Optimal. Leaf size=25 \[ \frac{2 \left (a x^2+b x^5\right )^{3/2}}{9 b x^3} \]
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Rubi [A] time = 0.0105747, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {1588} \[ \frac{2 \left (a x^2+b x^5\right )^{3/2}}{9 b x^3} \]
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin{align*} \int x \sqrt{a x^2+b x^5} \, dx &=\frac{2 \left (a x^2+b x^5\right )^{3/2}}{9 b x^3}\\ \end{align*}
Mathematica [A] time = 0.0023885, size = 25, normalized size = 1. \[ \frac{2 \left (x^2 \left (a+b x^3\right )\right )^{3/2}}{9 b x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 29, normalized size = 1.2 \begin{align*}{\frac{2\,b{x}^{3}+2\,a}{9\,bx}\sqrt{b{x}^{5}+a{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13338, size = 19, normalized size = 0.76 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{9 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.937805, size = 58, normalized size = 2.32 \begin{align*} \frac{2 \, \sqrt{b x^{5} + a x^{2}}{\left (b x^{3} + a\right )}}{9 \, 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 x \sqrt{x^{2} \left (a + b x^{3}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17149, size = 36, normalized size = 1.44 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{9 \, b} - \frac{2 \, a^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{9 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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