Optimal. Leaf size=23 \[ \frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{7 a} \]
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Rubi [A] time = 0.0057093, 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} \[ \frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{7 a} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^{5/2} x^{5/2} \, dx &=\frac{2 \left (a+\frac{b}{x}\right )^{7/2} x^{7/2}}{7 a}\\ \end{align*}
Mathematica [A] time = 0.0165905, size = 23, normalized size = 1. \[ \frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{7 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 1.1 \begin{align*}{\frac{2\,ax+2\,b}{7\,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 = 0.953487, size = 23, normalized size = 1. \begin{align*} \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} x^{\frac{7}{2}}}{7 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.49879, size = 101, normalized size = 4.39 \begin{align*} \frac{2 \,{\left (a^{3} x^{3} + 3 \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{7 \, a} \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 [B] time = 1.20944, size = 81, normalized size = 3.52 \begin{align*} \frac{2 \,{\left (15 \,{\left (a x + b\right )}^{\frac{7}{2}} - 42 \,{\left (a x + b\right )}^{\frac{5}{2}} b + 70 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2} + 14 \,{\left (3 \,{\left (a x + b\right )}^{\frac{5}{2}} - 5 \,{\left (a x + b\right )}^{\frac{3}{2}} b\right )} b\right )}}{105 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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