Optimal. Leaf size=152 \[ \frac{a x^4 \sqrt{c-\frac{c}{a^2 x^2}}}{3 \sqrt{1-a^2 x^2}}-\frac{3 x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 \sqrt{1-a^2 x^2}}+\frac{4 x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-a^2 x^2}}-\frac{4 x \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{a^2 \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.240186, antiderivative size = 152, 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} \[ \frac{a x^4 \sqrt{c-\frac{c}{a^2 x^2}}}{3 \sqrt{1-a^2 x^2}}-\frac{3 x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 \sqrt{1-a^2 x^2}}+\frac{4 x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-a^2 x^2}}-\frac{4 x \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{a^2 \sqrt{1-a^2 x^2}} \]
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 = 0.0462516, size = 64, normalized size = 0.42 \[ \frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left (-\frac{4 \log (a x+1)}{a^2}+\frac{a x^3}{3}+\frac{4 x}{a}-\frac{3 x^2}{2}\right )}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.139, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \sqrt{c - \frac{c}{a^{2} x^{2}}} x^{2}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.45616, size = 846, normalized size = 5.57 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{2} x^{2} + 1\right )}^{\frac{3}{2}} \sqrt{c - \frac{c}{a^{2} x^{2}}} x^{2}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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