Optimal. Leaf size=81 \[ -\frac{\sqrt{a+c x^2} (4 a B-3 A c x)}{6 c^2}-\frac{a A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}+\frac{B x^2 \sqrt{a+c x^2}}{3 c} \]
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Rubi [A] time = 0.0396998, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {833, 780, 217, 206} \[ -\frac{\sqrt{a+c x^2} (4 a B-3 A c x)}{6 c^2}-\frac{a A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}+\frac{B x^2 \sqrt{a+c x^2}}{3 c} \]
Antiderivative was successfully verified.
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Rule 833
Rule 780
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^2 (A+B x)}{\sqrt{a+c x^2}} \, dx &=\frac{B x^2 \sqrt{a+c x^2}}{3 c}+\frac{\int \frac{x (-2 a B+3 A c x)}{\sqrt{a+c x^2}} \, dx}{3 c}\\ &=\frac{B x^2 \sqrt{a+c x^2}}{3 c}-\frac{(4 a B-3 A c x) \sqrt{a+c x^2}}{6 c^2}-\frac{(a A) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{2 c}\\ &=\frac{B x^2 \sqrt{a+c x^2}}{3 c}-\frac{(4 a B-3 A c x) \sqrt{a+c x^2}}{6 c^2}-\frac{(a A) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{2 c}\\ &=\frac{B x^2 \sqrt{a+c x^2}}{3 c}-\frac{(4 a B-3 A c x) \sqrt{a+c x^2}}{6 c^2}-\frac{a A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0346598, size = 64, normalized size = 0.79 \[ \frac{\sqrt{a+c x^2} (c x (3 A+2 B x)-4 a B)-3 a A \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{6 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 75, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}B}{3\,c}\sqrt{c{x}^{2}+a}}-{\frac{2\,aB}{3\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{Ax}{2\,c}\sqrt{c{x}^{2}+a}}-{\frac{aA}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{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 = 1.66548, size = 320, normalized size = 3.95 \begin{align*} \left [\frac{3 \, A a \sqrt{c} \log \left (-2 \, c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (2 \, B c x^{2} + 3 \, A c x - 4 \, B a\right )} \sqrt{c x^{2} + a}}{12 \, c^{2}}, \frac{3 \, A a \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (2 \, B c x^{2} + 3 \, A c x - 4 \, B a\right )} \sqrt{c x^{2} + a}}{6 \, c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.19506, size = 94, normalized size = 1.16 \begin{align*} \frac{A \sqrt{a} x \sqrt{1 + \frac{c x^{2}}{a}}}{2 c} - \frac{A a \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 c^{\frac{3}{2}}} + B \left (\begin{cases} - \frac{2 a \sqrt{a + c x^{2}}}{3 c^{2}} + \frac{x^{2} \sqrt{a + c x^{2}}}{3 c} & \text{for}\: c \neq 0 \\\frac{x^{4}}{4 \sqrt{a}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1662, size = 82, normalized size = 1.01 \begin{align*} \frac{1}{6} \, \sqrt{c x^{2} + a}{\left ({\left (\frac{2 \, B x}{c} + \frac{3 \, A}{c}\right )} x - \frac{4 \, B a}{c^{2}}\right )} + \frac{A a \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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