\[ \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx \]
Optimal antiderivative \[ -\frac {11 a^{3} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x^{4}}{30 \left (-a x +1\right )^{3}}+\frac {57 a^{6} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x^{7}}{16 \left (-a x +1\right )^{3} \left (a x +1\right )^{3}}-\frac {41 a^{5} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x^{6}}{24 \left (-a x +1\right )^{3} \left (a x +1\right )^{2}}-\frac {57 a^{4} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x^{5}}{80 \left (-a x +1\right )^{3} \left (a x +1\right )}+\frac {13 a^{2} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x^{3} \left (a x +1\right )}{40 \left (-a x +1\right )^{3}}-\frac {a \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x^{2} \left (a x +1\right )}{15 \left (-a x +1\right )^{2}}-\frac {\left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x \left (a x +1\right )}{6 \left (-a x +1\right )}-\frac {2 a^{6} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x^{7} \arcsin \left (a x \right )}{\left (-a x +1\right )^{\frac {7}{2}} \left (a x +1\right )^{\frac {7}{2}}}-\frac {25 a^{6} \left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {7}{2}} x^{7} \arctanh \left (\sqrt {-a x +1}\, \sqrt {a x +1}\right )}{16 \left (-a x +1\right )^{\frac {7}{2}} \left (a x +1\right )^{\frac {7}{2}}} \]
command
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{120} \, {\left (\frac {375 \, c^{\frac {7}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {240 \, c^{\frac {7}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a {\left | a \right |}} - \frac {120 \, \sqrt {a^{2} c x^{2} - c} c^{3} \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {105 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{11} c^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 1440 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{10} a c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{9} c^{5} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 4320 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{8} a c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 150 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} c^{6} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 7360 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a c^{\frac {13}{2}} \mathrm {sgn}\left (x\right ) + 150 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} c^{7} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 6720 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {15}{2}} \mathrm {sgn}\left (x\right ) - 595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{8} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 2976 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {17}{2}} \mathrm {sgn}\left (x\right ) - 105 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{9} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 736 \, a c^{\frac {19}{2}} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{6} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________