\[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx \]
Optimal antiderivative \[ \frac {8 a^{2} c^{3} \left (1-\frac {1}{a^{2} x^{2}}\right )^{\frac {3}{2}} x^{3}}{15 \left (-a c x +c \right )^{\frac {3}{2}}}+\frac {2 a^{2} c^{2} \left (1-\frac {1}{a^{2} x^{2}}\right )^{\frac {3}{2}} x^{3}}{5 \sqrt {-a c x +c}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(-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
\[ \frac {2 \, {\left (\frac {8 \, \sqrt {2} \sqrt {-c} c}{\mathrm {sgn}\left (c\right )} - \frac {3 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} + 10 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c}{c \mathrm {sgn}\left (-a c x - c\right )}\right )}}{15 \, a} \]