Optimal. Leaf size=45 \[ -\frac{6 a^2 b}{\sqrt{x}}+2 a^3 \sqrt{x}-\frac{2 a b^2}{x^{3/2}}-\frac{2 b^3}{5 x^{5/2}} \]
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Rubi [A] time = 0.0131865, antiderivative size = 45, 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} \[ -\frac{6 a^2 b}{\sqrt{x}}+2 a^3 \sqrt{x}-\frac{2 a b^2}{x^{3/2}}-\frac{2 b^3}{5 x^{5/2}} \]
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}{\sqrt{x}} \, dx &=\int \frac{(b+a x)^3}{x^{7/2}} \, dx\\ &=\int \left (\frac{b^3}{x^{7/2}}+\frac{3 a b^2}{x^{5/2}}+\frac{3 a^2 b}{x^{3/2}}+\frac{a^3}{\sqrt{x}}\right ) \, dx\\ &=-\frac{2 b^3}{5 x^{5/2}}-\frac{2 a b^2}{x^{3/2}}-\frac{6 a^2 b}{\sqrt{x}}+2 a^3 \sqrt{x}\\ \end{align*}
Mathematica [A] time = 0.0102485, size = 39, normalized size = 0.87 \[ \frac{2 \left (-15 a^2 b x^2+5 a^3 x^3-5 a b^2 x-b^3\right )}{5 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 36, normalized size = 0.8 \begin{align*}{\frac{10\,{a}^{3}{x}^{3}-30\,{a}^{2}b{x}^{2}-10\,xa{b}^{2}-2\,{b}^{3}}{5}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981312, size = 47, normalized size = 1.04 \begin{align*} 2 \, a^{3} \sqrt{x} - \frac{6 \, a^{2} b}{\sqrt{x}} - \frac{2 \, a b^{2}}{x^{\frac{3}{2}}} - \frac{2 \, b^{3}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73576, size = 78, normalized size = 1.73 \begin{align*} \frac{2 \,{\left (5 \, a^{3} x^{3} - 15 \, a^{2} b x^{2} - 5 \, a b^{2} x - b^{3}\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.773901, size = 44, normalized size = 0.98 \begin{align*} 2 a^{3} \sqrt{x} - \frac{6 a^{2} b}{\sqrt{x}} - \frac{2 a b^{2}}{x^{\frac{3}{2}}} - \frac{2 b^{3}}{5 x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09222, size = 46, normalized size = 1.02 \begin{align*} 2 \, a^{3} \sqrt{x} - \frac{2 \,{\left (15 \, a^{2} b x^{2} + 5 \, a b^{2} x + b^{3}\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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