Optimal. Leaf size=15 \[ \frac{1}{b \sqrt{a+\frac{b}{x^2}}} \]
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Rubi [A] time = 0.0061414, antiderivative size = 15, 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} \[ \frac{1}{b \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x^3} \, dx &=\frac{1}{b \sqrt{a+\frac{b}{x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0063079, size = 15, normalized size = 1. \[ \frac{1}{b \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 28, normalized size = 1.9 \begin{align*}{\frac{a{x}^{2}+b}{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 = 0.97695, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{\sqrt{a + \frac{b}{x^{2}}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.44293, size = 58, normalized size = 3.87 \begin{align*} \frac{x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a b x^{2} + b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.85188, size = 26, normalized size = 1.73 \begin{align*} \begin{cases} \frac{1}{b \sqrt{a + \frac{b}{x^{2}}}} & \text{for}\: b \neq 0 \\- \frac{1}{2 a^{\frac{3}{2}} x^{2}} & \text{otherwise} \end{cases} \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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