Optimal. Leaf size=227 \[ \frac{x^4 \sqrt{c-a^2 c x^2}}{5 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 x^3 \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 x^2 \sqrt{c-a^2 c x^2}}{3 a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2 x \sqrt{c-a^2 c x^2}}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2}}{a^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-a^2 c x^2} \log (a x+1)}{a^5 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.261407, antiderivative size = 227, 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} \[ \frac{x^4 \sqrt{c-a^2 c x^2}}{5 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 x^3 \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 x^2 \sqrt{c-a^2 c x^2}}{3 a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2 x \sqrt{c-a^2 c x^2}}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2}}{a^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-a^2 c x^2} \log (a x+1)}{a^5 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
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^3 \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^4 \, dx}{\sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \frac{x^3 (-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^3}-\frac{4 x}{a^2}+\frac{4 x^2}{a}-3 x^3+a x^4-\frac{4}{a^3 (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^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2 x \sqrt{c-a^2 c x^2}}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 x^2 \sqrt{c-a^2 c x^2}}{3 a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 x^3 \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^4 \sqrt{c-a^2 c x^2}}{5 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-a^2 c x^2} \log (1+a x)}{a^5 \sqrt{1-\frac{1}{a^2 x^2}} x}\\ \end{align*}
Mathematica [A] time = 0.0570865, size = 87, normalized size = 0.38 \[ \frac{\sqrt{c-a^2 c x^2} \left (-\frac{2 x^2}{a^2}+\frac{4 x}{a^3}-\frac{4 \log (a x+1)}{a^4}+\frac{a x^5}{5}+\frac{4 x^3}{3 a}-\frac{3 x^4}{4}\right )}{a x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.127, size = 92, normalized size = 0.4 \begin{align*} -{\frac{ \left ( -12\,{x}^{5}{a}^{5}+45\,{x}^{4}{a}^{4}-80\,{x}^{3}{a}^{3}+120\,{a}^{2}{x}^{2}-240\,ax+240\,\ln \left ( ax+1 \right ) \right ) \left ( ax+1 \right ) }{60\,{a}^{4} \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^{3} \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 = 1.64446, size = 142, normalized size = 0.63 \begin{align*} \frac{{\left (12 \, a^{5} x^{5} - 45 \, a^{4} x^{4} + 80 \, a^{3} x^{3} - 120 \, a^{2} x^{2} + 240 \, a x - 240 \, \log \left (a x + 1\right )\right )} \sqrt{-a^{2} c}}{60 \, a^{5}} \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^{3} \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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