\[ \int \frac {1}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {b^{\frac {2}{3}} \ln \! \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{3 a^{\frac {2}{3}} \left (-a d +b c \right )}-\frac {d^{\frac {2}{3}} \ln \! \left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right )}{3 c^{\frac {2}{3}} \left (-a d +b c \right )}-\frac {b^{\frac {2}{3}} \ln \! \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{6 a^{\frac {2}{3}} \left (-a d +b c \right )}+\frac {d^{\frac {2}{3}} \ln \! \left (c^{\frac {2}{3}}-c^{\frac {1}{3}} d^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right )}{6 c^{\frac {2}{3}} \left (-a d +b c \right )}-\frac {b^{\frac {2}{3}} \arctan \! \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{3 a^{\frac {2}{3}} \left (-a d +b c \right )}+\frac {d^{\frac {2}{3}} \arctan \! \left (\frac {\left (c^{\frac {1}{3}}-2 d^{\frac {1}{3}} x \right ) \sqrt {3}}{3 c^{\frac {1}{3}}}\right ) \sqrt {3}}{3 c^{\frac {2}{3}} \left (-a d +b c \right )} \]
command
integrate(1/(b*x**3+a)/(d*x**3+c),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \operatorname {RootSum} {\left (t^{3} \left (27 a^{5} d^{3} - 81 a^{4} b c d^{2} + 81 a^{3} b^{2} c^{2} d - 27 a^{2} b^{3} c^{3}\right ) + b^{2}, \left ( t \mapsto t \log {\left (x + \frac {81 t^{4} a^{7} c^{2} d^{5} - 243 t^{4} a^{6} b c^{3} d^{4} + 162 t^{4} a^{5} b^{2} c^{4} d^{3} + 162 t^{4} a^{4} b^{3} c^{5} d^{2} - 243 t^{4} a^{3} b^{4} c^{6} d + 81 t^{4} a^{2} b^{5} c^{7} - 3 t a^{4} d^{4} + 3 t a^{3} b c d^{3} + 3 t a b^{3} c^{3} d - 3 t b^{4} c^{4}}{a^{2} b d^{3} + b^{3} c^{2} d} \right )} \right )\right )} + \operatorname {RootSum} {\left (t^{3} \left (27 a^{3} c^{2} d^{3} - 81 a^{2} b c^{3} d^{2} + 81 a b^{2} c^{4} d - 27 b^{3} c^{5}\right ) - d^{2}, \left ( t \mapsto t \log {\left (x + \frac {81 t^{4} a^{7} c^{2} d^{5} - 243 t^{4} a^{6} b c^{3} d^{4} + 162 t^{4} a^{5} b^{2} c^{4} d^{3} + 162 t^{4} a^{4} b^{3} c^{5} d^{2} - 243 t^{4} a^{3} b^{4} c^{6} d + 81 t^{4} a^{2} b^{5} c^{7} - 3 t a^{4} d^{4} + 3 t a^{3} b c d^{3} + 3 t a b^{3} c^{3} d - 3 t b^{4} c^{4}}{a^{2} b d^{3} + b^{3} c^{2} d} \right )} \right )\right )} \]