Optimal. Leaf size=16 \[ \frac{2}{b \sqrt{a+\frac{b}{x}}} \]
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Rubi [A] time = 0.0056079, antiderivative size = 16, 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} \[ \frac{2}{b \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^2} \, dx &=\frac{2}{b \sqrt{a+\frac{b}{x}}}\\ \end{align*}
Mathematica [A] time = 0.0085634, size = 16, normalized size = 1. \[ \frac{2}{b \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 1.6 \begin{align*} 2\,{\frac{ax+b}{bx} \left ({\frac{ax+b}{x}} \right ) ^{-3/2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980123, size = 19, normalized size = 1.19 \begin{align*} \frac{2}{\sqrt{a + \frac{b}{x}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46794, size = 50, normalized size = 3.12 \begin{align*} \frac{2 \, x \sqrt{\frac{a x + b}{x}}}{a b x + b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.69579, size = 20, normalized size = 1.25 \begin{align*} \begin{cases} \frac{2}{b \sqrt{a + \frac{b}{x}}} & \text{for}\: b \neq 0 \\- \frac{1}{a^{\frac{3}{2}} x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24935, size = 19, normalized size = 1.19 \begin{align*} \frac{2}{\sqrt{a + \frac{b}{x}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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