Optimal. Leaf size=25 \[ \frac{2 \sqrt{a x^2+b x^5}}{3 b x} \]
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Rubi [A] time = 0.016886, 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 = {1588} \[ \frac{2 \sqrt{a x^2+b x^5}}{3 b x} \]
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{a x^2+b x^5}} \, dx &=\frac{2 \sqrt{a x^2+b x^5}}{3 b x}\\ \end{align*}
Mathematica [A] time = 0.0077292, size = 25, normalized size = 1. \[ \frac{2 \sqrt{x^2 \left (a+b x^3\right )}}{3 b x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 27, normalized size = 1.1 \begin{align*}{\frac{ \left ( 2\,b{x}^{3}+2\,a \right ) x}{3\,b}{\frac{1}{\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 = 1.1064, size = 19, normalized size = 0.76 \begin{align*} \frac{2 \, \sqrt{b x^{3} + a}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.796274, size = 42, normalized size = 1.68 \begin{align*} \frac{2 \, \sqrt{b x^{5} + a x^{2}}}{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{x^{3}}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{b x^{5} + a x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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