Optimal. Leaf size=21 \[ -\frac{2 x}{b \sqrt{a x^2+b x^3}} \]
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Rubi [A] time = 0.0175054, antiderivative size = 21, 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} \[ -\frac{2 x}{b \sqrt{a x^2+b x^3}} \]
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a x^2+b x^3\right )^{3/2}} \, dx &=-\frac{2 x}{b \sqrt{a x^2+b x^3}}\\ \end{align*}
Mathematica [A] time = 0.0059717, size = 19, normalized size = 0.9 \[ -\frac{2 x}{b \sqrt{x^2 (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 27, normalized size = 1.3 \begin{align*} -2\,{\frac{ \left ( bx+a \right ){x}^{3}}{b \left ( b{x}^{3}+a{x}^{2} \right ) ^{3/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07839, size = 16, normalized size = 0.76 \begin{align*} -\frac{2}{\sqrt{b x + a} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.83839, size = 57, normalized size = 2.71 \begin{align*} -\frac{2 \, \sqrt{b x^{3} + a x^{2}}}{b^{2} x^{2} + a 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}}{\left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18193, size = 50, normalized size = 2.38 \begin{align*} \frac{2}{{\left (\sqrt{a}{\left (\sqrt{\frac{b}{x} + \frac{a}{x^{2}}} - \frac{\sqrt{a}}{x}\right )} - b\right )} \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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