\[ \int \frac {(f+g x)^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (d g +e f \right )^{2}}{16 d^{4} e^{3} \left (-e x +d \right )}-\frac {\left (-d g +e f \right )^{2}}{12 d^{2} e^{3} \left (e x +d \right )^{3}}+\frac {d^{2} g^{2}-e^{2} f^{2}}{8 d^{3} e^{3} \left (e x +d \right )^{2}}-\frac {\left (-d g +3 e f \right ) \left (d g +e f \right )}{16 d^{4} e^{3} \left (e x +d \right )}+\frac {f \left (d g +e f \right ) \arctanh \left (\frac {e x}{d}\right )}{4 d^{5} e^{2}} \]
command
integrate((g*x+f)^2/(e*x+d)^2/(-e^2*x^2+d^2)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (d f g + f^{2} e\right )} e^{\left (-2\right )} \log \left ({\left | -\frac {2 \, d}{x e + d} + 1 \right |}\right )}{8 \, d^{5}} + \frac {{\left (d^{2} g^{2} + 2 \, d f g e + f^{2} e^{2}\right )} e^{\left (-3\right )}}{32 \, d^{5} {\left (\frac {2 \, d}{x e + d} - 1\right )}} + \frac {{\left (\frac {3 \, d^{4} g^{2} e^{3}}{x e + d} + \frac {6 \, d^{5} g^{2} e^{3}}{{\left (x e + d\right )}^{2}} - \frac {4 \, d^{6} g^{2} e^{3}}{{\left (x e + d\right )}^{3}} - \frac {6 \, d^{3} f g e^{4}}{x e + d} + \frac {8 \, d^{5} f g e^{4}}{{\left (x e + d\right )}^{3}} - \frac {9 \, d^{2} f^{2} e^{5}}{x e + d} - \frac {6 \, d^{3} f^{2} e^{5}}{{\left (x e + d\right )}^{2}} - \frac {4 \, d^{4} f^{2} e^{5}}{{\left (x e + d\right )}^{3}}\right )} e^{\left (-6\right )}}{48 \, d^{6}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________