Optimal. Leaf size=153 \[ -\frac{a x^4 \sqrt{c-\frac{c}{a^2 x^2}}}{3 \sqrt{1-a^2 x^2}}-\frac{3 x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 \sqrt{1-a^2 x^2}}-\frac{4 x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-a^2 x^2}}-\frac{4 x \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{a^2 \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.2323, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6160, 6150, 77} \[ -\frac{a x^4 \sqrt{c-\frac{c}{a^2 x^2}}}{3 \sqrt{1-a^2 x^2}}-\frac{3 x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 \sqrt{1-a^2 x^2}}-\frac{4 x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-a^2 x^2}}-\frac{4 x \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{a^2 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 77
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^2 \, dx &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int e^{3 \tanh ^{-1}(a x)} x \sqrt{1-a^2 x^2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{x (1+a x)^2}{1-a x} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \left (-\frac{4}{a}-3 x-a x^2-\frac{4}{a (-1+a x)}\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} x^2}{a \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{c-\frac{c}{a^2 x^2}} x^3}{2 \sqrt{1-a^2 x^2}}-\frac{a \sqrt{c-\frac{c}{a^2 x^2}} x^4}{3 \sqrt{1-a^2 x^2}}-\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} x \log (1-a x)}{a^2 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0424301, size = 65, normalized size = 0.42 \[ \frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left (-\frac{4 \log (1-a x)}{a^2}-\frac{a x^3}{3}-\frac{4 x}{a}-\frac{3 x^2}{2}\right )}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.141, size = 78, normalized size = 0.5 \begin{align*}{\frac{x \left ( 2\,{x}^{3}{a}^{3}+9\,{a}^{2}{x}^{2}+24\,ax+24\,\ln \left ( ax-1 \right ) \right ) }{ \left ( 6\,{a}^{2}{x}^{2}-6 \right ){a}^{2}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.32941, size = 232, normalized size = 1.52 \begin{align*} \frac{1}{6} \, a^{3}{\left (\frac{2 \,{\left (i \, a^{2} \sqrt{c} x^{3} + 3 i \, \sqrt{c} x\right )}}{a^{5}} - \frac{3 i \, \sqrt{c} \log \left (a x + 1\right )}{a^{6}} + \frac{3 i \, \sqrt{c} \log \left (a x - 1\right )}{a^{6}}\right )} - \frac{3}{2} \, a^{2}{\left (-\frac{i \, \sqrt{c} x^{2}}{a^{3}} - \frac{i \, \sqrt{c} \log \left (a x + 1\right )}{a^{5}} - \frac{i \, \sqrt{c} \log \left (a x - 1\right )}{a^{5}}\right )} - \frac{3}{2} \, a{\left (-\frac{2 i \, \sqrt{c} x}{a^{3}} + \frac{i \, \sqrt{c} \log \left (a x + 1\right )}{a^{4}} - \frac{i \, \sqrt{c} \log \left (a x - 1\right )}{a^{4}}\right )} + \frac{i \, \sqrt{c} \log \left (a x + 1\right )}{2 \, a^{3}} + \frac{i \, \sqrt{c} \log \left (a x - 1\right )}{2 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.52007, size = 846, normalized size = 5.53 \begin{align*} \left [\frac{12 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{-c} \log \left (\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x -{\left (a^{5} x^{5} - 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - 4 \, a^{2} x^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right ) +{\left (2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 24 \, a^{2} x^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \,{\left (a^{5} x^{2} - a^{3}\right )}}, \frac{24 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{c} \arctan \left (\frac{{\left (a^{2} x^{2} - 2 \, a x + 2\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} - 2 \, a^{2} c x^{2} - a c x + 2 \, c}\right ) +{\left (2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 24 \, a^{2} x^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \,{\left (a^{5} x^{2} - a^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \sqrt{- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}} x^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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