Optimal. Leaf size=47 \[ 6 a^2 b \sqrt{x}+\frac{2}{3} a^3 x^{3/2}-\frac{6 a b^2}{\sqrt{x}}-\frac{2 b^3}{3 x^{3/2}} \]
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Rubi [A] time = 0.0130861, 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} \[ 6 a^2 b \sqrt{x}+\frac{2}{3} a^3 x^{3/2}-\frac{6 a b^2}{\sqrt{x}}-\frac{2 b^3}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^3 \sqrt{x} \, dx &=\int \frac{(b+a x)^3}{x^{5/2}} \, dx\\ &=\int \left (\frac{b^3}{x^{5/2}}+\frac{3 a b^2}{x^{3/2}}+\frac{3 a^2 b}{\sqrt{x}}+a^3 \sqrt{x}\right ) \, dx\\ &=-\frac{2 b^3}{3 x^{3/2}}-\frac{6 a b^2}{\sqrt{x}}+6 a^2 b \sqrt{x}+\frac{2}{3} a^3 x^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0099883, size = 38, normalized size = 0.81 \[ \frac{2 \left (9 a^2 b x^2+a^3 x^3-9 a b^2 x-b^3\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 35, normalized size = 0.7 \begin{align*}{\frac{2\,{a}^{3}{x}^{3}+18\,{a}^{2}b{x}^{2}-18\,xa{b}^{2}-2\,{b}^{3}}{3}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971368, size = 49, normalized size = 1.04 \begin{align*} -\frac{6 \, a b^{2}}{\sqrt{x}} + \frac{2}{3} \,{\left (a^{3} + \frac{9 \, a^{2} b}{x}\right )} x^{\frac{3}{2}} - \frac{2 \, b^{3}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69702, size = 74, normalized size = 1.57 \begin{align*} \frac{2 \,{\left (a^{3} x^{3} + 9 \, a^{2} b x^{2} - 9 \, a b^{2} x - b^{3}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.757705, size = 46, normalized size = 0.98 \begin{align*} \frac{2 a^{3} x^{\frac{3}{2}}}{3} + 6 a^{2} b \sqrt{x} - \frac{6 a b^{2}}{\sqrt{x}} - \frac{2 b^{3}}{3 x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09305, size = 46, normalized size = 0.98 \begin{align*} \frac{2}{3} \, a^{3} x^{\frac{3}{2}} + 6 \, a^{2} b \sqrt{x} - \frac{2 \,{\left (9 \, a b^{2} x + b^{3}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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