\[ \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {6}{5 a \,c^{2} \left (c -\frac {c}{a x}\right )^{\frac {5}{2}}}+\frac {11}{6 a \,c^{3} \left (c -\frac {c}{a x}\right )^{\frac {3}{2}}}-\frac {x}{c^{2} \left (c -\frac {c}{a x}\right )^{\frac {5}{2}}}-\frac {5 \arctanh \! \left (\frac {\sqrt {c -\frac {c}{a x}}}{\sqrt {c}}\right )}{a \,c^{\frac {9}{2}}}-\frac {\arctanh \! \left (\frac {\sqrt {c -\frac {c}{a x}}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}}{8 a \,c^{\frac {9}{2}}}+\frac {21}{4 a \,c^{4} \sqrt {c -\frac {c}{a x}}} \]
command
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(9/2),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
\[ \frac {1}{120} \, a c {\left (\frac {2 \, {\left (12 \, c^{2} + \frac {50 \, {\left (a c x - c\right )} c}{a x} + \frac {255 \, {\left (a c x - c\right )}^{2}}{a^{2} x^{2}}\right )} x^{2}}{{\left (a c x - c\right )}^{2} c^{5} \sqrt {\frac {a c x - c}{a x}}} + \frac {15 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a c x - c}{a x}}}{2 \, \sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{5}} + \frac {600 \, \arctan \left (\frac {\sqrt {\frac {a c x - c}{a x}}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{5}} - \frac {120 \, \sqrt {\frac {a c x - c}{a x}}}{a^{2} {\left (c - \frac {a c x - c}{a x}\right )} c^{5}}\right )} \]