\[ \int \frac {(f+g x)^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {\left (d g +e f \right )^{2}}{64 d^{5} e^{3} \left (-e x +d \right )^{2}}+\frac {\left (d g +e f \right ) \left (d g +5 e f \right )}{64 d^{6} e^{3} \left (-e x +d \right )}-\frac {\left (-d g +e f \right )^{2}}{32 d^{3} e^{3} \left (e x +d \right )^{4}}-\frac {\left (-d g +e f \right ) \left (d g +3 e f \right )}{48 d^{4} e^{3} \left (e x +d \right )^{3}}+\frac {d^{2} g^{2}-3 e^{2} f^{2}}{32 d^{5} e^{3} \left (e x +d \right )^{2}}+\frac {d^{2} g^{2}-2 d e f g -5 e^{2} f^{2}}{32 d^{6} e^{3} \left (e x +d \right )}+\frac {\left (-d^{2} g^{2}+10 d e f g +15 e^{2} f^{2}\right ) \arctanh \left (\frac {e x}{d}\right )}{64 d^{7} e^{3}} \]
command
integrate((g*x+f)^2/(e*x+d)^2/(-e^2*x^2+d^2)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (d^{2} g^{2} - 10 \, d f g e - 15 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | -\frac {2 \, d}{x e + d} + 1 \right |}\right )}{128 \, d^{7}} - \frac {{\left (3 \, d^{2} g^{2} + 14 \, d f g e + 11 \, f^{2} e^{2} - \frac {8 \, {\left (d^{3} g^{2} e + 4 \, d^{2} f g e^{2} + 3 \, d f^{2} e^{3}\right )} e^{\left (-1\right )}}{x e + d}\right )} e^{\left (-3\right )}}{256 \, d^{7} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{2}} + \frac {{\left (\frac {3 \, d^{8} g^{2} e^{9}}{x e + d} + \frac {3 \, d^{9} g^{2} e^{9}}{{\left (x e + d\right )}^{2}} + \frac {2 \, d^{10} g^{2} e^{9}}{{\left (x e + d\right )}^{3}} - \frac {3 \, d^{11} g^{2} e^{9}}{{\left (x e + d\right )}^{4}} - \frac {6 \, d^{7} f g e^{10}}{x e + d} + \frac {4 \, d^{9} f g e^{10}}{{\left (x e + d\right )}^{3}} + \frac {6 \, d^{10} f g e^{10}}{{\left (x e + d\right )}^{4}} - \frac {15 \, d^{6} f^{2} e^{11}}{x e + d} - \frac {9 \, d^{7} f^{2} e^{11}}{{\left (x e + d\right )}^{2}} - \frac {6 \, d^{8} f^{2} e^{11}}{{\left (x e + d\right )}^{3}} - \frac {3 \, d^{9} f^{2} e^{11}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-12\right )}}{96 \, d^{12}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________