Optimal. Leaf size=57 \[ \frac{1}{3} a^2 x^2 \sqrt{c x^2}+\frac{1}{2} a b x^3 \sqrt{c x^2}+\frac{1}{5} b^2 x^4 \sqrt{c x^2} \]
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Rubi [A] time = 0.0134847, antiderivative size = 57, 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} \[ \frac{1}{3} a^2 x^2 \sqrt{c x^2}+\frac{1}{2} a b x^3 \sqrt{c x^2}+\frac{1}{5} b^2 x^4 \sqrt{c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int x \sqrt{c x^2} (a+b x)^2 \, dx &=\frac{\sqrt{c x^2} \int x^2 (a+b x)^2 \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (a^2 x^2+2 a b x^3+b^2 x^4\right ) \, dx}{x}\\ &=\frac{1}{3} a^2 x^2 \sqrt{c x^2}+\frac{1}{2} a b x^3 \sqrt{c x^2}+\frac{1}{5} b^2 x^4 \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.0052016, size = 35, normalized size = 0.61 \[ \frac{1}{30} x^2 \sqrt{c x^2} \left (10 a^2+15 a b x+6 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.6 \begin{align*}{\frac{{x}^{2} \left ( 6\,{b}^{2}{x}^{2}+15\,abx+10\,{a}^{2} \right ) }{30}\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 = 1.50877, size = 76, normalized size = 1.33 \begin{align*} \frac{1}{30} \,{\left (6 \, b^{2} x^{4} + 15 \, a b x^{3} + 10 \, a^{2} x^{2}\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.347242, size = 60, normalized size = 1.05 \begin{align*} \frac{a^{2} \sqrt{c} x^{2} \sqrt{x^{2}}}{3} + \frac{a b \sqrt{c} x^{3} \sqrt{x^{2}}}{2} + \frac{b^{2} \sqrt{c} x^{4} \sqrt{x^{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05721, size = 47, normalized size = 0.82 \begin{align*} \frac{1}{30} \,{\left (6 \, b^{2} x^{5} \mathrm{sgn}\left (x\right ) + 15 \, a b x^{4} \mathrm{sgn}\left (x\right ) + 10 \, a^{2} x^{3} \mathrm{sgn}\left (x\right )\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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