\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx \]
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="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (a^{3} n^{2} + 6 \, a^{3} n + 8 \, a^{3}\right )} x^{3} - {\left (a^{2} n^{2} + 14 \, a^{2} n + 24 \, a^{2}\right )} x^{2} + n^{2} - {\left (a n^{2} + 6 \, a n - 24 \, a\right )} x + 14 \, n + 56\right )} {\left (-a c x + c\right )}^{\frac {1}{2} \, n + 2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a n^{3} + 12 \, a n^{2} + {\left (a^{3} n^{3} + 12 \, a^{3} n^{2} + 44 \, a^{3} n + 48 \, a^{3}\right )} x^{2} + 44 \, a n - 2 \, {\left (a^{2} n^{3} + 12 \, a^{2} n^{2} + 44 \, a^{2} n + 48 \, a^{2}\right )} x + 48 \, a} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (-a c x + c\right )}^{\frac {1}{2} \, n + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}, x\right ) \]