Optimal. Leaf size=18 \[ -\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b} \]
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Rubi [A] time = 0.0054849, 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} \[ -\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int \frac{\sqrt{a+\frac{b}{x}}}{x^2} \, dx &=-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0063915, size = 18, normalized size = 1. \[ -\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 1.4 \begin{align*} -{\frac{2\,ax+2\,b}{3\,bx}\sqrt{{\frac{ax+b}{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00931, size = 19, normalized size = 1.06 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67137, size = 54, normalized size = 3. \begin{align*} -\frac{2 \,{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{3 \, b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.10398, size = 41, normalized size = 2.28 \begin{align*} - \frac{2 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}}{3 b} - \frac{2 \sqrt{a} \sqrt{1 + \frac{b}{a x}}}{3 x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12869, size = 112, normalized size = 6.22 \begin{align*} \frac{2 \,{\left (3 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a \mathrm{sgn}\left (x\right ) + 3 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b \mathrm{sgn}\left (x\right ) + b^{2} \mathrm{sgn}\left (x\right )\right )}}{3 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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