Optimal. Leaf size=25 \[ \frac{2 \left (a x^2+b x^3\right )^{3/2}}{3 b x^3} \]
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Rubi [A] time = 0.0358451, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2014} \[ \frac{2 \left (a x^2+b x^3\right )^{3/2}}{3 b x^3} \]
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin{align*} \int \frac{\sqrt{a x^2+b x^3}}{x} \, dx &=\frac{2 \left (a x^2+b x^3\right )^{3/2}}{3 b x^3}\\ \end{align*}
Mathematica [A] time = 0.0090109, size = 23, normalized size = 0.92 \[ \frac{2 \left (x^2 (a+b x)\right )^{3/2}}{3 b x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 27, normalized size = 1.1 \begin{align*}{\frac{2\,bx+2\,a}{3\,bx}\sqrt{b{x}^{3}+a{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00775, size = 16, normalized size = 0.64 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.823993, size = 55, normalized size = 2.2 \begin{align*} \frac{2 \, \sqrt{b x^{3} + a x^{2}}{\left (b x + a\right )}}{3 \, b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} \left (a + b x\right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17417, size = 34, normalized size = 1.36 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{3 \, b} - \frac{2 \, a^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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