Optimal. Leaf size=18 \[ -\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{5 b} \]
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Rubi [A] time = 0.0060208, antiderivative size = 18, 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 = {261} \[ -\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{x^3} \, dx &=-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{5 b}\\ \end{align*}
Mathematica [A] time = 0.009633, size = 28, normalized size = 1.56 \[ -\frac{\left (a+\frac{b}{x^2}\right )^{3/2} \left (a x^2+b\right )}{5 b x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 29, normalized size = 1.6 \begin{align*} -{\frac{a{x}^{2}+b}{5\,b{x}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989785, size = 19, normalized size = 1.06 \begin{align*} -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}}}{5 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.50641, size = 86, normalized size = 4.78 \begin{align*} -\frac{{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{5 \, b x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.13738, size = 68, normalized size = 3.78 \begin{align*} - \frac{a^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{5 b} - \frac{2 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{5 x^{2}} - \frac{\sqrt{a} b \sqrt{1 + \frac{b}{a x^{2}}}}{5 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21479, size = 124, normalized size = 6.89 \begin{align*} \frac{2 \,{\left (5 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{8} a^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + 10 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{4} a^{\frac{5}{2}} b^{2} \mathrm{sgn}\left (x\right ) + a^{\frac{5}{2}} b^{4} \mathrm{sgn}\left (x\right )\right )}}{5 \,{\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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