Optimal. Leaf size=21 \[ \frac{x^3}{3 a \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0045606, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ \frac{x^3}{3 a \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac{x^3}{3 a \left (a+b x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0049984, size = 21, normalized size = 1. \[ \frac{x^3}{3 a \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 18, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.02234, size = 46, normalized size = 2.19 \begin{align*} -\frac{x}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} + \frac{x}{3 \, \sqrt{b x^{2} + a} a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.2413, size = 77, normalized size = 3.67 \begin{align*} \frac{\sqrt{b x^{2} + a} x^{3}}{3 \,{\left (a b^{2} x^{4} + 2 \, a^{2} b x^{2} + a^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.707849, size = 44, normalized size = 2.1 \begin{align*} \frac{x^{3}}{3 a^{\frac{5}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{3}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.04626, size = 23, normalized size = 1.1 \begin{align*} \frac{x^{3}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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