\[ \int \frac {1}{x^3 \left (a+b x^3+c x^6\right )} \, dx \]
Optimal antiderivative \[ -\frac {1}{2 a \,x^{2}}-\frac {c^{\frac {2}{3}} \ln \! \left (2^{\frac {1}{3}} c^{\frac {1}{3}} x +\left (b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}\right ) \left (1+\frac {b}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {2}{3}}}{6 a \left (b -\sqrt {-4 a c +b^{2}}\right )^{\frac {2}{3}}}+\frac {c^{\frac {2}{3}} \ln \! \left (2^{\frac {2}{3}} c^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} c^{\frac {1}{3}} x \left (b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}+\left (b -\sqrt {-4 a c +b^{2}}\right )^{\frac {2}{3}}\right ) \left (1+\frac {b}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {2}{3}}}{12 a \left (b -\sqrt {-4 a c +b^{2}}\right )^{\frac {2}{3}}}+\frac {c^{\frac {2}{3}} \arctan \! \left (\frac {\left (1-\frac {2 \,2^{\frac {1}{3}} c^{\frac {1}{3}} x}{\left (b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \left (1+\frac {b}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {2}{3}} \sqrt {3}}{6 a \left (b -\sqrt {-4 a c +b^{2}}\right )^{\frac {2}{3}}}-\frac {c^{\frac {2}{3}} \ln \! \left (2^{\frac {1}{3}} c^{\frac {1}{3}} x +\left (b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}\right ) \left (1-\frac {b}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {2}{3}}}{6 a \left (b +\sqrt {-4 a c +b^{2}}\right )^{\frac {2}{3}}}+\frac {c^{\frac {2}{3}} \ln \! \left (2^{\frac {2}{3}} c^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} c^{\frac {1}{3}} x \left (b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}+\left (b +\sqrt {-4 a c +b^{2}}\right )^{\frac {2}{3}}\right ) \left (1-\frac {b}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {2}{3}}}{12 a \left (b +\sqrt {-4 a c +b^{2}}\right )^{\frac {2}{3}}}+\frac {c^{\frac {2}{3}} \arctan \! \left (\frac {\left (1-\frac {2 \,2^{\frac {1}{3}} c^{\frac {1}{3}} x}{\left (b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \left (1-\frac {b}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {2}{3}} \sqrt {3}}{6 a \left (b +\sqrt {-4 a c +b^{2}}\right )^{\frac {2}{3}}} \]
command
integrate(1/x**3/(c*x**6+b*x**3+a),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \operatorname {RootSum} {\left (t^{6} \left (46656 a^{8} c^{3} - 34992 a^{7} b^{2} c^{2} + 8748 a^{6} b^{4} c - 729 a^{5} b^{6}\right ) + t^{3} \left (- 432 a^{4} c^{4} + 1512 a^{3} b^{2} c^{3} - 1107 a^{2} b^{4} c^{2} + 297 a b^{6} c - 27 b^{8}\right ) + c^{5}, \left ( t \mapsto t \log {\left (x + \frac {- 2592 t^{4} a^{8} c^{3} + 2592 t^{4} a^{7} b^{2} c^{2} - 810 t^{4} a^{6} b^{4} c + 81 t^{4} a^{5} b^{6} + 12 t a^{4} c^{4} - 75 t a^{3} b^{2} c^{3} + 78 t a^{2} b^{4} c^{2} - 27 t a b^{6} c + 3 t b^{8}}{5 a^{2} b c^{4} - 5 a b^{3} c^{3} + b^{5} c^{2}} \right )} \right )\right )} - \frac {1}{2 a x^{2}} \]