\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx \]
Optimal antiderivative \[ \frac {\left (-a \,e^{2}+c \,d^{2}\right )^{3}}{7 e^{4} \left (e x +d \right )^{7}}-\frac {c d \left (-a \,e^{2}+c \,d^{2}\right )^{2}}{2 e^{4} \left (e x +d \right )^{6}}+\frac {3 c^{2} d^{2} \left (-a \,e^{2}+c \,d^{2}\right )}{5 e^{4} \left (e x +d \right )^{5}}-\frac {c^{3} d^{3}}{4 e^{4} \left (e x +d \right )^{4}} \]
command
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**11,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {- 20 a^{3} e^{6} - 10 a^{2} c d^{2} e^{4} - 4 a c^{2} d^{4} e^{2} - c^{3} d^{6} - 35 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (- 84 a c^{2} d^{2} e^{4} - 21 c^{3} d^{4} e^{2}\right ) + x \left (- 70 a^{2} c d e^{5} - 28 a c^{2} d^{3} e^{3} - 7 c^{3} d^{5} e\right )}{140 d^{7} e^{4} + 980 d^{6} e^{5} x + 2940 d^{5} e^{6} x^{2} + 4900 d^{4} e^{7} x^{3} + 4900 d^{3} e^{8} x^{4} + 2940 d^{2} e^{9} x^{5} + 980 d e^{10} x^{6} + 140 e^{11} x^{7}} \]