\[ \int \frac {1}{a+b x^4+c x^8} \, dx \]
Optimal antiderivative \[ \frac {c^{\frac {3}{4}} \arctan \! \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} x}{\left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{2 \left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}} \sqrt {-4 a c +b^{2}}}+\frac {c^{\frac {3}{4}} \arctanh \! \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} x}{\left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{2 \left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}} \sqrt {-4 a c +b^{2}}}-\frac {c^{\frac {3}{4}} \arctan \! \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} x}{\left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{2 \sqrt {-4 a c +b^{2}}\, \left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}}}-\frac {c^{\frac {3}{4}} \arctanh \! \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} x}{\left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{2 \sqrt {-4 a c +b^{2}}\, \left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}}} \]
command
integrate(1/(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
\[ \operatorname {RootSum} {\left (t^{8} \left (16777216 a^{7} c^{4} - 16777216 a^{6} b^{2} c^{3} + 6291456 a^{5} b^{4} c^{2} - 1048576 a^{4} b^{6} c + 65536 a^{3} b^{8}\right ) + t^{4} \left (- 12288 a^{3} b c^{3} + 10240 a^{2} b^{3} c^{2} - 2816 a b^{5} c + 256 b^{7}\right ) + c^{3}, \left ( t \mapsto t \log {\left (x + \frac {16384 t^{5} a^{5} b c^{2} - 8192 t^{5} a^{4} b^{3} c + 1024 t^{5} a^{3} b^{5} + 8 t a^{2} c^{2} - 16 t a b^{2} c + 4 t b^{4}}{a c^{2} - b^{2} c} \right )} \right )\right )} \]