Optimal. Leaf size=21 \[ \frac{x^3 \left (a+\frac{b}{x^2}\right )^{3/2}}{3 a} \]
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Rubi [A] time = 0.0057577, 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} \[ \frac{x^3 \left (a+\frac{b}{x^2}\right )^{3/2}}{3 a} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \sqrt{a+\frac{b}{x^2}} x^2 \, dx &=\frac{\left (a+\frac{b}{x^2}\right )^{3/2} x^3}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0049841, size = 26, normalized size = 1.24 \[ \frac{x \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )}{3 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 27, normalized size = 1.3 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) x}{3\,a}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00758, size = 23, normalized size = 1.1 \begin{align*} \frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{3}}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48493, size = 58, normalized size = 2.76 \begin{align*} \frac{{\left (a x^{3} + b x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.750546, size = 41, normalized size = 1.95 \begin{align*} \frac{\sqrt{b} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3} + \frac{b^{\frac{3}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.72396, size = 36, normalized size = 1.71 \begin{align*} \frac{{\left (a x^{2} + b\right )}^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{3 \, a} - \frac{b^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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