Optimal. Leaf size=55 \[ \frac{1}{2} a^2 x \sqrt{c x^2}+\frac{2}{3} a b x^2 \sqrt{c x^2}+\frac{1}{4} b^2 x^3 \sqrt{c x^2} \]
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Rubi [A] time = 0.0136003, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {15, 43} \[ \frac{1}{2} a^2 x \sqrt{c x^2}+\frac{2}{3} a b x^2 \sqrt{c x^2}+\frac{1}{4} b^2 x^3 \sqrt{c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \sqrt{c x^2} (a+b x)^2 \, dx &=\frac{\sqrt{c x^2} \int x (a+b x)^2 \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx}{x}\\ &=\frac{1}{2} a^2 x \sqrt{c x^2}+\frac{2}{3} a b x^2 \sqrt{c x^2}+\frac{1}{4} b^2 x^3 \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.0063803, size = 33, normalized size = 0.6 \[ \frac{1}{12} x \sqrt{c x^2} \left (6 a^2+8 a b x+3 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 30, normalized size = 0.6 \begin{align*}{\frac{x \left ( 3\,{b}^{2}{x}^{2}+8\,abx+6\,{a}^{2} \right ) }{12}\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 = 1.54565, size = 70, normalized size = 1.27 \begin{align*} \frac{1}{12} \,{\left (3 \, b^{2} x^{3} + 8 \, a b x^{2} + 6 \, a^{2} 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 = 0.258771, size = 60, normalized size = 1.09 \begin{align*} \frac{a^{2} \sqrt{c} x \sqrt{x^{2}}}{2} + \frac{2 a b \sqrt{c} x^{2} \sqrt{x^{2}}}{3} + \frac{b^{2} \sqrt{c} x^{3} \sqrt{x^{2}}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07227, size = 47, normalized size = 0.85 \begin{align*} \frac{1}{12} \,{\left (3 \, b^{2} x^{4} \mathrm{sgn}\left (x\right ) + 8 \, a b x^{3} \mathrm{sgn}\left (x\right ) + 6 \, a^{2} 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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