\[ \int \frac {x^9}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {x^{2}}{2 b d}+\frac {a^{\frac {3}{2}} \arctan \! \left (\frac {x^{2} \sqrt {b}}{\sqrt {a}}\right )}{2 b^{\frac {3}{2}} \left (-a d +b c \right )}-\frac {c^{\frac {3}{2}} \arctan \! \left (\frac {x^{2} \sqrt {d}}{\sqrt {c}}\right )}{2 d^{\frac {3}{2}} \left (-a d +b c \right )} \]
command
integrate(x**9/(b*x**4+a)/(d*x**4+c),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ - \frac {\sqrt {- \frac {a^{3}}{b^{3}}} \log {\left (x^{2} + \frac {- \frac {a^{4} d^{4} \sqrt {- \frac {a^{3}}{b^{3}}}}{a d - b c} - \frac {a^{3} b^{3} d^{6} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{2} b^{4} c d^{5} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a b^{5} c^{2} d^{4} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{6} c^{3} d^{3} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{4} c^{4} \sqrt {- \frac {a^{3}}{b^{3}}}}{a d - b c}}{a^{3} c d^{2} + a^{2} b c^{2} d + a b^{2} c^{3}} \right )}}{4 \left (a d - b c\right )} + \frac {\sqrt {- \frac {a^{3}}{b^{3}}} \log {\left (x^{2} + \frac {\frac {a^{4} d^{4} \sqrt {- \frac {a^{3}}{b^{3}}}}{a d - b c} + \frac {a^{3} b^{3} d^{6} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{2} b^{4} c d^{5} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a b^{5} c^{2} d^{4} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{6} c^{3} d^{3} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{4} c^{4} \sqrt {- \frac {a^{3}}{b^{3}}}}{a d - b c}}{a^{3} c d^{2} + a^{2} b c^{2} d + a b^{2} c^{3}} \right )}}{4 \left (a d - b c\right )} - \frac {\sqrt {- \frac {c^{3}}{d^{3}}} \log {\left (x^{2} + \frac {- \frac {a^{4} d^{4} \sqrt {- \frac {c^{3}}{d^{3}}}}{a d - b c} - \frac {a^{3} b^{3} d^{6} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{2} b^{4} c d^{5} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a b^{5} c^{2} d^{4} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{6} c^{3} d^{3} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{4} c^{4} \sqrt {- \frac {c^{3}}{d^{3}}}}{a d - b c}}{a^{3} c d^{2} + a^{2} b c^{2} d + a b^{2} c^{3}} \right )}}{4 \left (a d - b c\right )} + \frac {\sqrt {- \frac {c^{3}}{d^{3}}} \log {\left (x^{2} + \frac {\frac {a^{4} d^{4} \sqrt {- \frac {c^{3}}{d^{3}}}}{a d - b c} + \frac {a^{3} b^{3} d^{6} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{2} b^{4} c d^{5} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a b^{5} c^{2} d^{4} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{6} c^{3} d^{3} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{4} c^{4} \sqrt {- \frac {c^{3}}{d^{3}}}}{a d - b c}}{a^{3} c d^{2} + a^{2} b c^{2} d + a b^{2} c^{3}} \right )}}{4 \left (a d - b c\right )} + \frac {x^{2}}{2 b d} \]