\[ \int \frac {(d+e x)^{10}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx \]
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, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {20 \, {\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{7} d^{7}} - \frac {c^{6} d^{12} + 3 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 110 \, a^{3} c^{3} d^{6} e^{6} + 195 \, a^{4} c^{2} d^{4} e^{8} - 141 \, a^{5} c d^{2} e^{10} + 37 \, a^{6} e^{12} + 45 \, {\left (c^{6} d^{10} e^{2} - 4 \, a c^{5} d^{8} e^{4} + 6 \, a^{2} c^{4} d^{6} e^{6} - 4 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 9 \, {\left (c^{6} d^{11} e + 5 \, a c^{5} d^{9} e^{3} - 30 \, a^{2} c^{4} d^{7} e^{5} + 50 \, a^{3} c^{3} d^{5} e^{7} - 35 \, a^{4} c^{2} d^{3} e^{9} + 9 \, a^{5} c d e^{11}\right )} x}{3 \, {\left (c d x + a e\right )}^{3} c^{7} d^{7}} + \frac {c^{8} d^{8} x^{3} e^{6} + 9 \, c^{8} d^{9} x^{2} e^{5} + 45 \, c^{8} d^{10} x e^{4} - 6 \, a c^{7} d^{7} x^{2} e^{7} - 72 \, a c^{7} d^{8} x e^{6} + 30 \, a^{2} c^{6} d^{6} x e^{8}}{3 \, c^{12} d^{12}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________