\[ \int \frac {c+d x^3+e x^6+f x^9}{x^5 \left (a+b x^3\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {c}{4 a^{2} x^{4}}+\frac {-a d +2 b c}{a^{3} x}+\frac {\left (-a^{3} f +a^{2} b e -a \,b^{2} d +b^{3} c \right ) x^{2}}{3 a^{3} b \left (b \,x^{3}+a \right )}-\frac {\left (2 a^{3} f +a^{2} b e -4 a \,b^{2} d +7 b^{3} c \right ) \ln \! \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{9 a^{\frac {10}{3}} b^{\frac {5}{3}}}+\frac {\left (2 a^{3} f +a^{2} b e -4 a \,b^{2} d +7 b^{3} c \right ) \ln \! \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{18 a^{\frac {10}{3}} b^{\frac {5}{3}}}-\frac {\left (2 a^{3} f +a^{2} b e -4 a \,b^{2} d +7 b^{3} c \right ) \arctan \! \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{9 a^{\frac {10}{3}} b^{\frac {5}{3}}} \]
command
integrate((f*x**9+e*x**6+d*x**3+c)/x**5/(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
\[ \operatorname {RootSum} {\left (729 t^{3} a^{10} b^{5} + 8 a^{9} f^{3} + 12 a^{8} b e f^{2} - 48 a^{7} b^{2} d f^{2} + 6 a^{7} b^{2} e^{2} f + 84 a^{6} b^{3} c f^{2} - 48 a^{6} b^{3} d e f + a^{6} b^{3} e^{3} + 84 a^{5} b^{4} c e f + 96 a^{5} b^{4} d^{2} f - 12 a^{5} b^{4} d e^{2} - 336 a^{4} b^{5} c d f + 21 a^{4} b^{5} c e^{2} + 48 a^{4} b^{5} d^{2} e + 294 a^{3} b^{6} c^{2} f - 168 a^{3} b^{6} c d e - 64 a^{3} b^{6} d^{3} + 147 a^{2} b^{7} c^{2} e + 336 a^{2} b^{7} c d^{2} - 588 a b^{8} c^{2} d + 343 b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {81 t^{2} a^{7} b^{3}}{4 a^{6} f^{2} + 4 a^{5} b e f - 16 a^{4} b^{2} d f + a^{4} b^{2} e^{2} + 28 a^{3} b^{3} c f - 8 a^{3} b^{3} d e + 14 a^{2} b^{4} c e + 16 a^{2} b^{4} d^{2} - 56 a b^{5} c d + 49 b^{6} c^{2}} + x \right )} \right )\right )} + \frac {- 3 a^{2} b c + x^{6} \left (- 4 a^{3} f + 4 a^{2} b e - 16 a b^{2} d + 28 b^{3} c\right ) + x^{3} \left (- 12 a^{2} b d + 21 a b^{2} c\right )}{12 a^{4} b x^{4} + 12 a^{3} b^{2} x^{7}} \]