\[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx \]
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)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ x \left (- \frac {4 a e^{6}}{c^{5} d^{5}} + \frac {5 e^{4}}{c^{4} d^{3}}\right ) + \frac {47 a^{5} e^{10} - 130 a^{4} c d^{2} e^{8} + 110 a^{3} c^{2} d^{4} e^{6} - 20 a^{2} c^{3} d^{6} e^{4} - 5 a c^{4} d^{8} e^{2} - 2 c^{5} d^{10} + x^{2} \left (60 a^{3} c^{2} d^{2} e^{8} - 180 a^{2} c^{3} d^{4} e^{6} + 180 a c^{4} d^{6} e^{4} - 60 c^{5} d^{8} e^{2}\right ) + x \left (105 a^{4} c d e^{9} - 300 a^{3} c^{2} d^{3} e^{7} + 270 a^{2} c^{3} d^{5} e^{5} - 60 a c^{4} d^{7} e^{3} - 15 c^{5} d^{9} e\right )}{6 a^{3} c^{6} d^{6} e^{3} + 18 a^{2} c^{7} d^{7} e^{2} x + 18 a c^{8} d^{8} e x^{2} + 6 c^{9} d^{9} x^{3}} + \frac {e^{5} x^{2}}{2 c^{4} d^{4}} + \frac {10 e^{3} \left (a e^{2} - c d^{2}\right )^{2} \log {\left (a e + c d x \right )}}{c^{6} d^{6}} \]