Optimal. Leaf size=186 \[ \frac{x^3 \sqrt{c-a^2 c x^2}}{4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x^2 \sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 x \sqrt{c-a^2 c x^2}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-a^2 c x^2}}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (a x+1)}{a^4 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.257982, antiderivative size = 186, 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 = {6192, 6193, 88} \[ \frac{x^3 \sqrt{c-a^2 c x^2}}{4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x^2 \sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 x \sqrt{c-a^2 c x^2}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-a^2 c x^2}}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (a x+1)}{a^4 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 88
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt{c-a^2 c x^2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int e^{-3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}} x^3 \, dx}{\sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \frac{x^2 (-1+a x)^2}{1+a x} \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \left (-\frac{4}{a^2}+\frac{4 x}{a}-3 x^2+a x^3+\frac{4}{a^2 (1+a x)}\right ) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=-\frac{4 \sqrt{c-a^2 c x^2}}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 x \sqrt{c-a^2 c x^2}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x^2 \sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^3 \sqrt{c-a^2 c x^2}}{4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (1+a x)}{a^4 \sqrt{1-\frac{1}{a^2 x^2}} x}\\ \end{align*}
Mathematica [A] time = 0.0510912, size = 72, normalized size = 0.39 \[ \frac{\sqrt{c-a^2 c x^2} \left (a x \left (a^3 x^3-4 a^2 x^2+8 a x-16\right )+16 \log (a x+1)\right )}{4 a^4 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.132, size = 83, normalized size = 0.5 \begin{align*}{\frac{ \left ({x}^{4}{a}^{4}-4\,{x}^{3}{a}^{3}+8\,{a}^{2}{x}^{2}-16\,ax+16\,\ln \left ( ax+1 \right ) \right ) \left ( ax+1 \right ) }{4\,{a}^{3} \left ( ax-1 \right ) ^{2}}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \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 \sqrt{-a^{2} c x^{2} + c} x^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56866, size = 112, normalized size = 0.6 \begin{align*} \frac{{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 8 \, a^{2} x^{2} - 16 \, a x + 16 \, \log \left (a x + 1\right )\right )} \sqrt{-a^{2} c}}{4 \, a^{4}} \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 \sqrt{-a^{2} c x^{2} + c} x^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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