\[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx \]
Optimal antiderivative \[ -\frac {8 \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} \left (-a c x +c \right )^{\frac {7}{2}}}{21 a \left (1-\frac {1}{a x}\right )^{\frac {7}{2}}}-\frac {568 \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} \left (-a c x +c \right )^{\frac {7}{2}}}{315 a^{3} \left (1-\frac {1}{a x}\right )^{\frac {7}{2}} x^{2}}+\frac {48 \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} \left (-a c x +c \right )^{\frac {7}{2}}}{35 a^{2} \left (1-\frac {1}{a x}\right )^{\frac {7}{2}} x}+\frac {2 \left (a -\frac {1}{x}\right )^{3} \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} x \left (-a c x +c \right )^{\frac {7}{2}}}{9 a^{3} \left (1-\frac {1}{a x}\right )^{\frac {7}{2}}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(-a*c*x+c)^(7/2),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
\[ \frac {2 \, {\left (\frac {256 \, \sqrt {2} \sqrt {-c} c^{3}}{\mathrm {sgn}\left (c\right )} - \frac {35 \, {\left (a c x + c\right )}^{4} \sqrt {-a c x - c} - 270 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} c + 756 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} c^{2} + 840 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c^{3}}{c \mathrm {sgn}\left (-a c x - c\right )}\right )}}{315 \, a} \]