Optimal. Leaf size=163 \[ -\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{a^{3/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{a \sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.187324, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6176, 6181, 94, 93, 206} \[ -\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{a^{3/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{a \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 94
Rule 93
Rule 206
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a 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}} \sqrt{x} \, 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^{5/2} \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 )^{3/2} x \sqrt{c-a c x}}{3 \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 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{4 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{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 \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^2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{4 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{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 \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^2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{4 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{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 \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^{3/2} \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0599873, size = 105, normalized size = 0.64 \[ \frac{2 \sqrt{c-a c x} \left (\sqrt{a} \sqrt{\frac{1}{a x}+1} (a x+7)-6 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{3 a^{3/2} \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.194, size = 107, normalized size = 0.7 \begin{align*} -{\frac{2\,ax-2}{ \left ( 3\,ax+3 \right ) a}\sqrt{-c \left ( ax-1 \right ) } \left ( 6\,\sqrt{c}\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) -xa\sqrt{-c \left ( ax+1 \right ) }-7\,\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}}{\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 = 1.59122, size = 598, normalized size = 3.67 \begin{align*} \left [\frac{2 \,{\left (3 \, \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 (a^{2} x^{2} + 8 \, a x + 7\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{3 \,{\left (a^{2} x - a\right )}}, -\frac{2 \,{\left (6 \, \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 (a^{2} x^{2} + 8 \, a x + 7\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{3 \,{\left (a^{2} x - a\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 = 1.17755, size = 144, normalized size = 0.88 \begin{align*} -\frac{12 i \, \sqrt{2} \sqrt{-c} \arctan \left (-i\right ) - 16 \, \sqrt{2} \sqrt{-c}}{3 \, a \mathrm{sgn}\left (c\right )} - \frac{2 \,{\left (6 \, \sqrt{2} c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right ) +{\left (-a c x - c\right )}^{\frac{3}{2}} - 6 \, \sqrt{-a c x - c} c\right )}}{3 \, a c \mathrm{sgn}\left (-a c x - c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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