\[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^4} \, dx \]
Optimal antiderivative \[ -\frac {\left (2 c x +b \right ) \left (e x +d \right )^{4}}{3 \left (-4 a c +b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{3}}+\frac {\left (e x +d \right )^{3} \left (5 b c d -2 b^{2} e -2 a c e +5 c \left (-b e +2 c d \right ) x \right )}{3 \left (-4 a c +b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{2}}-\frac {2 \left (5 c^{2} d^{2}+b^{2} e^{2}-c e \left (-a e +5 b d \right )\right ) \left (e x +d \right ) \left (b d -2 a e +\left (-b e +2 c d \right ) x \right )}{\left (-4 a c +b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )}+\frac {8 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (5 c^{2} d^{2}+b^{2} e^{2}-c e \left (-a e +5 b d \right )\right ) \arctanh \! \left (\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}\right )}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}} \]
command
integrate((e*x+d)**4/(c*x**2+b*x+a)**4,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {output too large to display} \]