Optimal. Leaf size=23 \[ -\frac {2}{3 a x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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} \[ -\frac {2}{3 a x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{5/2}} \, dx &=-\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 1.00 \[ -\frac {2}{3 a x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.98, size = 38, normalized size = 1.65 \[ -\frac {2 \, \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{3} x^{2} + 2 \, a^{2} b x + a b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 21, normalized size = 0.91 \[ -\frac {2}{3 \, {\left (a x + b\right )}^{\frac {3}{2}} a} + \frac {2}{3 \, a b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 1.09 \[ -\frac {2 \left (a x +b \right )}{3 \left (\frac {a x +b}{x}\right )^{\frac {5}{2}} a \,x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 17, normalized size = 0.74 \[ -\frac {2}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.49, size = 38, normalized size = 1.65 \[ -\frac {2\,\sqrt {x}\,\sqrt {a+\frac {b}{x}}}{3\,\left (a^3\,x^2+2\,a^2\,b\,x+a\,b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 20.02, size = 42, normalized size = 1.83 \[ - \frac {2}{3 a^{2} \sqrt {b} x \sqrt {\frac {a x}{b} + 1} + 3 a b^{\frac {3}{2}} \sqrt {\frac {a x}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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