\[ \int \frac {(f+g x)^2}{(d+e x) \left (d^2-e^2 x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {\left (d g +e f \right )^{2}}{32 d^{4} e^{3} \left (-e x +d \right )^{2}}+\frac {f \left (d g +e f \right )}{8 d^{5} e^{2} \left (-e x +d \right )}-\frac {\left (-d g +e f \right )^{2}}{24 d^{3} e^{3} \left (e x +d \right )^{3}}-\frac {\left (-d g +e f \right ) \left (d g +3 e f \right )}{32 d^{4} e^{3} \left (e x +d \right )^{2}}+\frac {d^{2} g^{2}-3 e^{2} f^{2}}{16 d^{5} e^{3} \left (e x +d \right )}+\frac {\left (-d^{2} g^{2}+2 d e f g +5 e^{2} f^{2}\right ) \arctanh \left (\frac {e x}{d}\right )}{16 d^{6} e^{3}} \]
command
integrate((g*x+f)^2/(e*x+d)/(-e^2*x^2+d^2)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (d^{2} g^{2} - 2 \, d f g e - 5 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right )}{32 \, d^{6}} + \frac {{\left (d^{2} g^{2} - 2 \, d f g e - 5 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e - d \right |}\right )}{32 \, d^{6}} + \frac {{\left (4 \, d^{7} g^{2} + 16 \, d^{6} f g e - 8 \, d^{5} f^{2} e^{2} + 3 \, {\left (d^{3} g^{2} e^{4} - 2 \, d^{2} f g e^{5} - 5 \, d f^{2} e^{6}\right )} x^{4} + 3 \, {\left (d^{4} g^{2} e^{3} - 2 \, d^{3} f g e^{4} - 5 \, d^{2} f^{2} e^{5}\right )} x^{3} - 5 \, {\left (d^{5} g^{2} e^{2} - 2 \, d^{4} f g e^{3} - 5 \, d^{3} f^{2} e^{4}\right )} x^{2} + {\left (7 \, d^{6} g^{2} e + 10 \, d^{5} f g e^{2} + 25 \, d^{4} f^{2} e^{3}\right )} x\right )} e^{\left (-3\right )}}{48 \, {\left (x e + d\right )}^{3} {\left (x e - d\right )}^{2} d^{6}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________