\[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {c}{2 a^{2} x^{2}}+\frac {f x}{b^{2}}-\frac {\left (-a^{3} f +a^{2} b e -a \,b^{2} d +b^{3} c \right ) x}{3 a^{2} b^{2} \left (b \,x^{3}+a \right )}-\frac {\left (4 a^{3} f -a^{2} b e -2 a \,b^{2} d +5 b^{3} c \right ) \ln \! \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{9 a^{\frac {8}{3}} b^{\frac {7}{3}}}+\frac {\left (4 a^{3} f -a^{2} b e -2 a \,b^{2} d +5 b^{3} c \right ) \ln \! \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{18 a^{\frac {8}{3}} b^{\frac {7}{3}}}+\frac {\left (4 a^{3} f -a^{2} b e -2 a \,b^{2} d +5 b^{3} c \right ) \arctan \! \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{9 a^{\frac {8}{3}} b^{\frac {7}{3}}} \]
command
integrate((f*x**9+e*x**6+d*x**3+c)/x**3/(b*x**3+a)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {- 3 a b^{2} c + x^{3} \left (2 a^{3} f - 2 a^{2} b e + 2 a b^{2} d - 5 b^{3} c\right )}{6 a^{3} b^{2} x^{2} + 6 a^{2} b^{3} x^{5}} + \operatorname {RootSum} {\left (729 t^{3} a^{8} b^{7} + 64 a^{9} f^{3} - 48 a^{8} b e f^{2} - 96 a^{7} b^{2} d f^{2} + 12 a^{7} b^{2} e^{2} f + 240 a^{6} b^{3} c f^{2} + 48 a^{6} b^{3} d e f - a^{6} b^{3} e^{3} - 120 a^{5} b^{4} c e f + 48 a^{5} b^{4} d^{2} f - 6 a^{5} b^{4} d e^{2} - 240 a^{4} b^{5} c d f + 15 a^{4} b^{5} c e^{2} - 12 a^{4} b^{5} d^{2} e + 300 a^{3} b^{6} c^{2} f + 60 a^{3} b^{6} c d e - 8 a^{3} b^{6} d^{3} - 75 a^{2} b^{7} c^{2} e + 60 a^{2} b^{7} c d^{2} - 150 a b^{8} c^{2} d + 125 b^{9} c^{3}, \left ( t \mapsto t \log {\left (- \frac {9 t a^{3} b^{2}}{4 a^{3} f - a^{2} b e - 2 a b^{2} d + 5 b^{3} c} + x \right )} \right )\right )} + \frac {f x}{b^{2}} \]