Optimal. Leaf size=51 \[ -\frac{6 a^2 b}{5 x^{5/2}}-\frac{2 a^3}{3 x^{3/2}}-\frac{6 a b^2}{7 x^{7/2}}-\frac{2 b^3}{9 x^{9/2}} \]
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Rubi [A] time = 0.0133382, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {263, 43} \[ -\frac{6 a^2 b}{5 x^{5/2}}-\frac{2 a^3}{3 x^{3/2}}-\frac{6 a b^2}{7 x^{7/2}}-\frac{2 b^3}{9 x^{9/2}} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^3}{x^{5/2}} \, dx &=\int \frac{(b+a x)^3}{x^{11/2}} \, dx\\ &=\int \left (\frac{b^3}{x^{11/2}}+\frac{3 a b^2}{x^{9/2}}+\frac{3 a^2 b}{x^{7/2}}+\frac{a^3}{x^{5/2}}\right ) \, dx\\ &=-\frac{2 b^3}{9 x^{9/2}}-\frac{6 a b^2}{7 x^{7/2}}-\frac{6 a^2 b}{5 x^{5/2}}-\frac{2 a^3}{3 x^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0112361, size = 39, normalized size = 0.76 \[ -\frac{2 \left (189 a^2 b x^2+105 a^3 x^3+135 a b^2 x+35 b^3\right )}{315 x^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 36, normalized size = 0.7 \begin{align*} -{\frac{210\,{a}^{3}{x}^{3}+378\,{a}^{2}b{x}^{2}+270\,xa{b}^{2}+70\,{b}^{3}}{315}{x}^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997705, size = 47, normalized size = 0.92 \begin{align*} -\frac{2 \, a^{3}}{3 \, x^{\frac{3}{2}}} - \frac{6 \, a^{2} b}{5 \, x^{\frac{5}{2}}} - \frac{6 \, a b^{2}}{7 \, x^{\frac{7}{2}}} - \frac{2 \, b^{3}}{9 \, x^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79762, size = 93, normalized size = 1.82 \begin{align*} -\frac{2 \,{\left (105 \, a^{3} x^{3} + 189 \, a^{2} b x^{2} + 135 \, a b^{2} x + 35 \, b^{3}\right )}}{315 \, x^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.06731, size = 51, normalized size = 1. \begin{align*} - \frac{2 a^{3}}{3 x^{\frac{3}{2}}} - \frac{6 a^{2} b}{5 x^{\frac{5}{2}}} - \frac{6 a b^{2}}{7 x^{\frac{7}{2}}} - \frac{2 b^{3}}{9 x^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1057, size = 47, normalized size = 0.92 \begin{align*} -\frac{2 \,{\left (105 \, a^{3} x^{3} + 189 \, a^{2} b x^{2} + 135 \, a b^{2} x + 35 \, b^{3}\right )}}{315 \, x^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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