Optimal. Leaf size=151 \[ \frac{x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 x \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.286942, antiderivative size = 151, 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 = {6197, 6193, 77} \[ \frac{x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 x \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6197
Rule 6193
Rule 77
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^2 \, dx &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int e^{-3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}} x^2 \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int \frac{x (-1+a x)^2}{1+a x} \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int \left (\frac{4}{a}-3 x+a x^2-\frac{4}{a (1+a x)}\right ) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} x}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 \sqrt{c-\frac{c}{a^2 x^2}} x^2}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^3}{3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} \log (1+a x)}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0434261, size = 62, normalized size = 0.41 \[ \frac{\sqrt{c-\frac{c}{a^2 x^2}} \left (a x \left (2 a^2 x^2-9 a x+24\right )-24 \log (a x+1)\right )}{6 a^3 \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.237, size = 82, normalized size = 0.5 \begin{align*} -{\frac{ \left ( -2\,{x}^{3}{a}^{3}+9\,{a}^{2}{x}^{2}-24\,ax+24\,\ln \left ( ax+1 \right ) \right ) x \left ( ax+1 \right ) }{6\,{a}^{2} \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}} \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{c - \frac{c}{a^{2} x^{2}}} 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 = 1.67213, size = 97, normalized size = 0.64 \begin{align*} \frac{{\left (2 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 24 \, a x - 24 \, \log \left (a x + 1\right )\right )} \sqrt{a^{2} c}}{6 \, 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{c - \frac{c}{a^{2} x^{2}}} 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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