\[ \int \frac {x^4 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (3 a^{2} f -2 a b e +b^{2} d \right ) x^{2}}{2 b^{4}}+\frac {\left (-2 a f +b e \right ) x^{5}}{5 b^{3}}+\frac {f \,x^{8}}{8 b^{2}}-\frac {\left (-a^{3} f +a^{2} b e -a \,b^{2} d +b^{3} c \right ) x^{2}}{3 b^{4} \left (b \,x^{3}+a \right )}-\frac {\left (-11 a^{3} f +8 a^{2} b e -5 a \,b^{2} d +2 b^{3} c \right ) \ln \! \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{9 a^{\frac {1}{3}} b^{\frac {14}{3}}}+\frac {\left (-11 a^{3} f +8 a^{2} b e -5 a \,b^{2} d +2 b^{3} c \right ) \ln \! \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{18 a^{\frac {1}{3}} b^{\frac {14}{3}}}-\frac {\left (-11 a^{3} f +8 a^{2} b e -5 a \,b^{2} d +2 b^{3} c \right ) \arctan \! \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{9 a^{\frac {1}{3}} b^{\frac {14}{3}}} \]
command
integrate(x**4*(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
\[ x^{5} \left (- \frac {2 a f}{5 b^{3}} + \frac {e}{5 b^{2}}\right ) + x^{2} \left (\frac {3 a^{2} f}{2 b^{4}} - \frac {a e}{b^{3}} + \frac {d}{2 b^{2}}\right ) + \frac {x^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 a b^{4} + 3 b^{5} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a b^{14} - 1331 a^{9} f^{3} + 2904 a^{8} b e f^{2} - 1815 a^{7} b^{2} d f^{2} - 2112 a^{7} b^{2} e^{2} f + 726 a^{6} b^{3} c f^{2} + 2640 a^{6} b^{3} d e f + 512 a^{6} b^{3} e^{3} - 1056 a^{5} b^{4} c e f - 825 a^{5} b^{4} d^{2} f - 960 a^{5} b^{4} d e^{2} + 660 a^{4} b^{5} c d f + 384 a^{4} b^{5} c e^{2} + 600 a^{4} b^{5} d^{2} e - 132 a^{3} b^{6} c^{2} f - 480 a^{3} b^{6} c d e - 125 a^{3} b^{6} d^{3} + 96 a^{2} b^{7} c^{2} e + 150 a^{2} b^{7} c d^{2} - 60 a b^{8} c^{2} d + 8 b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {81 t^{2} a b^{9}}{121 a^{6} f^{2} - 176 a^{5} b e f + 110 a^{4} b^{2} d f + 64 a^{4} b^{2} e^{2} - 44 a^{3} b^{3} c f - 80 a^{3} b^{3} d e + 32 a^{2} b^{4} c e + 25 a^{2} b^{4} d^{2} - 20 a b^{5} c d + 4 b^{6} c^{2}} + x \right )} \right )\right )} + \frac {f x^{8}}{8 b^{2}} \]