Optimal. Leaf size=38 \[ -\frac{6 a^2 b}{\sqrt{x}}-\frac{a^3}{x}+3 a b^2 \log (x)+2 b^3 \sqrt{x} \]
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Rubi [A] time = 0.0195794, antiderivative size = 38, 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 = {266, 43} \[ -\frac{6 a^2 b}{\sqrt{x}}-\frac{a^3}{x}+3 a b^2 \log (x)+2 b^3 \sqrt{x} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^3}{x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (b^3+\frac{a^3}{x^3}+\frac{3 a^2 b}{x^2}+\frac{3 a b^2}{x}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^3}{x}-\frac{6 a^2 b}{\sqrt{x}}+2 b^3 \sqrt{x}+3 a b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0226885, size = 38, normalized size = 1. \[ -\frac{6 a^2 b}{\sqrt{x}}-\frac{a^3}{x}+3 a b^2 \log (x)+2 b^3 \sqrt{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 35, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{x}}+3\,a{b}^{2}\ln \left ( x \right ) -6\,{\frac{b{a}^{2}}{\sqrt{x}}}+2\,{b}^{3}\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98843, size = 47, normalized size = 1.24 \begin{align*} 3 \, a b^{2} \log \left (x\right ) + 2 \, b^{3} \sqrt{x} - \frac{6 \, a^{2} b \sqrt{x} + a^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48879, size = 86, normalized size = 2.26 \begin{align*} \frac{6 \, a b^{2} x \log \left (\sqrt{x}\right ) - a^{3} + 2 \,{\left (b^{3} x - 3 \, a^{2} b\right )} \sqrt{x}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.575448, size = 36, normalized size = 0.95 \begin{align*} - \frac{a^{3}}{x} - \frac{6 a^{2} b}{\sqrt{x}} + 3 a b^{2} \log{\left (x \right )} + 2 b^{3} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11535, size = 49, normalized size = 1.29 \begin{align*} 3 \, a b^{2} \log \left ({\left | x \right |}\right ) + 2 \, b^{3} \sqrt{x} - \frac{6 \, a^{2} b \sqrt{x} + a^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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