Optimal. Leaf size=18 \[ -\frac {2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b} \]
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Rubi [A] time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \[ -\frac {2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.03, size = 35, normalized size = 1.94 \[ -\frac {2 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {\frac {a x + b}{x}}}{5 \, b x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 145, normalized size = 8.06 \[ \frac {2 \, {\left (5 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} \mathrm {sgn}\relax (x) + 10 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b \mathrm {sgn}\relax (x) + 10 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{2} \mathrm {sgn}\relax (x) + 5 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{3} \mathrm {sgn}\relax (x) + b^{4} \mathrm {sgn}\relax (x)\right )}}{5 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 1.39 \[ -\frac {2 \left (a x +b \right ) \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}{5 b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 14, normalized size = 0.78 \[ -\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}}}{5 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.59, size = 24, normalized size = 1.33 \[ -\frac {2\,\sqrt {a+\frac {b}{x}}\,{\left (b+a\,x\right )}^2}{5\,b\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.36, size = 65, normalized size = 3.61 \[ - \frac {2 a^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x}}}{5 b} - \frac {4 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}}{5 x} - \frac {2 \sqrt {a} b \sqrt {1 + \frac {b}{a x}}}{5 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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