\[ \int \frac {x^6}{a+b x^4+c x^8} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \! \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} x}{\left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}} 2^{\frac {1}{4}}}{4 c^{\frac {3}{4}} \sqrt {-4 a c +b^{2}}}+\frac {\arctanh \! \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} x}{\left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}} 2^{\frac {1}{4}}}{4 c^{\frac {3}{4}} \sqrt {-4 a c +b^{2}}}+\frac {\arctan \! \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} x}{\left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}} 2^{\frac {1}{4}}}{4 c^{\frac {3}{4}} \sqrt {-4 a c +b^{2}}}-\frac {\arctanh \! \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} x}{\left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}} 2^{\frac {1}{4}}}{4 c^{\frac {3}{4}} \sqrt {-4 a c +b^{2}}} \]
command
integrate(x**6/(c*x**8+b*x**4+a),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \operatorname {RootSum} {\left (t^{8} \left (16777216 a^{4} c^{7} - 16777216 a^{3} b^{2} c^{6} + 6291456 a^{2} b^{4} c^{5} - 1048576 a b^{6} c^{4} + 65536 b^{8} c^{3}\right ) + t^{4} \left (- 12288 a^{3} b c^{3} + 10240 a^{2} b^{3} c^{2} - 2816 a b^{5} c + 256 b^{7}\right ) + a^{3}, \left ( t \mapsto t \log {\left (x + \frac {2097152 t^{7} a^{4} c^{7} - 2621440 t^{7} a^{3} b^{2} c^{6} + 1179648 t^{7} a^{2} b^{4} c^{5} - 229376 t^{7} a b^{6} c^{4} + 16384 t^{7} b^{8} c^{3} - 1280 t^{3} a^{3} b c^{3} + 1600 t^{3} a^{2} b^{3} c^{2} - 576 t^{3} a b^{5} c + 64 t^{3} b^{7}}{a^{3} c - a^{2} b^{2}} \right )} \right )\right )} \]