Optimal. Leaf size=38 \[ \frac{3 \left (a+b x^2\right )^{8/3}}{16 b^2}-\frac{3 a \left (a+b x^2\right )^{5/3}}{10 b^2} \]
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Rubi [A] time = 0.0234761, 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} \[ \frac{3 \left (a+b x^2\right )^{8/3}}{16 b^2}-\frac{3 a \left (a+b x^2\right )^{5/3}}{10 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \left (a+b x^2\right )^{2/3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x)^{2/3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^{2/3}}{b}+\frac{(a+b x)^{5/3}}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 a \left (a+b x^2\right )^{5/3}}{10 b^2}+\frac{3 \left (a+b x^2\right )^{8/3}}{16 b^2}\\ \end{align*}
Mathematica [A] time = 0.0127471, size = 28, normalized size = 0.74 \[ \frac{3 \left (a+b x^2\right )^{5/3} \left (5 b x^2-3 a\right )}{80 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-15\,b{x}^{2}+9\,a}{80\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.46676, size = 41, normalized size = 1.08 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{8}{3}}}{16 \, b^{2}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} a}{10 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76861, size = 81, normalized size = 2.13 \begin{align*} \frac{3 \,{\left (5 \, b^{2} x^{4} + 2 \, a b x^{2} - 3 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{80 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.737089, size = 66, normalized size = 1.74 \begin{align*} \begin{cases} - \frac{9 a^{2} \left (a + b x^{2}\right )^{\frac{2}{3}}}{80 b^{2}} + \frac{3 a x^{2} \left (a + b x^{2}\right )^{\frac{2}{3}}}{40 b} + \frac{3 x^{4} \left (a + b x^{2}\right )^{\frac{2}{3}}}{16} & \text{for}\: b \neq 0 \\\frac{a^{\frac{2}{3}} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.88715, size = 39, normalized size = 1.03 \begin{align*} \frac{3 \,{\left (5 \,{\left (b x^{2} + a\right )}^{\frac{8}{3}} - 8 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} a\right )}}{80 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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