\[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {c^{2} d^{2}}{\left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (c d x +a e \right )}-\frac {e}{2 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{2}}-\frac {2 c d e}{\left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (e x +d \right )}-\frac {3 c^{2} d^{2} e \ln \left (c d x +a e \right )}{\left (-a \,e^{2}+c \,d^{2}\right )^{4}}+\frac {3 c^{2} d^{2} e \ln \left (e x +d \right )}{\left (-a \,e^{2}+c \,d^{2}\right )^{4}} \]
command
integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {3 \, c^{3} d^{3} e \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{9} - 4 \, a c^{4} d^{7} e^{2} + 6 \, a^{2} c^{3} d^{5} e^{4} - 4 \, a^{3} c^{2} d^{3} e^{6} + a^{4} c d e^{8}} + \frac {3 \, c^{2} d^{2} e^{2} \log \left ({\left | x e + d \right |}\right )}{c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}} - \frac {2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (3 \, c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x}{2 \, {\left (c d^{2} - a e^{2}\right )}^{4} {\left (c d x + a e\right )} {\left (x e + d\right )}^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________