\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a e +b d \right )^{4}}{10 e^{5} \left (e x +d \right )^{10}}+\frac {4 b \left (-a e +b d \right )^{3}}{9 e^{5} \left (e x +d \right )^{9}}-\frac {3 b^{2} \left (-a e +b d \right )^{2}}{4 e^{5} \left (e x +d \right )^{8}}+\frac {4 b^{3} \left (-a e +b d \right )}{7 e^{5} \left (e x +d \right )^{7}}-\frac {b^{4}}{6 e^{5} \left (e x +d \right )^{6}} \]
command
integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**11,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {- 126 a^{4} e^{4} - 56 a^{3} b d e^{3} - 21 a^{2} b^{2} d^{2} e^{2} - 6 a b^{3} d^{3} e - b^{4} d^{4} - 210 b^{4} e^{4} x^{4} + x^{3} \left (- 720 a b^{3} e^{4} - 120 b^{4} d e^{3}\right ) + x^{2} \left (- 945 a^{2} b^{2} e^{4} - 270 a b^{3} d e^{3} - 45 b^{4} d^{2} e^{2}\right ) + x \left (- 560 a^{3} b e^{4} - 210 a^{2} b^{2} d e^{3} - 60 a b^{3} d^{2} e^{2} - 10 b^{4} d^{3} e\right )}{1260 d^{10} e^{5} + 12600 d^{9} e^{6} x + 56700 d^{8} e^{7} x^{2} + 151200 d^{7} e^{8} x^{3} + 264600 d^{6} e^{9} x^{4} + 317520 d^{5} e^{10} x^{5} + 264600 d^{4} e^{11} x^{6} + 151200 d^{3} e^{12} x^{7} + 56700 d^{2} e^{13} x^{8} + 12600 d e^{14} x^{9} + 1260 e^{15} x^{10}} \]