \text{for}\: b \neq 0 \\\frac{a^{\frac{2}{3}} x^{12}}{12} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1111, size = 77, normalized size = 0.96 \begin{align*} \frac{220 \,{\left (b x^{3} + a\right )}^{\frac{14}{3}} - 840 \,{\left (b x^{3} + a\right )}^{\frac{11}{3}} a + 1155 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} a^{2} - 616 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a^{3}}{3080 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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