\[ \int \frac {x \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (-2 a f +b e \right ) x^{2}}{2 b^{3}}+\frac {f \,x^{5}}{5 b^{2}}+\frac {\left (-a^{3} f +a^{2} b e -a \,b^{2} d +b^{3} c \right ) x^{2}}{3 a \,b^{3} \left (b \,x^{3}+a \right )}-\frac {\left (8 a^{3} f -5 a^{2} b e +2 a \,b^{2} d +b^{3} c \right ) \ln \! \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{9 a^{\frac {4}{3}} b^{\frac {11}{3}}}+\frac {\left (8 a^{3} f -5 a^{2} b e +2 a \,b^{2} d +b^{3} c \right ) \ln \! \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{18 a^{\frac {4}{3}} b^{\frac {11}{3}}}-\frac {\left (8 a^{3} f -5 a^{2} b e +2 a \,b^{2} d +b^{3} c \right ) \arctan \! \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{9 a^{\frac {4}{3}} b^{\frac {11}{3}}} \]
command
integrate(x*(f*x**9+e*x**6+d*x**3+c)/(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
\[ x^{2} \left (- \frac {a f}{b^{3}} + \frac {e}{2 b^{2}}\right ) + \frac {x^{2} \left (- a^{3} f + a^{2} b e - a b^{2} d + b^{3} c\right )}{3 a^{2} b^{3} + 3 a b^{4} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a^{4} b^{11} + 512 a^{9} f^{3} - 960 a^{8} b e f^{2} + 384 a^{7} b^{2} d f^{2} + 600 a^{7} b^{2} e^{2} f + 192 a^{6} b^{3} c f^{2} - 480 a^{6} b^{3} d e f - 125 a^{6} b^{3} e^{3} - 240 a^{5} b^{4} c e f + 96 a^{5} b^{4} d^{2} f + 150 a^{5} b^{4} d e^{2} + 96 a^{4} b^{5} c d f + 75 a^{4} b^{5} c e^{2} - 60 a^{4} b^{5} d^{2} e + 24 a^{3} b^{6} c^{2} f - 60 a^{3} b^{6} c d e + 8 a^{3} b^{6} d^{3} - 15 a^{2} b^{7} c^{2} e + 12 a^{2} b^{7} c d^{2} + 6 a b^{8} c^{2} d + b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {81 t^{2} a^{3} b^{7}}{64 a^{6} f^{2} - 80 a^{5} b e f + 32 a^{4} b^{2} d f + 25 a^{4} b^{2} e^{2} + 16 a^{3} b^{3} c f - 20 a^{3} b^{3} d e - 10 a^{2} b^{4} c e + 4 a^{2} b^{4} d^{2} + 4 a b^{5} c d + b^{6} c^{2}} + x \right )} \right )\right )} + \frac {f x^{5}}{5 b^{2}} \]