Optimal. Leaf size=47 \[ -\frac{2 a^2 b}{x^{3/2}}-\frac{2 a^3}{\sqrt{x}}-\frac{6 a b^2}{5 x^{5/2}}-\frac{2 b^3}{7 x^{7/2}} \]
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Rubi [A] time = 0.0126803, antiderivative size = 47, 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} \[ -\frac{2 a^2 b}{x^{3/2}}-\frac{2 a^3}{\sqrt{x}}-\frac{6 a b^2}{5 x^{5/2}}-\frac{2 b^3}{7 x^{7/2}} \]
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^{3/2}} \, dx &=\int \frac{(b+a x)^3}{x^{9/2}} \, dx\\ &=\int \left (\frac{b^3}{x^{9/2}}+\frac{3 a b^2}{x^{7/2}}+\frac{3 a^2 b}{x^{5/2}}+\frac{a^3}{x^{3/2}}\right ) \, dx\\ &=-\frac{2 b^3}{7 x^{7/2}}-\frac{6 a b^2}{5 x^{5/2}}-\frac{2 a^2 b}{x^{3/2}}-\frac{2 a^3}{\sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.011938, size = 39, normalized size = 0.83 \[ -\frac{2 \left (35 a^2 b x^2+35 a^3 x^3+21 a b^2 x+5 b^3\right )}{35 x^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 36, normalized size = 0.8 \begin{align*} -{\frac{70\,{a}^{3}{x}^{3}+70\,{a}^{2}b{x}^{2}+42\,xa{b}^{2}+10\,{b}^{3}}{35}{x}^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976587, size = 47, normalized size = 1. \begin{align*} -\frac{2 \, a^{3}}{\sqrt{x}} - \frac{2 \, a^{2} b}{x^{\frac{3}{2}}} - \frac{6 \, a b^{2}}{5 \, x^{\frac{5}{2}}} - \frac{2 \, b^{3}}{7 \, x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75888, size = 86, normalized size = 1.83 \begin{align*} -\frac{2 \,{\left (35 \, a^{3} x^{3} + 35 \, a^{2} b x^{2} + 21 \, a b^{2} x + 5 \, b^{3}\right )}}{35 \, x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.1246, size = 48, normalized size = 1.02 \begin{align*} - \frac{2 a^{3}}{\sqrt{x}} - \frac{2 a^{2} b}{x^{\frac{3}{2}}} - \frac{6 a b^{2}}{5 x^{\frac{5}{2}}} - \frac{2 b^{3}}{7 x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11551, size = 47, normalized size = 1. \begin{align*} -\frac{2 \,{\left (35 \, a^{3} x^{3} + 35 \, a^{2} b x^{2} + 21 \, a b^{2} x + 5 \, b^{3}\right )}}{35 \, x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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