\[ \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (-a x +1\right )^{2}}{a^{2} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {5}{2}} x}+\frac {2 \left (-a x +1\right )^{3}}{5 a^{3} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {5}{2}} x^{2}}-\frac {2 \left (-a x +1\right )^{3} \left (a x +1\right )}{15 a^{4} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {5}{2}} x^{3}}+\frac {2 \left (-a x +1\right )^{3} \left (a x +1\right )^{2} \left (13 a x +28\right )}{15 a^{6} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {5}{2}} x^{5}}+\frac {2 \left (-a x +1\right )^{\frac {5}{2}} \left (a x +1\right )^{\frac {5}{2}} \arcsin \! \left (a x \right )}{a^{6} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {5}{2}} x^{5}} \]
command
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^(5/2),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
\[ -\frac {4 \, \arctan \left (\frac {\sqrt {c - \frac {2 \, c}{a x + 1}}}{\sqrt {-c}}\right )}{a \sqrt {-c} c^{2} \mathrm {sgn}\left (-\frac {1}{a x + 1} + 1\right )} + \frac {8 \, c - \frac {17 \, c}{a x + 1}}{4 \, {\left ({\left (c - \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} - \sqrt {c - \frac {2 \, c}{a x + 1}} c\right )} a c^{2} \mathrm {sgn}\left (-\frac {1}{a x + 1} + 1\right )} - \frac {3 \, a^{4} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{\frac {5}{2}} c^{20} + 35 \, a^{4} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} c^{21} + 345 \, a^{4} \sqrt {c - \frac {2 \, c}{a x + 1}} c^{22}}{120 \, a^{5} c^{25} \mathrm {sgn}\left (-\frac {1}{a x + 1} + 1\right )} \]