\[ \int \frac {1}{(d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx \]
Optimal antiderivative \[ -\frac {1}{2 a e \left (e x +d \right )^{2}}-\frac {b \ln \! \left (e x +d \right )}{a^{2} e}+\frac {b \ln \! \left (a +b \left (e x +d \right )^{2}+c \left (e x +d \right )^{4}\right )}{4 a^{2} e}-\frac {\left (-2 a c +b^{2}\right ) \arctanh \! \left (\frac {b +2 c \left (e x +d \right )^{2}}{\sqrt {-4 a c +b^{2}}}\right )}{2 a^{2} e \sqrt {-4 a c +b^{2}}} \]
command
integrate(1/(e*x+d)**3/(a+b*(e*x+d)**2+c*(e*x+d)**4),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \left (\frac {b}{4 a^{2} e} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e \left (4 a c - b^{2}\right )}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a^{3} c e \left (\frac {b}{4 a^{2} e} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} e \left (\frac {b}{4 a^{2} e} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e \left (4 a c - b^{2}\right )}\right ) + 3 a b c + 2 a c^{2} d^{2} - b^{3} - b^{2} c d^{2}}{2 a c^{2} e^{2} - b^{2} c e^{2}} \right )} + \left (\frac {b}{4 a^{2} e} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e \left (4 a c - b^{2}\right )}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a^{3} c e \left (\frac {b}{4 a^{2} e} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} e \left (\frac {b}{4 a^{2} e} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e \left (4 a c - b^{2}\right )}\right ) + 3 a b c + 2 a c^{2} d^{2} - b^{3} - b^{2} c d^{2}}{2 a c^{2} e^{2} - b^{2} c e^{2}} \right )} - \frac {1}{2 a d^{2} e + 4 a d e^{2} x + 2 a e^{3} x^{2}} - \frac {b \log {\left (\frac {d}{e} + x \right )}}{a^{2} e} \]