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