\[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a \,e^{2}+c \,d^{2}\right )^{3}}{3 c^{4} d^{4} \left (c d x +a e \right )^{3}}-\frac {3 e \left (-a \,e^{2}+c \,d^{2}\right )^{2}}{2 c^{4} d^{4} \left (c d x +a e \right )^{2}}-\frac {3 e^{2} \left (-a \,e^{2}+c \,d^{2}\right )}{c^{4} d^{4} \left (c d x +a e \right )}+\frac {e^{3} \ln \left (c d x +a e \right )}{c^{4} d^{4}} \]
command
integrate((e*x+d)^7/(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
\[ \frac {e^{3} \log \left ({\left | c d x + a e \right |}\right )}{c^{4} d^{4}} - \frac {18 \, {\left (c^{2} d^{3} e^{2} - a c d e^{4}\right )} x^{2} + 9 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 3 \, a^{2} e^{5}\right )} x + \frac {2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6}}{c d}}{6 \, {\left (c d x + a e\right )}^{3} c^{3} d^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________