Optimal. Leaf size=33 \[ \frac{1}{2} a x \sqrt{c x^2}+\frac{1}{3} b x^2 \sqrt{c x^2} \]
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Rubi [A] time = 0.0078348, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {15, 43} \[ \frac{1}{2} a x \sqrt{c x^2}+\frac{1}{3} b x^2 \sqrt{c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \sqrt{c x^2} (a+b x) \, dx &=\frac{\sqrt{c x^2} \int x (a+b x) \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (a x+b x^2\right ) \, dx}{x}\\ &=\frac{1}{2} a x \sqrt{c x^2}+\frac{1}{3} b x^2 \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.0029974, size = 22, normalized size = 0.67 \[ \frac{1}{6} x \sqrt{c x^2} (3 a+2 b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 19, normalized size = 0.6 \begin{align*}{\frac{x \left ( 2\,bx+3\,a \right ) }{6}\sqrt{c{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57154, size = 47, normalized size = 1.42 \begin{align*} \frac{1}{6} \,{\left (2 \, b x^{2} + 3 \, a x\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.193054, size = 34, normalized size = 1.03 \begin{align*} \frac{a \sqrt{c} x \sqrt{x^{2}}}{2} + \frac{b \sqrt{c} x^{2} \sqrt{x^{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04885, size = 30, normalized size = 0.91 \begin{align*} \frac{1}{6} \,{\left (2 \, b x^{3} \mathrm{sgn}\left (x\right ) + 3 \, a x^{2} \mathrm{sgn}\left (x\right )\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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