Optimal. Leaf size=211 \[ \frac{4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{a^2 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{a^{5/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x^2 \left (\frac{1}{a x}+1\right )^{5/2} \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{2 x \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.236304, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6176, 6181, 96, 94, 93, 206} \[ \frac{4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{a^2 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{a^{5/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x^2 \left (\frac{1}{a x}+1\right )^{5/2} \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{2 x \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 96
Rule 94
Rule 93
Rule 206
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} x \sqrt{c-a c x} \, dx &=\frac{\sqrt{c-a c x} \int e^{3 \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{\left (1+\frac{x}{a}\right )^{3/2}}{x^{7/2} \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \left (1+\frac{1}{a x}\right )^{5/2} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x^{5/2} \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{a \sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{5/2} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{\left (2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^{3/2} \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{4 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{a^2 \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{5/2} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{\left (4 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a^3 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{4 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{a^2 \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{5/2} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{\left (8 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 x^2}{a}} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{a x}}}\right )}{a^3 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{4 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{a^2 \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{5/2} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{a^{5/2} \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0740351, size = 114, normalized size = 0.54 \[ \frac{2 \sqrt{c-a c x} \left (\sqrt{a} \sqrt{\frac{1}{a x}+1} \left (3 a^2 x^2+11 a x+38\right )-30 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{15 a^{5/2} \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.178, size = 125, normalized size = 0.6 \begin{align*} -{\frac{2\,ax-2}{ \left ( 15\,ax+15 \right ){a}^{2}}\sqrt{-c \left ( ax-1 \right ) } \left ( -3\,{x}^{2}{a}^{2}\sqrt{-c \left ( ax+1 \right ) }+30\,\sqrt{c}\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) -11\,xa\sqrt{-c \left ( ax+1 \right ) }-38\,\sqrt{-c \left ( ax+1 \right ) } \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}} \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{\sqrt{-a c x + c} x}{\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 = 1.63742, size = 655, normalized size = 3.1 \begin{align*} \left [\frac{2 \,{\left (15 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) +{\left (3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 49 \, a x + 38\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{15 \,{\left (a^{3} x - a^{2}\right )}}, -\frac{2 \,{\left (30 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) -{\left (3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 49 \, a x + 38\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{15 \,{\left (a^{3} x - a^{2}\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 [C] time = 1.2015, size = 180, normalized size = 0.85 \begin{align*} -\frac{60 i \, \sqrt{2} \sqrt{-c} \arctan \left (-i\right ) - 104 \, \sqrt{2} \sqrt{-c}}{15 \, a^{2} \mathrm{sgn}\left (c\right )} - \frac{2 \,{\left (30 \, \sqrt{2} c^{\frac{5}{2}} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right ) - 3 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} + 5 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c - 30 \, \sqrt{-a c x - c} c^{2}\right )}}{15 \, a^{2} c^{2} \mathrm{sgn}\left (-a c x - c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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