\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx \]
Optimal antiderivative \[ \frac {2 \,{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (a x +1\right ) \left (-a c x +c \right )^{\frac {n}{2}}}{a \left (2+n \right )} \]
command
integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^(1/2*n),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {2 \, {\left (a \left (-c\right )^{\frac {1}{2} \, n} x + \left (-c\right )^{\frac {1}{2} \, n}\right )} {\left (a x + 1\right )}^{\frac {1}{2} \, n}}{a {\left (n + 2\right )}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int {\left (-a c x + c\right )}^{\frac {1}{2} \, n} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]________________________________________________________________________________________