\[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx \]
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)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {8 c \sqrt {- a c x + c}}{5 a \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}} + \frac {8 \left (- a c x + c\right )^{\frac {3}{2}}}{5 a \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}} - \frac {2 \left (- a c x + c\right )^{\frac {5}{2}}}{5 a c \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________