Optimal. Leaf size=21 \[ -\frac{2}{a \sqrt{x} \sqrt{a+\frac{b}{x}}} \]
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Rubi [A] time = 0.0056761, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ -\frac{2}{a \sqrt{x} \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^{3/2}} \, dx &=-\frac{2}{a \sqrt{a+\frac{b}{x}} \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0128592, size = 21, normalized size = 1. \[ -\frac{2}{a \sqrt{x} \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 25, normalized size = 1.2 \begin{align*} -2\,{\frac{ax+b}{a{x}^{3/2}} \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 = 0.964417, size = 23, normalized size = 1.1 \begin{align*} -\frac{2}{\sqrt{a + \frac{b}{x}} a \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44867, size = 59, normalized size = 2.81 \begin{align*} -\frac{2 \, \sqrt{x} \sqrt{\frac{a x + b}{x}}}{a^{2} x + a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.5344, size = 19, normalized size = 0.9 \begin{align*} - \frac{2}{a \sqrt{b} \sqrt{\frac{a x}{b} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2211, size = 28, normalized size = 1.33 \begin{align*} -\frac{2}{\sqrt{a x + b} a} + \frac{2}{a \sqrt{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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