\[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx \]
Optimal antiderivative \[ -\frac {c^{2} \left (-3 b e +5 c d \right ) x}{e^{6}}+\frac {c^{3} x^{2}}{2 e^{5}}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}{4 e^{7} \left (e x +d \right )^{4}}+\frac {\left (-b e +2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}{e^{7} \left (e x +d \right )^{3}}-\frac {3 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (5 c^{2} d^{2}+b^{2} e^{2}-c e \left (-a e +5 b d \right )\right )}{2 e^{7} \left (e x +d \right )^{2}}+\frac {\left (-b e +2 c d \right ) \left (10 c^{2} d^{2}+b^{2} e^{2}-2 c e \left (-3 a e +5 b d \right )\right )}{e^{7} \left (e x +d \right )}+\frac {3 c \left (5 c^{2} d^{2}+b^{2} e^{2}-c e \left (-a e +5 b d \right )\right ) \ln \! \left (e x +d \right )}{e^{7}} \]
command
integrate((c*x**2+b*x+a)**3/(e*x+d)**5,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {c^{3} x^{2}}{2 e^{5}} + \frac {3 c \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} + x \left (\frac {3 b c^{2}}{e^{5}} - \frac {5 c^{3} d}{e^{6}}\right ) + \frac {- a^{3} e^{6} - a^{2} b d e^{5} - a^{2} c d^{2} e^{4} - a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 25 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 25 b^{2} c d^{4} e^{2} - 77 b c^{2} d^{5} e + 57 c^{3} d^{6} + x^{3} \left (- 24 a b c e^{6} + 48 a c^{2} d e^{5} - 4 b^{3} e^{6} + 48 b^{2} c d e^{5} - 120 b c^{2} d^{2} e^{4} + 80 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 6 a^{2} c e^{6} - 6 a b^{2} e^{6} - 36 a b c d e^{5} + 108 a c^{2} d^{2} e^{4} - 6 b^{3} d e^{5} + 108 b^{2} c d^{2} e^{4} - 300 b c^{2} d^{3} e^{3} + 210 c^{3} d^{4} e^{2}\right ) + x \left (- 4 a^{2} b e^{6} - 4 a^{2} c d e^{5} - 4 a b^{2} d e^{5} - 24 a b c d^{2} e^{4} + 88 a c^{2} d^{3} e^{3} - 4 b^{3} d^{2} e^{4} + 88 b^{2} c d^{3} e^{3} - 260 b c^{2} d^{4} e^{2} + 188 c^{3} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} \]