\[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx \]
Optimal antiderivative \[ \frac {2 \sqrt {1+\frac {1}{a x}}\, \sqrt {-a c x +c}}{\sqrt {1-\frac {1}{a x}}}+\frac {2 \arcsinh \! \left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right ) \sqrt {\frac {1}{x}}\, \sqrt {-a c x +c}}{\sqrt {a}\, \sqrt {1-\frac {1}{a x}}}-\frac {4 \arctanh \! \left (\frac {\sqrt {2}\, \sqrt {\frac {1}{x}}}{\sqrt {a}\, \sqrt {1+\frac {1}{a x}}}\right ) \sqrt {2}\, \sqrt {\frac {1}{x}}\, \sqrt {-a c x +c}}{\sqrt {a}\, \sqrt {1-\frac {1}{a x}}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(1/2)/x,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
\[ -c {\left (\frac {4 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{\sqrt {c} \mathrm {sgn}\left (-a c x - c\right )} - \frac {2 \, \arctan \left (\frac {\sqrt {-a c x - c}}{\sqrt {c}}\right )}{\sqrt {c} \mathrm {sgn}\left (-a c x - c\right )} + \frac {-4 i \, \sqrt {2} \arctan \left (-i\right ) + 2 \, \sqrt {2} + 2 i \, \arctan \left (-i \, \sqrt {2}\right )}{\sqrt {-c} \mathrm {sgn}\left (c\right )} - \frac {2 \, \sqrt {-a c x - c}}{c \mathrm {sgn}\left (-a c x - c\right )}\right )} \]