Optimal. Leaf size=78 \[ \frac{3 a^2 \left (a+b x^3\right )^{4/3}}{4 b^4}-\frac{a^3 \sqrt [3]{a+b x^3}}{b^4}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^4}-\frac{3 a \left (a+b x^3\right )^{7/3}}{7 b^4} \]
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Rubi [A] time = 0.0444571, antiderivative size = 78, 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} \[ \frac{3 a^2 \left (a+b x^3\right )^{4/3}}{4 b^4}-\frac{a^3 \sqrt [3]{a+b x^3}}{b^4}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^4}-\frac{3 a \left (a+b x^3\right )^{7/3}}{7 b^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (a+b x^3\right )^{2/3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^{2/3}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 (a+b x)^{2/3}}+\frac{3 a^2 \sqrt [3]{a+b x}}{b^3}-\frac{3 a (a+b x)^{4/3}}{b^3}+\frac{(a+b x)^{7/3}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^3 \sqrt [3]{a+b x^3}}{b^4}+\frac{3 a^2 \left (a+b x^3\right )^{4/3}}{4 b^4}-\frac{3 a \left (a+b x^3\right )^{7/3}}{7 b^4}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^4}\\ \end{align*}
Mathematica [A] time = 0.0227493, size = 50, normalized size = 0.64 \[ \frac{\sqrt [3]{a+b x^3} \left (27 a^2 b x^3-81 a^3-18 a b^2 x^6+14 b^3 x^9\right )}{140 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-14\,{b}^{3}{x}^{9}+18\,a{b}^{2}{x}^{6}-27\,{a}^{2}b{x}^{3}+81\,{a}^{3}}{140\,{b}^{4}}\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 = 0.9893, size = 86, normalized size = 1.1 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{10}{3}}}{10 \, b^{4}} - \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a}{7 \, b^{4}} + \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a^{2}}{4 \, b^{4}} - \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{3}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4684, size = 109, normalized size = 1.4 \begin{align*} \frac{{\left (14 \, b^{3} x^{9} - 18 \, a b^{2} x^{6} + 27 \, a^{2} b x^{3} - 81 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{140 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.216, size = 92, normalized size = 1.18 \begin{align*} \begin{cases} - \frac{81 a^{3} \sqrt [3]{a + b x^{3}}}{140 b^{4}} + \frac{27 a^{2} x^{3} \sqrt [3]{a + b x^{3}}}{140 b^{3}} - \frac{9 a x^{6} \sqrt [3]{a + b x^{3}}}{70 b^{2}} + \frac{x^{9} \sqrt [3]{a + b x^{3}}}{10 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 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 = 1.11208, size = 77, normalized size = 0.99 \begin{align*} \frac{14 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}} - 60 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a + 105 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a^{2} - 140 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{3}}{140 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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