\[ \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \! \left (\frac {\sqrt {c -\frac {c}{a x}}}{\sqrt {c}}\right )}{a \,c^{\frac {3}{2}}}-\frac {\arctanh \! \left (\frac {\sqrt {c -\frac {c}{a x}}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}}{a \,c^{\frac {3}{2}}}-\frac {x \sqrt {c -\frac {c}{a x}}}{c^{2}} \]
command
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(3/2),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
\[ a c {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a c x - c}{a x}}}{2 \, \sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{2}} - \frac {\arctan \left (\frac {\sqrt {\frac {a c x - c}{a x}}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{2}} - \frac {\sqrt {\frac {a c x - c}{a x}}}{a^{2} {\left (c - \frac {a c x - c}{a x}\right )} c^{2}}\right )} \]