\[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (d g +e f \right )^{2}}{64 d^{6} e^{3} \left (-e x +d \right )}-\frac {\left (-d g +e f \right )^{2}}{20 d^{2} e^{3} \left (e x +d \right )^{5}}+\frac {d^{2} g^{2}-e^{2} f^{2}}{16 d^{3} e^{3} \left (e x +d \right )^{4}}-\frac {\left (-d g +3 e f \right ) \left (d g +e f \right )}{48 d^{4} e^{3} \left (e x +d \right )^{3}}-\frac {f \left (d g +e f \right )}{16 d^{5} e^{2} \left (e x +d \right )^{2}}-\frac {\left (d g +e f \right ) \left (d g +5 e f \right )}{64 d^{6} e^{3} \left (e x +d \right )}+\frac {\left (d g +e f \right ) \left (d g +3 e f \right ) \arctanh \left (\frac {e x}{d}\right )}{32 d^{7} e^{3}} \]
command
integrate((g*x+f)^2/(e*x+d)^4/(-e^2*x^2+d^2)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (d^{2} g^{2} + 4 \, d f g e + 3 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right )}{64 \, d^{7}} - \frac {{\left (d^{2} g^{2} + 4 \, d f g e + 3 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e - d \right |}\right )}{64 \, d^{7}} - \frac {{\left (16 \, d^{8} g^{2} - 32 \, d^{7} f g e - 144 \, d^{6} f^{2} e^{2} + 15 \, {\left (d^{3} g^{2} e^{5} + 4 \, d^{2} f g e^{6} + 3 \, d f^{2} e^{7}\right )} x^{5} + 60 \, {\left (d^{4} g^{2} e^{4} + 4 \, d^{3} f g e^{5} + 3 \, d^{2} f^{2} e^{6}\right )} x^{4} + 80 \, {\left (d^{5} g^{2} e^{3} + 4 \, d^{4} f g e^{4} + 3 \, d^{3} f^{2} e^{5}\right )} x^{3} + 20 \, {\left (d^{6} g^{2} e^{2} + 4 \, d^{5} f g e^{3} + 3 \, d^{4} f^{2} e^{4}\right )} x^{2} + {\left (49 \, d^{7} g^{2} e - 188 \, d^{6} f g e^{2} - 141 \, d^{5} f^{2} e^{3}\right )} x\right )} e^{\left (-3\right )}}{480 \, {\left (x e + d\right )}^{5} {\left (x e - d\right )} d^{7}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________