\[ \int \frac {x^4}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {a^{\frac {2}{3}} \ln \! \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{3 b^{\frac {2}{3}} \left (-a d +b c \right )}-\frac {c^{\frac {2}{3}} \ln \! \left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right )}{3 d^{\frac {2}{3}} \left (-a d +b c \right )}-\frac {a^{\frac {2}{3}} \ln \! \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{6 b^{\frac {2}{3}} \left (-a d +b c \right )}+\frac {c^{\frac {2}{3}} \ln \! \left (c^{\frac {2}{3}}-c^{\frac {1}{3}} d^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right )}{6 d^{\frac {2}{3}} \left (-a d +b c \right )}+\frac {a^{\frac {2}{3}} \arctan \! \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{3 b^{\frac {2}{3}} \left (-a d +b c \right )}-\frac {c^{\frac {2}{3}} \arctan \! \left (\frac {\left (c^{\frac {1}{3}}-2 d^{\frac {1}{3}} x \right ) \sqrt {3}}{3 c^{\frac {1}{3}}}\right ) \sqrt {3}}{3 d^{\frac {2}{3}} \left (-a d +b c \right )} \]
command
integrate(x**4/(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
\[ \operatorname {RootSum} {\left (t^{3} \left (27 a^{3} d^{5} - 81 a^{2} b c d^{4} + 81 a b^{2} c^{2} d^{3} - 27 b^{3} c^{3} d^{2}\right ) - c^{2}, \left ( t \mapsto t \log {\left (x + \frac {243 t^{5} a^{6} b^{2} d^{8} - 1458 t^{5} a^{5} b^{3} c d^{7} + 3645 t^{5} a^{4} b^{4} c^{2} d^{6} - 4860 t^{5} a^{3} b^{5} c^{3} d^{5} + 3645 t^{5} a^{2} b^{6} c^{4} d^{4} - 1458 t^{5} a b^{7} c^{5} d^{3} + 243 t^{5} b^{8} c^{6} d^{2} + 9 t^{2} a^{5} d^{5} - 18 t^{2} a^{4} b c d^{4} + 9 t^{2} a^{3} b^{2} c^{2} d^{3} + 9 t^{2} a^{2} b^{3} c^{3} d^{2} - 18 t^{2} a b^{4} c^{4} d + 9 t^{2} b^{5} c^{5}}{a^{3} c d^{2} + a b^{2} c^{3}} \right )} \right )\right )} + \operatorname {RootSum} {\left (t^{3} \left (27 a^{3} b^{2} d^{3} - 81 a^{2} b^{3} c d^{2} + 81 a b^{4} c^{2} d - 27 b^{5} c^{3}\right ) + a^{2}, \left ( t \mapsto t \log {\left (x + \frac {243 t^{5} a^{6} b^{2} d^{8} - 1458 t^{5} a^{5} b^{3} c d^{7} + 3645 t^{5} a^{4} b^{4} c^{2} d^{6} - 4860 t^{5} a^{3} b^{5} c^{3} d^{5} + 3645 t^{5} a^{2} b^{6} c^{4} d^{4} - 1458 t^{5} a b^{7} c^{5} d^{3} + 243 t^{5} b^{8} c^{6} d^{2} + 9 t^{2} a^{5} d^{5} - 18 t^{2} a^{4} b c d^{4} + 9 t^{2} a^{3} b^{2} c^{2} d^{3} + 9 t^{2} a^{2} b^{3} c^{3} d^{2} - 18 t^{2} a b^{4} c^{4} d + 9 t^{2} b^{5} c^{5}}{a^{3} c d^{2} + a b^{2} c^{3}} \right )} \right )\right )} \]