\[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^8} \, dx \]
Optimal antiderivative \[ -\frac {d^{3} \left (-b e +c d \right )^{3}}{7 e^{7} \left (e x +d \right )^{7}}+\frac {d^{2} \left (-b e +c d \right )^{2} \left (-b e +2 c d \right )}{2 e^{7} \left (e x +d \right )^{6}}-\frac {3 d \left (-b e +c d \right ) \left (b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{5 e^{7} \left (e x +d \right )^{5}}+\frac {\left (-b e +2 c d \right ) \left (b^{2} e^{2}-10 b c d e +10 c^{2} d^{2}\right )}{4 e^{7} \left (e x +d \right )^{4}}-\frac {c \left (b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{e^{7} \left (e x +d \right )^{3}}+\frac {3 c^{2} \left (-b e +2 c d \right )}{2 e^{7} \left (e x +d \right )^{2}}-\frac {c^{3}}{e^{7} \left (e x +d \right )} \]
command
integrate((c*x**2+b*x)**3/(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
\[ \frac {- b^{3} d^{3} e^{3} - 4 b^{2} c d^{4} e^{2} - 10 b c^{2} d^{5} e - 20 c^{3} d^{6} - 140 c^{3} e^{6} x^{6} + x^{5} \left (- 210 b c^{2} e^{6} - 420 c^{3} d e^{5}\right ) + x^{4} \left (- 140 b^{2} c e^{6} - 350 b c^{2} d e^{5} - 700 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 35 b^{3} e^{6} - 140 b^{2} c d e^{5} - 350 b c^{2} d^{2} e^{4} - 700 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 21 b^{3} d e^{5} - 84 b^{2} c d^{2} e^{4} - 210 b c^{2} d^{3} e^{3} - 420 c^{3} d^{4} e^{2}\right ) + x \left (- 7 b^{3} d^{2} e^{4} - 28 b^{2} c d^{3} e^{3} - 70 b c^{2} d^{4} e^{2} - 140 c^{3} d^{5} e\right )}{140 d^{7} e^{7} + 980 d^{6} e^{8} x + 2940 d^{5} e^{9} x^{2} + 4900 d^{4} e^{10} x^{3} + 4900 d^{3} e^{11} x^{4} + 2940 d^{2} e^{12} x^{5} + 980 d e^{13} x^{6} + 140 e^{14} x^{7}} \]