\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{10}} \, dx \]
Optimal antiderivative \[ \frac {\left (-a \,e^{2}+c \,d^{2}\right )^{3}}{6 e^{4} \left (e x +d \right )^{6}}-\frac {3 c d \left (-a \,e^{2}+c \,d^{2}\right )^{2}}{5 e^{4} \left (e x +d \right )^{5}}+\frac {3 c^{2} d^{2} \left (-a \,e^{2}+c \,d^{2}\right )}{4 e^{4} \left (e x +d \right )^{4}}-\frac {c^{3} d^{3}}{3 e^{4} \left (e x +d \right )^{3}} \]
command
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**10,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {- 10 a^{3} e^{6} - 6 a^{2} c d^{2} e^{4} - 3 a c^{2} d^{4} e^{2} - c^{3} d^{6} - 20 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (- 45 a c^{2} d^{2} e^{4} - 15 c^{3} d^{4} e^{2}\right ) + x \left (- 36 a^{2} c d e^{5} - 18 a c^{2} d^{3} e^{3} - 6 c^{3} d^{5} e\right )}{60 d^{6} e^{4} + 360 d^{5} e^{5} x + 900 d^{4} e^{6} x^{2} + 1200 d^{3} e^{7} x^{3} + 900 d^{2} e^{8} x^{4} + 360 d e^{9} x^{5} + 60 e^{10} x^{6}} \]