\[ \int \frac {d+e x^2}{a+b x^2+c x^4} \, dx \]
Optimal antiderivative \[ \frac {\arctan \! \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \left (e +\frac {-b e +2 c d}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{2 \sqrt {c}\, \sqrt {b -\sqrt {-4 a c +b^{2}}}}+\frac {\arctan \! \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}}\right ) \left (e +\frac {b e -2 c d}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{2 \sqrt {c}\, \sqrt {b +\sqrt {-4 a c +b^{2}}}} \]
command
integrate((e*x**2+d)/(c*x**4+b*x**2+a),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \operatorname {RootSum} {\left (t^{4} \left (256 a^{3} c^{3} - 128 a^{2} b^{2} c^{2} + 16 a b^{4} c\right ) + t^{2} \left (- 16 a^{2} b c e^{2} + 64 a^{2} c^{2} d e + 4 a b^{3} e^{2} - 16 a b^{2} c d e - 16 a b c^{2} d^{2} + 4 b^{3} c d^{2}\right ) + a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}, \left ( t \mapsto t \log {\left (x + \frac {64 t^{3} a^{3} c^{2} e - 16 t^{3} a^{2} b^{2} c e - 32 t^{3} a^{2} b c^{2} d + 8 t^{3} a b^{3} c d - 2 t a^{2} b e^{3} + 12 t a^{2} c d e^{2} - 6 t a b c d^{2} e - 4 t a c^{2} d^{3} + 2 t b^{2} c d^{3}}{a^{2} e^{4} - a b d e^{3} + b c d^{3} e - c^{2} d^{4}} \right )} \right )\right )} \]