\[ \int \frac {x \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx \]
Optimal antiderivative \[ \frac {2 a c e -b \left (a f +c d \right )-\left (-2 a c f +b^{2} f -b c e +2 c^{2} d \right ) x^{2}}{2 c \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {\left (2 a f -b e +2 c d \right ) \arctanh \! \left (\frac {2 c \,x^{2}+b}{\sqrt {-4 a c +b^{2}}}\right )}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}} \]
command
integrate(x*(f*x**4+e*x**2+d)/(c*x**4+b*x**2+a)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ - \frac {\sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) \log {\left (x^{2} + \frac {- 16 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) + 8 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) + 2 a b f - b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) - b^{2} e + 2 b c d}{4 a c f - 2 b c e + 4 c^{2} d} \right )}}{2} + \frac {\sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) \log {\left (x^{2} + \frac {16 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) - 8 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) + 2 a b f + b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) - b^{2} e + 2 b c d}{4 a c f - 2 b c e + 4 c^{2} d} \right )}}{2} + \frac {a b f - 2 a c e + b c d + x^{2} \left (- 2 a c f + b^{2} f - b c e + 2 c^{2} d\right )}{8 a^{2} c^{2} - 2 a b^{2} c + x^{4} \left (8 a c^{3} - 2 b^{2} c^{2}\right ) + x^{2} \left (8 a b c^{2} - 2 b^{3} c\right )} \]