Optimal. Leaf size=67 \[ \frac{1}{2} A x \sqrt{a+c x^2}+\frac{a A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}}+\frac{B \left (a+c x^2\right )^{3/2}}{3 c} \]
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Rubi [A] time = 0.0186641, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {641, 195, 217, 206} \[ \frac{1}{2} A x \sqrt{a+c x^2}+\frac{a A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}}+\frac{B \left (a+c x^2\right )^{3/2}}{3 c} \]
Antiderivative was successfully verified.
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Rule 641
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int (A+B x) \sqrt{a+c x^2} \, dx &=\frac{B \left (a+c x^2\right )^{3/2}}{3 c}+A \int \sqrt{a+c x^2} \, dx\\ &=\frac{1}{2} A x \sqrt{a+c x^2}+\frac{B \left (a+c x^2\right )^{3/2}}{3 c}+\frac{1}{2} (a A) \int \frac{1}{\sqrt{a+c x^2}} \, dx\\ &=\frac{1}{2} A x \sqrt{a+c x^2}+\frac{B \left (a+c x^2\right )^{3/2}}{3 c}+\frac{1}{2} (a A) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )\\ &=\frac{1}{2} A x \sqrt{a+c x^2}+\frac{B \left (a+c x^2\right )^{3/2}}{3 c}+\frac{a A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0526464, size = 67, normalized size = 1. \[ \frac{\sqrt{a+c x^2} (2 a B+c x (3 A+2 B x))+3 a A \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{6 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 53, normalized size = 0.8 \begin{align*}{\frac{B}{3\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Ax}{2}\sqrt{c{x}^{2}+a}}+{\frac{aA}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \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 = 1.64579, size = 316, normalized size = 4.72 \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 + 2 \, B a\right )} \sqrt{c x^{2} + a}}{12 \, c}, -\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 + 2 \, B a\right )} \sqrt{c x^{2} + a}}{6 \, c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.21722, size = 70, normalized size = 1.04 \begin{align*} \frac{A \sqrt{a} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{A a \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 \sqrt{c}} + B \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10577, size = 74, normalized size = 1.1 \begin{align*} -\frac{A a \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, \sqrt{c}} + \frac{1}{6} \, \sqrt{c x^{2} + a}{\left ({\left (2 \, B x + 3 \, A\right )} x + \frac{2 \, B a}{c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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