Optimal. Leaf size=37 \[ 6 a^2 b \sqrt{x}+a^3 \log (x)+3 a b^2 x+\frac{2}{3} b^3 x^{3/2} \]
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Rubi [A] time = 0.0175741, antiderivative size = 37, 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} \[ 6 a^2 b \sqrt{x}+a^3 \log (x)+3 a b^2 x+\frac{2}{3} b^3 x^{3/2} \]
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} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (3 a^2 b+\frac{a^3}{x}+3 a b^2 x+b^3 x^2\right ) \, dx,x,\sqrt{x}\right )\\ &=6 a^2 b \sqrt{x}+3 a b^2 x+\frac{2}{3} b^3 x^{3/2}+a^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.012699, size = 37, normalized size = 1. \[ 6 a^2 b \sqrt{x}+a^3 \log (x)+3 a b^2 x+\frac{2}{3} b^3 x^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 32, normalized size = 0.9 \begin{align*} 3\,xa{b}^{2}+{\frac{2\,{b}^{3}}{3}{x}^{{\frac{3}{2}}}}+{a}^{3}\ln \left ( x \right ) +6\,{a}^{2}b\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961255, size = 42, normalized size = 1.14 \begin{align*} \frac{2}{3} \, b^{3} x^{\frac{3}{2}} + 3 \, a b^{2} x + a^{3} \log \left (x\right ) + 6 \, a^{2} b \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47456, size = 86, normalized size = 2.32 \begin{align*} 3 \, a b^{2} x + 2 \, a^{3} \log \left (\sqrt{x}\right ) + \frac{2}{3} \,{\left (b^{3} x + 9 \, a^{2} b\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.305087, size = 37, normalized size = 1. \begin{align*} a^{3} \log{\left (x \right )} + 6 a^{2} b \sqrt{x} + 3 a b^{2} x + \frac{2 b^{3} x^{\frac{3}{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1018, size = 43, normalized size = 1.16 \begin{align*} \frac{2}{3} \, b^{3} x^{\frac{3}{2}} + 3 \, a b^{2} x + a^{3} \log \left ({\left | x \right |}\right ) + 6 \, a^{2} b \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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