Optimal. Leaf size=35 \[ \frac{a x^3}{2 \sqrt{c x^2}}+\frac{b x^4}{3 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0081311, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 43} \[ \frac{a x^3}{2 \sqrt{c x^2}}+\frac{b x^4}{3 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2 (a+b x)}{\sqrt{c x^2}} \, dx &=\frac{x \int x (a+b x) \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (a x+b x^2\right ) \, dx}{\sqrt{c x^2}}\\ &=\frac{a x^3}{2 \sqrt{c x^2}}+\frac{b x^4}{3 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0041718, size = 24, normalized size = 0.69 \[ \frac{x^3 (3 a+2 b x)}{6 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 21, normalized size = 0.6 \begin{align*}{\frac{{x}^{3} \left ( 2\,bx+3\,a \right ) }{6}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0442, size = 35, normalized size = 1. \begin{align*} \frac{\sqrt{c x^{2}} b x^{2}}{3 \, c} + \frac{a x^{2}}{2 \, \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69145, size = 50, normalized size = 1.43 \begin{align*} \frac{{\left (2 \, b x^{2} + 3 \, a x\right )} \sqrt{c x^{2}}}{6 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.526608, size = 36, normalized size = 1.03 \begin{align*} \frac{a x^{3}}{2 \sqrt{c} \sqrt{x^{2}}} + \frac{b x^{4}}{3 \sqrt{c} \sqrt{x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06585, size = 32, normalized size = 0.91 \begin{align*} \frac{1}{6} \, \sqrt{c x^{2}}{\left (\frac{2 \, b x}{c} + \frac{3 \, a}{c}\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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