\[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {c^{3} d^{3}}{\left (-a \,e^{2}+c \,d^{2}\right )^{4} \left (c d x +a e \right )}-\frac {e}{3 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{3}}-\frac {c d e}{\left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (e x +d \right )^{2}}-\frac {3 c^{2} d^{2} e}{\left (-a \,e^{2}+c \,d^{2}\right )^{4} \left (e x +d \right )}-\frac {4 c^{3} d^{3} e \ln \left (c d x +a e \right )}{\left (-a \,e^{2}+c \,d^{2}\right )^{5}}+\frac {4 c^{3} d^{3} e \ln \left (e x +d \right )}{\left (-a \,e^{2}+c \,d^{2}\right )^{5}} \]
command
integrate(1/(e*x+d)^2/(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
\[ -\frac {4 \, c^{3} d^{3} e^{2} \log \left ({\left | -c d + \frac {c d^{2}}{x e + d} - \frac {a e^{2}}{x e + d} \right |}\right )}{c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}} - \frac {c^{4} d^{4} e}{{\left (c d^{2} - a e^{2}\right )}^{5} {\left (c d - \frac {c d^{2}}{x e + d} + \frac {a e^{2}}{x e + d}\right )}} - \frac {\frac {9 \, c^{4} d^{6} e^{7}}{x e + d} + \frac {3 \, c^{4} d^{7} e^{7}}{{\left (x e + d\right )}^{2}} + \frac {c^{4} d^{8} e^{7}}{{\left (x e + d\right )}^{3}} - \frac {18 \, a c^{3} d^{4} e^{9}}{x e + d} - \frac {9 \, a c^{3} d^{5} e^{9}}{{\left (x e + d\right )}^{2}} - \frac {4 \, a c^{3} d^{6} e^{9}}{{\left (x e + d\right )}^{3}} + \frac {9 \, a^{2} c^{2} d^{2} e^{11}}{x e + d} + \frac {9 \, a^{2} c^{2} d^{3} e^{11}}{{\left (x e + d\right )}^{2}} + \frac {6 \, a^{2} c^{2} d^{4} e^{11}}{{\left (x e + d\right )}^{3}} - \frac {3 \, a^{3} c d e^{13}}{{\left (x e + d\right )}^{2}} - \frac {4 \, a^{3} c d^{2} e^{13}}{{\left (x e + d\right )}^{3}} + \frac {a^{4} e^{15}}{{\left (x e + d\right )}^{3}}}{3 \, {\left (c^{6} d^{12} e^{6} - 6 \, a c^{5} d^{10} e^{8} + 15 \, a^{2} c^{4} d^{8} e^{10} - 20 \, a^{3} c^{3} d^{6} e^{12} + 15 \, a^{4} c^{2} d^{4} e^{14} - 6 \, a^{5} c d^{2} e^{16} + a^{6} e^{18}\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________