\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (n^{2}+14 n +56\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{1+\frac {n}{2}} \left (-a c x +c \right )^{2+\frac {n}{2}}}{a \left (4+n \right ) \left (6+n \right )}+\frac {2 \left (n^{2}+14 n +56\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{1+\frac {n}{2}} \left (-a c x +c \right )^{2+\frac {n}{2}}}{a^{2} \left (6+n \right ) \left (n^{2}+6 n +8\right ) x}+\frac {\left (8+n \right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{1+\frac {n}{2}} x \left (-a c x +c \right )^{2+\frac {n}{2}}}{6+n}-\frac {\left (a -\frac {1}{x}\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{1+\frac {n}{2}} x \left (-a c x +c \right )^{2+\frac {n}{2}}}{a} \]
command
integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^(2+1/2*n),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (n^{2} + 6 \, n + 8\right )} a^{3} \left (-c\right )^{\frac {1}{2} \, n} c^{2} x^{3} - {\left (n^{2} + 14 \, n + 24\right )} a^{2} \left (-c\right )^{\frac {1}{2} \, n} c^{2} x^{2} - {\left (n^{2} + 6 \, n - 24\right )} a \left (-c\right )^{\frac {1}{2} \, n} c^{2} x + {\left (n^{2} + 14 \, n + 56\right )} \left (-c\right )^{\frac {1}{2} \, n} c^{2}\right )} {\left (a x + 1\right )}^{\frac {1}{2} \, n}}{{\left (n^{3} + 12 \, n^{2} + 44 \, n + 48\right )} a} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int {\left (-a c x + c\right )}^{\frac {1}{2} \, n + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]________________________________________________________________________________________