\[ \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {2 e^{3} \left (-b e +2 c d \right ) x}{c \left (-4 a c +b^{2}\right )}-\frac {\left (e x +d \right )^{4}}{2 \left (c \,x^{2}+b x +a \right )^{2}}-\frac {2 e \left (e x +d \right )^{2} \left (b d -2 a e +\left (-b e +2 c d \right ) x \right )}{\left (-4 a c +b^{2}\right ) \left (c \,x^{2}+b x +a \right )}+\frac {2 e \left (-b e +2 c d \right ) \left (2 c^{2} d^{2}-b^{2} e^{2}-2 c e \left (-3 a e +b d \right )\right ) \arctanh \! \left (\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}\right )}{c^{2} \left (-4 a c +b^{2}\right )^{\frac {3}{2}}}+\frac {e^{4} \ln \! \left (c \,x^{2}+b x +a \right )}{c^{2}} \]
command
integrate((2*c*x+b)*(e*x+d)**4/(c*x**2+b*x+a)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {output too large to display} \]