\[ \int \frac {(d+e x)^{10}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx \]
Optimal antiderivative \[ \frac {e^{4} \left (10 a^{2} e^{4}-24 a c \,d^{2} e^{2}+15 c^{2} d^{4}\right ) x}{c^{6} d^{6}}+\frac {e^{5} \left (-2 a \,e^{2}+3 c \,d^{2}\right ) x^{2}}{c^{5} d^{5}}+\frac {e^{6} x^{3}}{3 c^{4} d^{4}}-\frac {\left (-a \,e^{2}+c \,d^{2}\right )^{6}}{3 c^{7} d^{7} \left (c d x +a e \right )^{3}}-\frac {3 e \left (-a \,e^{2}+c \,d^{2}\right )^{5}}{c^{7} d^{7} \left (c d x +a e \right )^{2}}-\frac {15 e^{2} \left (-a \,e^{2}+c \,d^{2}\right )^{4}}{c^{7} d^{7} \left (c d x +a e \right )}+\frac {20 e^{3} \left (-a \,e^{2}+c \,d^{2}\right )^{3} \ln \! \left (c d x +a e \right )}{c^{7} d^{7}} \]
command
integrate((e*x+d)**10/(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
\[ x^{2} \left (- \frac {2 a e^{7}}{c^{5} d^{5}} + \frac {3 e^{5}}{c^{4} d^{3}}\right ) + x \left (\frac {10 a^{2} e^{8}}{c^{6} d^{6}} - \frac {24 a e^{6}}{c^{5} d^{4}} + \frac {15 e^{4}}{c^{4} d^{2}}\right ) + \frac {- 37 a^{6} e^{12} + 141 a^{5} c d^{2} e^{10} - 195 a^{4} c^{2} d^{4} e^{8} + 110 a^{3} c^{3} d^{6} e^{6} - 15 a^{2} c^{4} d^{8} e^{4} - 3 a c^{5} d^{10} e^{2} - c^{6} d^{12} + x^{2} \left (- 45 a^{4} c^{2} d^{2} e^{10} + 180 a^{3} c^{3} d^{4} e^{8} - 270 a^{2} c^{4} d^{6} e^{6} + 180 a c^{5} d^{8} e^{4} - 45 c^{6} d^{10} e^{2}\right ) + x \left (- 81 a^{5} c d e^{11} + 315 a^{4} c^{2} d^{3} e^{9} - 450 a^{3} c^{3} d^{5} e^{7} + 270 a^{2} c^{4} d^{7} e^{5} - 45 a c^{5} d^{9} e^{3} - 9 c^{6} d^{11} e\right )}{3 a^{3} c^{7} d^{7} e^{3} + 9 a^{2} c^{8} d^{8} e^{2} x + 9 a c^{9} d^{9} e x^{2} + 3 c^{10} d^{10} x^{3}} + \frac {e^{6} x^{3}}{3 c^{4} d^{4}} - \frac {20 e^{3} \left (a e^{2} - c d^{2}\right )^{3} \log {\left (a e + c d x \right )}}{c^{7} d^{7}} \]