\[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx \]
Optimal antiderivative \[ \frac {2 \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} x \sqrt {-a c x +c}}{3 \sqrt {1-\frac {1}{a x}}}+\frac {4 \sqrt {1+\frac {1}{a x}}\, \sqrt {-a c x +c}}{a \sqrt {1-\frac {1}{a x}}}-\frac {4 \arctanh \! \left (\frac {\sqrt {2}\, \sqrt {\frac {1}{x}}}{\sqrt {a}\, \sqrt {1+\frac {1}{a x}}}\right ) \sqrt {2}\, \sqrt {\frac {1}{x}}\, \sqrt {-a c x +c}}{a^{\frac {3}{2}} \sqrt {1-\frac {1}{a x}}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {\frac {12 i \, \sqrt {2} \sqrt {-c} \arctan \left (-i\right ) - 16 \, \sqrt {2} \sqrt {-c}}{\mathrm {sgn}\left (c\right )} + \frac {2 \, {\left (6 \, \sqrt {2} c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right ) + {\left (-a c x - c\right )}^{\frac {3}{2}} - 6 \, \sqrt {-a c x - c} c\right )}}{c \mathrm {sgn}\left (-a c x - c\right )}}{3 \, a} \]