\[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^9} \, dx \]
Optimal antiderivative \[ -\frac {\left (a \,e^{2}+c \,d^{2}\right )^{3}}{8 e^{7} \left (e x +d \right )^{8}}+\frac {6 c d \left (a \,e^{2}+c \,d^{2}\right )^{2}}{7 e^{7} \left (e x +d \right )^{7}}-\frac {c \left (a \,e^{2}+c \,d^{2}\right ) \left (a \,e^{2}+5 c \,d^{2}\right )}{2 e^{7} \left (e x +d \right )^{6}}+\frac {4 c^{2} d \left (3 a \,e^{2}+5 c \,d^{2}\right )}{5 e^{7} \left (e x +d \right )^{5}}-\frac {3 c^{2} \left (a \,e^{2}+5 c \,d^{2}\right )}{4 e^{7} \left (e x +d \right )^{4}}+\frac {2 c^{3} d}{e^{7} \left (e x +d \right )^{3}}-\frac {c^{3}}{2 e^{7} \left (e x +d \right )^{2}} \]
command
integrate((c*x**2+a)**3/(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
\[ \frac {- 35 a^{3} e^{6} - 5 a^{2} c d^{2} e^{4} - 3 a c^{2} d^{4} e^{2} - 5 c^{3} d^{6} - 280 c^{3} d e^{5} x^{5} - 140 c^{3} e^{6} x^{6} + x^{4} \left (- 210 a c^{2} e^{6} - 350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 168 a c^{2} d e^{5} - 280 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 140 a^{2} c e^{6} - 84 a c^{2} d^{2} e^{4} - 140 c^{3} d^{4} e^{2}\right ) + x \left (- 40 a^{2} c d e^{5} - 24 a c^{2} d^{3} e^{3} - 40 c^{3} d^{5} e\right )}{280 d^{8} e^{7} + 2240 d^{7} e^{8} x + 7840 d^{6} e^{9} x^{2} + 15680 d^{5} e^{10} x^{3} + 19600 d^{4} e^{11} x^{4} + 15680 d^{3} e^{12} x^{5} + 7840 d^{2} e^{13} x^{6} + 2240 d e^{14} x^{7} + 280 e^{15} x^{8}} \]