\[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^8} \, dx \]
Optimal antiderivative \[ -\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}{7 e^{5} \left (e x +d \right )^{7}}+\frac {\left (-b e +2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )}{3 e^{5} \left (e x +d \right )^{6}}+\frac {-6 c^{2} d^{2}-b^{2} e^{2}+2 c e \left (-a e +3 b d \right )}{5 e^{5} \left (e x +d \right )^{5}}+\frac {c \left (-b e +2 c d \right )}{2 e^{5} \left (e x +d \right )^{4}}-\frac {c^{2}}{3 e^{5} \left (e x +d \right )^{3}} \]
command
integrate((c*x**2+b*x+a)**2/(e*x+d)**8,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {- 30 a^{2} e^{4} - 10 a b d e^{3} - 4 a c d^{2} e^{2} - 2 b^{2} d^{2} e^{2} - 3 b c d^{3} e - 2 c^{2} d^{4} - 70 c^{2} e^{4} x^{4} + x^{3} \left (- 105 b c e^{4} - 70 c^{2} d e^{3}\right ) + x^{2} \left (- 84 a c e^{4} - 42 b^{2} e^{4} - 63 b c d e^{3} - 42 c^{2} d^{2} e^{2}\right ) + x \left (- 70 a b e^{4} - 28 a c d e^{3} - 14 b^{2} d e^{3} - 21 b c d^{2} e^{2} - 14 c^{2} d^{3} e\right )}{210 d^{7} e^{5} + 1470 d^{6} e^{6} x + 4410 d^{5} e^{7} x^{2} + 7350 d^{4} e^{8} x^{3} + 7350 d^{3} e^{9} x^{4} + 4410 d^{2} e^{10} x^{5} + 1470 d e^{11} x^{6} + 210 e^{12} x^{7}} \]