\[ \int \frac {x^2}{\left (a+b x^2+c x^4\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {x \left (2 c \,x^{2}+b \right )}{2 \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {\arctan \! \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {c}\, \left (2 b -\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}}{2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \sqrt {b -\sqrt {-4 a c +b^{2}}}}-\frac {\arctan \! \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {c}\, \left (2 b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}}{2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \sqrt {b +\sqrt {-4 a c +b^{2}}}} \]
command
integrate(x**2/(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
\[ \frac {b x + 2 c x^{3}}{8 a^{2} c - 2 a b^{2} + x^{4} \left (8 a c^{2} - 2 b^{2} c\right ) + x^{2} \left (8 a b c - 2 b^{3}\right )} + \operatorname {RootSum} {\left (t^{4} \left (1048576 a^{7} c^{6} - 1572864 a^{6} b^{2} c^{5} + 983040 a^{5} b^{4} c^{4} - 327680 a^{4} b^{6} c^{3} + 61440 a^{3} b^{8} c^{2} - 6144 a^{2} b^{10} c + 256 a b^{12}\right ) + t^{2} \left (- 12288 a^{4} b c^{4} + 8192 a^{3} b^{3} c^{3} - 1536 a^{2} b^{5} c^{2} + 16 b^{9}\right ) + 16 a^{2} c^{3} + 24 a b^{2} c^{2} + 9 b^{4} c, \left ( t \mapsto t \log {\left (x + \frac {16384 t^{3} a^{5} c^{4} - 8192 t^{3} a^{4} b^{2} c^{3} + 512 t^{3} a^{2} b^{6} c - 64 t^{3} a b^{8} - 128 t a^{2} b c^{2} - 16 t a b^{3} c - 4 t b^{5}}{4 a c^{2} + 3 b^{2} c} \right )} \right )\right )} \]