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 = 0.005742, antiderivative size = 23, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0166424, size = 23, normalized size = 1. \[ -\frac{2}{3 a x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 25, normalized size = 1.1 \begin{align*} -{\frac{2\,ax+2\,b}{3\,a} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973414, size = 23, normalized size = 1. \begin{align*} -\frac{2}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.49085, size = 84, normalized size = 3.65 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16383, size = 28, normalized size = 1.22 \begin{align*} -\frac{2}{3 \,{\left (a x + b\right )}^{\frac{3}{2}} a} + \frac{2}{3 \, a b^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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