Optimal. Leaf size=104 \[ -\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}}{5 a c}-\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}}{5 a}-\frac{8 c x \sqrt{1-\frac{1}{a^2 x^2}}}{5 a \sqrt{c-a c x}} \]
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Rubi [A] time = 0.193505, antiderivative size = 137, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {6176, 6181, 78, 45, 37} \[ \frac{12 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{5 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{2 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{6 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{5 a \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Rule 6176
Rule 6181
Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} x \sqrt{c-a c x} \, dx &=\frac{\sqrt{c-a c x} \int e^{-\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} x^{3/2} \, dx}{\sqrt{1-\frac{1}{a x}} \sqrt{x}}\\ &=-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{x}{a}}{x^{7/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{\left (9 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{5/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{5 a \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{6 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{5 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{\left (6 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{5 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{12 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{5 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{6 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{5 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0307322, size = 58, normalized size = 0.56 \[ \frac{2 \sqrt{\frac{1}{a x}+1} \left (a^2 x^2-3 a x+6\right ) \sqrt{c-a c x}}{5 a^2 \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.04, size = 55, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ({a}^{2}{x}^{2}-3\,ax+6 \right ) }{5\,{a}^{2} \left ( ax-1 \right ) }\sqrt{-acx+c}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12581, size = 93, normalized size = 0.89 \begin{align*} \frac{2 \,{\left (a^{3} \sqrt{-c} x^{3} - 2 \, a^{2} \sqrt{-c} x^{2} + 3 \, a \sqrt{-c} x + 6 \, \sqrt{-c}\right )}{\left (a x - 1\right )}}{5 \,{\left (a^{3} x - a^{2}\right )} \sqrt{a x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60257, size = 130, normalized size = 1.25 \begin{align*} \frac{2 \,{\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + 3 \, a x + 6\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{5 \,{\left (a^{3} x - a^{2}\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 [A] time = 1.16256, size = 120, normalized size = 1.15 \begin{align*} \frac{2 \,{\left (\frac{4 \, \sqrt{2} \sqrt{-c} c^{2}}{a} - \frac{{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} + 5 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c + 10 \, \sqrt{-a c x - c} c^{2}}{a}\right )}{\left | c \right |} \mathrm{sgn}\left (a x + 1\right )}{5 \, a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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