Optimal. Leaf size=25 \[ \frac{2 \left (a x^2+b x^3\right )^{5/2}}{5 b x^5} \]
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Rubi [A] time = 0.0406793, 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} \[ \frac{2 \left (a x^2+b x^3\right )^{5/2}}{5 b x^5} \]
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin{align*} \int \frac{\left (a x^2+b x^3\right )^{3/2}}{x^3} \, dx &=\frac{2 \left (a x^2+b x^3\right )^{5/2}}{5 b x^5}\\ \end{align*}
Mathematica [A] time = 0.0128377, size = 23, normalized size = 0.92 \[ \frac{2 \left (x^2 (a+b x)\right )^{5/2}}{5 b x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 27, normalized size = 1.1 \begin{align*}{\frac{2\,bx+2\,a}{5\,b{x}^{3}} \left ( b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988703, size = 38, normalized size = 1.52 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a}}{5 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.811251, size = 77, normalized size = 3.08 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x^{3} + a x^{2}}}{5 \, 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{\left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24223, size = 70, normalized size = 2.8 \begin{align*} -\frac{2 \, a^{\frac{5}{2}} \mathrm{sgn}\left (x\right )}{5 \, b} + \frac{2 \,{\left (5 \,{\left (b x + a\right )}^{\frac{3}{2}} a \mathrm{sgn}\left (x\right ) +{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} \mathrm{sgn}\left (x\right )\right )}}{15 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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