Optimal. Leaf size=18 \[ -\frac {2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b} \]
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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} \[ -\frac {2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b} \]
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*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \[ -\frac {2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 24, normalized size = 1.33 \[ -\frac {2 \, {\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}}{3 \, b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 83, normalized size = 4.61 \[ \frac {2 \, {\left (3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a \mathrm {sgn}\relax (x) + 3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b \mathrm {sgn}\relax (x) + b^{2} \mathrm {sgn}\relax (x)\right )}}{3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3}} \]
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 \[ -\frac {2 \left (a x +b \right ) \sqrt {\frac {a x +b}{x}}}{3 b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 14, normalized size = 0.78 \[ -\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 22, normalized size = 1.22 \[ -\frac {2\,\sqrt {a+\frac {b}{x}}\,\left (b+a\,x\right )}{3\,b\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.94, size = 41, normalized size = 2.28 \[ - \frac {2 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}}{3 b} - \frac {2 \sqrt {a} \sqrt {1 + \frac {b}{a x}}}{3 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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