\[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx \]
Optimal antiderivative \[ \frac {a \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} \sqrt {-a c x +c}}{4 x^{3} \sqrt {1-\frac {1}{a x}}}+\frac {11 a^{2} \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} \sqrt {-a c x +c}}{24 x^{2} \sqrt {1-\frac {1}{a x}}}+\frac {21 a^{3} \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} \sqrt {-a c x +c}}{32 x \sqrt {1-\frac {1}{a x}}}+\frac {107 a^{3} \sqrt {1+\frac {1}{a x}}\, \sqrt {-a c x +c}}{64 x \sqrt {1-\frac {1}{a x}}}+\frac {363 a^{\frac {7}{2}} \arcsinh \! \left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right ) \sqrt {\frac {1}{x}}\, \sqrt {-a c x +c}}{64 \sqrt {1-\frac {1}{a x}}}-\frac {4 a^{\frac {7}{2}} \arctanh \! \left (\frac {\sqrt {2}\, \sqrt {\frac {1}{x}}}{\sqrt {a}\, \sqrt {1+\frac {1}{a x}}}\right ) \sqrt {2}\, \sqrt {\frac {1}{x}}\, \sqrt {-a c x +c}}{\sqrt {1-\frac {1}{a x}}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(1/2)/x^5,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
\[ -\frac {\frac {768 \, \sqrt {2} a^{5} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{\mathrm {sgn}\left (-a c x - c\right )} - \frac {1089 \, a^{5} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x - c}}{\sqrt {c}}\right )}{\mathrm {sgn}\left (-a c x - c\right )} + \frac {768 i \, \sqrt {2} a^{5} \sqrt {-c} \arctan \left (-i\right ) - 1089 i \, a^{5} \sqrt {-c} \arctan \left (-i \, \sqrt {2}\right ) - 845 \, \sqrt {2} a^{5} \sqrt {-c}}{\mathrm {sgn}\left (c\right )} - \frac {447 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} a^{5} c - 1127 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} a^{5} c^{2} - 1049 \, {\left (-a c x - c\right )}^{\frac {3}{2}} a^{5} c^{3} - 321 \, \sqrt {-a c x - c} a^{5} c^{4}}{a^{4} c^{4} x^{4} \mathrm {sgn}\left (-a c x - c\right )}}{192 \, a} \]