Optimal. Leaf size=36 \[ \frac{2 x^3 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b \left (c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0173357, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {368, 261} \[ \frac{2 x^3 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 368
Rule 261
Rubi steps
\begin{align*} \int x^2 \sqrt{a+b \left (c x^2\right )^{3/2}} \, dx &=\frac{x^3 \operatorname{Subst}\left (\int x^2 \sqrt{a+b x^3} \, dx,x,\sqrt{c x^2}\right )}{\left (c x^2\right )^{3/2}}\\ &=\frac{2 x^3 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b \left (c x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0105376, size = 36, normalized size = 1. \[ \frac{2 x^3 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 29, normalized size = 0.8 \begin{align*}{\frac{2\,{x}^{3}}{9\,b} \left ( a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}} \right ) ^{{\frac{3}{2}}} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\frac{2 \,{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{3}{2}}{\left (c - \sqrt{c}\right )}}{9 \, b c^{\frac{3}{2}}}}{c + 1} + \frac{{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{3}{2}}}{3 \,{\left (c^{2} + c\right )} b \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28523, size = 99, normalized size = 2.75 \begin{align*} \frac{2 \,{\left (b c^{2} x^{4} + \sqrt{c x^{2}} a\right )} \sqrt{\sqrt{c x^{2}} b c x^{2} + a}}{9 \, b c^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17058, size = 27, normalized size = 0.75 \begin{align*} \frac{2 \,{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{3}{2}}}{9 \, b c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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