Optimal. Leaf size=18 \[ -\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b} \]
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Rubi [A] time = 0.0059618, 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} \[ -\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^{3/2}}{x^2} \, dx &=-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b}\\ \end{align*}
Mathematica [A] time = 0.0085436, size = 18, normalized size = 1. \[ -\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b} \]
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}{5\,bx} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.38087, size = 19, normalized size = 1.06 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}}}{5 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73605, size = 78, normalized size = 4.33 \begin{align*} -\frac{2 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{5 \, b x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.998756, size = 65, normalized size = 3.61 \begin{align*} - \frac{2 a^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x}}}{5 b} - \frac{4 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}}{5 x} - \frac{2 \sqrt{a} b \sqrt{1 + \frac{b}{a x}}}{5 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17028, size = 196, normalized size = 10.89 \begin{align*} \frac{2 \,{\left (5 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} \mathrm{sgn}\left (x\right ) + 10 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b \mathrm{sgn}\left (x\right ) + 10 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{2} \mathrm{sgn}\left (x\right ) + 5 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{3} \mathrm{sgn}\left (x\right ) + b^{4} \mathrm{sgn}\left (x\right )\right )}}{5 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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