\[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx \]
Optimal antiderivative \[ \frac {e^{4} x}{c^{4} d^{4}}-\frac {\left (-a \,e^{2}+c \,d^{2}\right )^{4}}{3 c^{5} d^{5} \left (c d x +a e \right )^{3}}-\frac {2 e \left (-a \,e^{2}+c \,d^{2}\right )^{3}}{c^{5} d^{5} \left (c d x +a e \right )^{2}}-\frac {6 e^{2} \left (-a \,e^{2}+c \,d^{2}\right )^{2}}{c^{5} d^{5} \left (c d x +a e \right )}+\frac {4 e^{3} \left (-a \,e^{2}+c \,d^{2}\right ) \ln \left (c d x +a e \right )}{c^{5} d^{5}} \]
command
integrate((e*x+d)^8/(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
\[ \frac {x e^{4}}{c^{4} d^{4}} + \frac {4 \, {\left (c d^{2} e^{3} - a e^{5}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{5}} - \frac {c^{4} d^{8} + 2 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 22 \, a^{3} c d^{2} e^{6} + 13 \, a^{4} e^{8} + 18 \, {\left (c^{4} d^{6} e^{2} - 2 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 6 \, {\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3} - 9 \, a^{2} c^{2} d^{3} e^{5} + 5 \, a^{3} c d e^{7}\right )} x}{3 \, {\left (c d x + a e\right )}^{3} c^{5} d^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________