Optimal. Leaf size=19 \[ -\frac{1}{a x \sqrt{a+\frac{b}{x^2}}} \]
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Rubi [A] time = 0.0065681, antiderivative size = 19, 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} \[ -\frac{1}{a x \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x^2} \, dx &=-\frac{1}{a \sqrt{a+\frac{b}{x^2}} x}\\ \end{align*}
Mathematica [A] time = 0.00497, size = 19, normalized size = 1. \[ -\frac{1}{a x \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 29, normalized size = 1.5 \begin{align*} -{\frac{a{x}^{2}+b}{a{x}^{3}} \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 = 0.989726, size = 23, normalized size = 1.21 \begin{align*} -\frac{1}{\sqrt{a + \frac{b}{x^{2}}} a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4721, size = 57, normalized size = 3. \begin{align*} -\frac{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a^{2} x^{2} + a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.987411, size = 20, normalized size = 1.05 \begin{align*} - \frac{1}{a \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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