\[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx \]
Optimal antiderivative \[ -\frac {c}{8 a \,x^{8}}+\frac {-a d +b c}{5 a^{2} x^{5}}+\frac {-a^{2} e +a b d -b^{2} c}{2 a^{3} x^{2}}-\frac {\left (-a^{3} f +a^{2} b e -a \,b^{2} d +b^{3} c \right ) \ln \! \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{3 a^{\frac {11}{3}} b^{\frac {1}{3}}}+\frac {\left (-a^{3} f +a^{2} b e -a \,b^{2} d +b^{3} c \right ) \ln \! \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{6 a^{\frac {11}{3}} b^{\frac {1}{3}}}+\frac {\left (-a^{3} f +a^{2} b e -a \,b^{2} d +b^{3} c \right ) \arctan \! \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{3 a^{\frac {11}{3}} b^{\frac {1}{3}}} \]
command
integrate((f*x**9+e*x**6+d*x**3+c)/x**9/(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
\[ \operatorname {RootSum} {\left (27 t^{3} a^{11} b - a^{9} f^{3} + 3 a^{8} b e f^{2} - 3 a^{7} b^{2} d f^{2} - 3 a^{7} b^{2} e^{2} f + 3 a^{6} b^{3} c f^{2} + 6 a^{6} b^{3} d e f + a^{6} b^{3} e^{3} - 6 a^{5} b^{4} c e f - 3 a^{5} b^{4} d^{2} f - 3 a^{5} b^{4} d e^{2} + 6 a^{4} b^{5} c d f + 3 a^{4} b^{5} c e^{2} + 3 a^{4} b^{5} d^{2} e - 3 a^{3} b^{6} c^{2} f - 6 a^{3} b^{6} c d e - a^{3} b^{6} d^{3} + 3 a^{2} b^{7} c^{2} e + 3 a^{2} b^{7} c d^{2} - 3 a b^{8} c^{2} d + b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {3 t a^{4}}{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac {- 5 a^{2} c + x^{6} \left (- 20 a^{2} e + 20 a b d - 20 b^{2} c\right ) + x^{3} \left (- 8 a^{2} d + 8 a b c\right )}{40 a^{3} x^{8}} \]