\[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx \]
Optimal antiderivative \[ -\frac {\left (-d g +e f \right )^{2}}{8 d \,e^{3} \left (e x +d \right )^{4}}-\frac {\left (-d g +e f \right ) \left (3 d g +e f \right )}{12 d^{2} e^{3} \left (e x +d \right )^{3}}-\frac {\left (d g +e f \right )^{2}}{16 d^{3} e^{3} \left (e x +d \right )^{2}}-\frac {\left (d g +e f \right )^{2}}{16 d^{4} e^{3} \left (e x +d \right )}+\frac {\left (d g +e f \right )^{2} \arctanh \left (\frac {e x}{d}\right )}{16 d^{5} e^{3}} \]
command
integrate((g*x+f)^2/(e*x+d)^4/(-e^2*x^2+d^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (d^{2} g^{2} + 2 \, d f g e + f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right )}{32 \, d^{5}} - \frac {{\left (d^{2} g^{2} + 2 \, d f g e + f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e - d \right |}\right )}{32 \, d^{5}} - \frac {{\left (8 \, d^{5} f g e + 16 \, d^{4} f^{2} e^{2} + 3 \, {\left (d^{3} g^{2} e^{3} + 2 \, d^{2} f g e^{4} + d f^{2} e^{5}\right )} x^{3} + 12 \, {\left (d^{4} g^{2} e^{2} + 2 \, d^{3} f g e^{3} + d^{2} f^{2} e^{4}\right )} x^{2} + {\left (3 \, d^{5} g^{2} e + 38 \, d^{4} f g e^{2} + 19 \, d^{3} f^{2} e^{3}\right )} x\right )} e^{\left (-3\right )}}{48 \, {\left (x e + d\right )}^{4} d^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________