\[ \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (-4 a^{3} f +3 a^{2} b e -2 a \,b^{2} d +b^{3} c \right ) x^{2}}{2 b^{5}}+\frac {\left (3 a^{2} f -2 a b e +b^{2} d \right ) x^{5}}{5 b^{4}}+\frac {\left (-2 a f +b e \right ) x^{8}}{8 b^{3}}+\frac {f \,x^{11}}{11 b^{2}}+\frac {a \left (-a^{3} f +a^{2} b e -a \,b^{2} d +b^{3} c \right ) x^{2}}{3 b^{5} \left (b \,x^{3}+a \right )}+\frac {a^{\frac {2}{3}} \left (-14 a^{3} f +11 a^{2} b e -8 a \,b^{2} d +5 b^{3} c \right ) \ln \! \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{9 b^{\frac {17}{3}}}-\frac {a^{\frac {2}{3}} \left (-14 a^{3} f +11 a^{2} b e -8 a \,b^{2} d +5 b^{3} c \right ) \ln \! \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{18 b^{\frac {17}{3}}}+\frac {a^{\frac {2}{3}} \left (-14 a^{3} f +11 a^{2} b e -8 a \,b^{2} d +5 b^{3} c \right ) \arctan \! \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{9 b^{\frac {17}{3}}} \]
command
integrate(x**7*(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
\[ x^{8} \left (- \frac {a f}{4 b^{3}} + \frac {e}{8 b^{2}}\right ) + x^{5} \left (\frac {3 a^{2} f}{5 b^{4}} - \frac {2 a e}{5 b^{3}} + \frac {d}{5 b^{2}}\right ) + x^{2} \left (- \frac {2 a^{3} f}{b^{5}} + \frac {3 a^{2} e}{2 b^{4}} - \frac {a d}{b^{3}} + \frac {c}{2 b^{2}}\right ) + \frac {x^{2} \left (- a^{4} f + a^{3} b e - a^{2} b^{2} d + a b^{3} c\right )}{3 a b^{5} + 3 b^{6} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} b^{17} + 2744 a^{11} f^{3} - 6468 a^{10} b e f^{2} + 4704 a^{9} b^{2} d f^{2} + 5082 a^{9} b^{2} e^{2} f - 2940 a^{8} b^{3} c f^{2} - 7392 a^{8} b^{3} d e f - 1331 a^{8} b^{3} e^{3} + 4620 a^{7} b^{4} c e f + 2688 a^{7} b^{4} d^{2} f + 2904 a^{7} b^{4} d e^{2} - 3360 a^{6} b^{5} c d f - 1815 a^{6} b^{5} c e^{2} - 2112 a^{6} b^{5} d^{2} e + 1050 a^{5} b^{6} c^{2} f + 2640 a^{5} b^{6} c d e + 512 a^{5} b^{6} d^{3} - 825 a^{4} b^{7} c^{2} e - 960 a^{4} b^{7} c d^{2} + 600 a^{3} b^{8} c^{2} d - 125 a^{2} b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {81 t^{2} b^{11}}{196 a^{7} f^{2} - 308 a^{6} b e f + 224 a^{5} b^{2} d f + 121 a^{5} b^{2} e^{2} - 140 a^{4} b^{3} c f - 176 a^{4} b^{3} d e + 110 a^{3} b^{4} c e + 64 a^{3} b^{4} d^{2} - 80 a^{2} b^{5} c d + 25 a b^{6} c^{2}} + x \right )} \right )\right )} + \frac {f x^{11}}{11 b^{2}} \]