\[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx \]
Optimal antiderivative \[ \frac {e^{4} \left (-4 a \,e^{2}+5 c \,d^{2}\right ) x}{c^{5} d^{5}}+\frac {e^{5} x^{2}}{2 c^{4} d^{4}}-\frac {\left (-a \,e^{2}+c \,d^{2}\right )^{5}}{3 c^{6} d^{6} \left (c d x +a e \right )^{3}}-\frac {5 e \left (-a \,e^{2}+c \,d^{2}\right )^{4}}{2 c^{6} d^{6} \left (c d x +a e \right )^{2}}-\frac {10 e^{2} \left (-a \,e^{2}+c \,d^{2}\right )^{3}}{c^{6} d^{6} \left (c d x +a e \right )}+\frac {10 e^{3} \left (-a \,e^{2}+c \,d^{2}\right )^{2} \ln \left (c d x +a e \right )}{c^{6} d^{6}} \]
command
integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {10 \, {\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{6} d^{6}} - \frac {2 \, c^{5} d^{10} + 5 \, a c^{4} d^{8} e^{2} + 20 \, a^{2} c^{3} d^{6} e^{4} - 110 \, a^{3} c^{2} d^{4} e^{6} + 130 \, a^{4} c d^{2} e^{8} - 47 \, a^{5} e^{10} + 60 \, {\left (c^{5} d^{8} e^{2} - 3 \, a c^{4} d^{6} e^{4} + 3 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 15 \, {\left (c^{5} d^{9} e + 4 \, a c^{4} d^{7} e^{3} - 18 \, a^{2} c^{3} d^{5} e^{5} + 20 \, a^{3} c^{2} d^{3} e^{7} - 7 \, a^{4} c d e^{9}\right )} x}{6 \, {\left (c d x + a e\right )}^{3} c^{6} d^{6}} + \frac {c^{4} d^{4} x^{2} e^{5} + 10 \, c^{4} d^{5} x e^{4} - 8 \, a c^{3} d^{3} x e^{6}}{2 \, c^{8} d^{8}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________