\[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {c^{3} d^{3}}{2 \left (-a \,e^{2}+c \,d^{2}\right )^{4} \left (c d x +a e \right )^{2}}+\frac {4 c^{3} d^{3} e}{\left (-a \,e^{2}+c \,d^{2}\right )^{5} \left (c d x +a e \right )}+\frac {e^{2}}{3 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (e x +d \right )^{3}}+\frac {3 c d \,e^{2}}{2 \left (-a \,e^{2}+c \,d^{2}\right )^{4} \left (e x +d \right )^{2}}+\frac {6 c^{2} d^{2} e^{2}}{\left (-a \,e^{2}+c \,d^{2}\right )^{5} \left (e x +d \right )}+\frac {10 c^{3} d^{3} e^{2} \ln \left (c d x +a e \right )}{\left (-a \,e^{2}+c \,d^{2}\right )^{6}}-\frac {10 c^{3} d^{3} e^{2} \ln \left (e x +d \right )}{\left (-a \,e^{2}+c \,d^{2}\right )^{6}} \]
command
integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {10 \, c^{4} d^{4} e^{2} \log \left ({\left | c d x + a e \right |}\right )}{c^{7} d^{13} - 6 \, a c^{6} d^{11} e^{2} + 15 \, a^{2} c^{5} d^{9} e^{4} - 20 \, a^{3} c^{4} d^{7} e^{6} + 15 \, a^{4} c^{3} d^{5} e^{8} - 6 \, a^{5} c^{2} d^{3} e^{10} + a^{6} c d e^{12}} - \frac {10 \, c^{3} d^{3} e^{3} \log \left ({\left | x e + d \right |}\right )}{c^{6} d^{12} e - 6 \, a c^{5} d^{10} e^{3} + 15 \, a^{2} c^{4} d^{8} e^{5} - 20 \, a^{3} c^{3} d^{6} e^{7} + 15 \, a^{4} c^{2} d^{4} e^{9} - 6 \, a^{5} c d^{2} e^{11} + a^{6} e^{13}} - \frac {3 \, c^{5} d^{10} - 30 \, a c^{4} d^{8} e^{2} - 20 \, a^{2} c^{3} d^{6} e^{4} + 60 \, a^{3} c^{2} d^{4} e^{6} - 15 \, a^{4} c d^{2} e^{8} + 2 \, a^{5} e^{10} - 60 \, {\left (c^{5} d^{6} e^{4} - a c^{4} d^{4} e^{6}\right )} x^{4} - 30 \, {\left (5 \, c^{5} d^{7} e^{3} - 2 \, a c^{4} d^{5} e^{5} - 3 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{3} - 10 \, {\left (11 \, c^{5} d^{8} e^{2} + 12 \, a c^{4} d^{6} e^{4} - 21 \, a^{2} c^{3} d^{4} e^{6} - 2 \, a^{3} c^{2} d^{2} e^{8}\right )} x^{2} - 5 \, {\left (3 \, c^{5} d^{9} e + 32 \, a c^{4} d^{7} e^{3} - 24 \, a^{2} c^{3} d^{5} e^{5} - 12 \, a^{3} c^{2} d^{3} e^{7} + a^{4} c d e^{9}\right )} x}{6 \, {\left (c d^{2} - a e^{2}\right )}^{6} {\left (c d x + a e\right )}^{2} {\left (x e + d\right )}^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________