\[ \int \frac {1}{\left (a+b x^2+c x^4\right )^2} \, dx \]
Optimal antiderivative \[ \frac {x \left (b c \,x^{2}-2 a c +b^{2}\right )}{2 a \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {\arctan \! \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {c}\, \left (b^{2}-12 a c +b \sqrt {-4 a c +b^{2}}\right ) \sqrt {2}}{4 a \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \sqrt {b -\sqrt {-4 a c +b^{2}}}}-\frac {\arctan \! \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {c}\, \left (b^{2}-12 a c -b \sqrt {-4 a c +b^{2}}\right ) \sqrt {2}}{4 a \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \sqrt {b +\sqrt {-4 a c +b^{2}}}} \]
command
integrate(1/(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
\[ \frac {- b c x^{3} + x \left (2 a c - b^{2}\right )}{8 a^{3} c - 2 a^{2} b^{2} + x^{4} \left (8 a^{2} c^{2} - 2 a b^{2} c\right ) + x^{2} \left (8 a^{2} b c - 2 a b^{3}\right )} + \operatorname {RootSum} {\left (t^{4} \left (1048576 a^{9} c^{6} - 1572864 a^{8} b^{2} c^{5} + 983040 a^{7} b^{4} c^{4} - 327680 a^{6} b^{6} c^{3} + 61440 a^{5} b^{8} c^{2} - 6144 a^{4} b^{10} c + 256 a^{3} b^{12}\right ) + t^{2} \left (- 61440 a^{5} b c^{5} + 61440 a^{4} b^{3} c^{4} - 24064 a^{3} b^{5} c^{3} + 4608 a^{2} b^{7} c^{2} - 432 a b^{9} c + 16 b^{11}\right ) + 1296 a^{2} c^{5} - 360 a b^{2} c^{4} + 25 b^{4} c^{3}, \left ( t \mapsto t \log {\left (x + \frac {32768 t^{3} a^{7} b c^{4} - 28672 t^{3} a^{6} b^{3} c^{3} + 9216 t^{3} a^{5} b^{5} c^{2} - 1280 t^{3} a^{4} b^{7} c + 64 t^{3} a^{3} b^{9} + 1728 t a^{4} c^{4} - 2304 t a^{3} b^{2} c^{3} + 740 t a^{2} b^{4} c^{2} - 92 t a b^{6} c + 4 t b^{8}}{324 a^{2} c^{4} - 81 a b^{2} c^{3} + 5 b^{4} c^{2}} \right )} \right )\right )} \]