\[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (a x +1\right ) \left (-a c x +c \right )^{\frac {3}{2}}}{5 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a} \]
command
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(3/2),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
\[ -\frac {2 \, {\left (\frac {4 \, \sqrt {2} \sqrt {-c} c}{\mathrm {sgn}\left (c\right )} + \frac {{\left (a c x + c\right )}^{2} \sqrt {-a c x - c}}{c \mathrm {sgn}\left (-a c x - c\right )}\right )}}{5 \, a} \]