\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a e +b d \right )^{4}}{8 e^{5} \left (e x +d \right )^{8}}+\frac {4 b \left (-a e +b d \right )^{3}}{7 e^{5} \left (e x +d \right )^{7}}-\frac {b^{2} \left (-a e +b d \right )^{2}}{e^{5} \left (e x +d \right )^{6}}+\frac {4 b^{3} \left (-a e +b d \right )}{5 e^{5} \left (e x +d \right )^{5}}-\frac {b^{4}}{4 e^{5} \left (e x +d \right )^{4}} \]
command
integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**9,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {- 35 a^{4} e^{4} - 20 a^{3} b d e^{3} - 10 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e - b^{4} d^{4} - 70 b^{4} e^{4} x^{4} + x^{3} \left (- 224 a b^{3} e^{4} - 56 b^{4} d e^{3}\right ) + x^{2} \left (- 280 a^{2} b^{2} e^{4} - 112 a b^{3} d e^{3} - 28 b^{4} d^{2} e^{2}\right ) + x \left (- 160 a^{3} b e^{4} - 80 a^{2} b^{2} d e^{3} - 32 a b^{3} d^{2} e^{2} - 8 b^{4} d^{3} e\right )}{280 d^{8} e^{5} + 2240 d^{7} e^{6} x + 7840 d^{6} e^{7} x^{2} + 15680 d^{5} e^{8} x^{3} + 19600 d^{4} e^{9} x^{4} + 15680 d^{3} e^{10} x^{5} + 7840 d^{2} e^{11} x^{6} + 2240 d e^{12} x^{7} + 280 e^{13} x^{8}} \]