\[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx \]
Optimal antiderivative \[ \frac {a \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} \sqrt {-a c x +c}}{3 x^{2} \sqrt {1-\frac {1}{a x}}}+\frac {3 a^{2} \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} \sqrt {-a c x +c}}{4 x \sqrt {1-\frac {1}{a x}}}+\frac {13 a^{2} \sqrt {1+\frac {1}{a x}}\, \sqrt {-a c x +c}}{8 x \sqrt {1-\frac {1}{a x}}}+\frac {45 a^{\frac {5}{2}} \arcsinh \! \left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right ) \sqrt {\frac {1}{x}}\, \sqrt {-a c x +c}}{8 \sqrt {1-\frac {1}{a x}}}-\frac {4 a^{\frac {5}{2}} \arctanh \! \left (\frac {\sqrt {2}\, \sqrt {\frac {1}{x}}}{\sqrt {a}\, \sqrt {1+\frac {1}{a x}}}\right ) \sqrt {2}\, \sqrt {\frac {1}{x}}\, \sqrt {-a c x +c}}{\sqrt {1-\frac {1}{a x}}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(1/2)/x^4,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
\[ -\frac {\frac {96 \, \sqrt {2} a^{4} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{\mathrm {sgn}\left (-a c x - c\right )} - \frac {135 \, a^{4} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x - c}}{\sqrt {c}}\right )}{\mathrm {sgn}\left (-a c x - c\right )} + \frac {96 i \, \sqrt {2} a^{4} \sqrt {-c} \arctan \left (-i\right ) - 135 i \, a^{4} \sqrt {-c} \arctan \left (-i \, \sqrt {2}\right ) - 91 \, \sqrt {2} a^{4} \sqrt {-c}}{\mathrm {sgn}\left (c\right )} - \frac {57 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} a^{4} c + 88 \, {\left (-a c x - c\right )}^{\frac {3}{2}} a^{4} c^{2} + 39 \, \sqrt {-a c x - c} a^{4} c^{3}}{a^{3} c^{3} x^{3} \mathrm {sgn}\left (-a c x - c\right )}}{24 \, a} \]