\[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^3} \, dx \]
Optimal antiderivative \[ \frac {x^{3} \left (b \,x^{2}+2 a \right )}{4 \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 x \left (4 a b +\left (4 a c +b^{2}\right ) x^{2}\right )}{8 \left (-4 a c +b^{2}\right )^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {3 \arctan \! \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \left (b^{2}+4 a c -\frac {b \left (12 a c +b^{2}\right )}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{16 \left (-4 a c +b^{2}\right )^{2} \sqrt {c}\, \sqrt {b -\sqrt {-4 a c +b^{2}}}}+\frac {3 \arctan \! \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}}\right ) \left (b^{2}+4 a c +\frac {b \left (12 a c +b^{2}\right )}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{16 \left (-4 a c +b^{2}\right )^{2} \sqrt {c}\, \sqrt {b +\sqrt {-4 a c +b^{2}}}} \]
command
integrate(x**6/(c*x**4+b*x**2+a)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {12 a^{2} b x + x^{7} \left (12 a c^{2} + 3 b^{2} c\right ) + x^{5} \left (16 a b c + 5 b^{3}\right ) + x^{3} \left (- 4 a^{2} c + 19 a b^{2}\right )}{128 a^{4} c^{2} - 64 a^{3} b^{2} c + 8 a^{2} b^{4} + x^{8} \left (128 a^{2} c^{4} - 64 a b^{2} c^{3} + 8 b^{4} c^{2}\right ) + x^{6} \left (256 a^{2} b c^{3} - 128 a b^{3} c^{2} + 16 b^{5} c\right ) + x^{4} \left (256 a^{3} c^{3} - 48 a b^{4} c + 8 b^{6}\right ) + x^{2} \left (256 a^{3} b c^{2} - 128 a^{2} b^{3} c + 16 a b^{5}\right )} + \operatorname {RootSum} {\left (t^{4} \left (68719476736 a^{10} c^{11} - 171798691840 a^{9} b^{2} c^{10} + 193273528320 a^{8} b^{4} c^{9} - 128849018880 a^{7} b^{6} c^{8} + 56371445760 a^{6} b^{8} c^{7} - 16911433728 a^{5} b^{10} c^{6} + 3523215360 a^{4} b^{12} c^{5} - 503316480 a^{3} b^{14} c^{4} + 47185920 a^{2} b^{16} c^{3} - 2621440 a b^{18} c^{2} + 65536 b^{20} c\right ) + t^{2} \left (- 188743680 a^{7} b c^{7} + 141557760 a^{6} b^{3} c^{6} - 2359296 a^{5} b^{5} c^{5} - 26542080 a^{4} b^{7} c^{4} + 9584640 a^{3} b^{9} c^{3} - 1290240 a^{2} b^{11} c^{2} + 46080 a b^{13} c + 2304 b^{15}\right ) + 20736 a^{5} c^{4} + 103680 a^{4} b^{2} c^{3} + 142560 a^{3} b^{4} c^{2} + 32400 a^{2} b^{6} c + 2025 a b^{8}, \left ( t \mapsto t \log {\left (x + \frac {33554432 t^{3} a^{6} c^{7} - 16777216 t^{3} a^{5} b^{2} c^{6} - 10485760 t^{3} a^{4} b^{4} c^{5} + 10485760 t^{3} a^{3} b^{6} c^{4} - 3276800 t^{3} a^{2} b^{8} c^{3} + 458752 t^{3} a b^{10} c^{2} - 24576 t^{3} b^{12} c - 64512 t a^{3} b c^{3} - 43776 t a^{2} b^{3} c^{2} - 21312 t a b^{5} c - 144 t b^{7}}{432 a^{2} c^{2} + 1080 a b^{2} c + 135 b^{4}} \right )} \right )\right )} \]