Optimal. Leaf size=38 \[ \frac{3 \left (a+b x^2\right )^{4/3}}{8 b^2}-\frac{3 a \sqrt [3]{a+b x^2}}{2 b^2} \]
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Rubi [A] time = 0.0235036, antiderivative size = 38, 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} \[ \frac{3 \left (a+b x^2\right )^{4/3}}{8 b^2}-\frac{3 a \sqrt [3]{a+b x^2}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x^2\right )^{2/3}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{2/3}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{2/3}}+\frac{\sqrt [3]{a+b x}}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 a \sqrt [3]{a+b x^2}}{2 b^2}+\frac{3 \left (a+b x^2\right )^{4/3}}{8 b^2}\\ \end{align*}
Mathematica [A] time = 0.0129794, size = 27, normalized size = 0.71 \[ \frac{3 \left (b x^2-3 a\right ) \sqrt [3]{a+b x^2}}{8 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-3\,b{x}^{2}+9\,a}{8\,{b}^{2}}\sqrt [3]{b{x}^{2}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.73467, size = 41, normalized size = 1.08 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}}}{8 \, b^{2}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74131, size = 55, normalized size = 1.45 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}}{\left (b x^{2} - 3 \, a\right )}}{8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.01786, size = 178, normalized size = 4.68 \begin{align*} - \frac{9 a^{\frac{10}{3}} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} + \frac{9 a^{\frac{10}{3}}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} - \frac{6 a^{\frac{7}{3}} b x^{2} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} + \frac{9 a^{\frac{7}{3}} b x^{2}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} + \frac{3 a^{\frac{4}{3}} b^{2} x^{4} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.1225, size = 36, normalized size = 0.95 \begin{align*} \frac{3 \,{\left ({\left (b x^{2} + a\right )}^{\frac{4}{3}} - 4 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a\right )}}{8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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