\[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx \]
Optimal antiderivative \[ -\frac {a \left (-a h +b e \right ) x}{b^{3}}+\frac {\left (-a f +b c \right ) x^{2}}{2 b^{2}}+\frac {\left (-a g +b d \right ) x^{3}}{3 b^{2}}+\frac {\left (-a h +b e \right ) x^{4}}{4 b^{2}}+\frac {f \,x^{5}}{5 b}+\frac {g \,x^{6}}{6 b}+\frac {h \,x^{7}}{7 b}+\frac {a^{\frac {2}{3}} \left (b^{\frac {2}{3}} \left (-a f +b c \right )+a^{\frac {2}{3}} \left (-a h +b e \right )\right ) \ln \! \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{3 b^{\frac {10}{3}}}-\frac {a^{\frac {2}{3}} \left (b^{\frac {2}{3}} \left (-a f +b c \right )+a^{\frac {2}{3}} \left (-a h +b e \right )\right ) \ln \! \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{6 b^{\frac {10}{3}}}-\frac {a \left (-a g +b d \right ) \ln \! \left (b \,x^{3}+a \right )}{3 b^{3}}+\frac {a^{\frac {2}{3}} \left (b^{\frac {5}{3}} c -a^{\frac {2}{3}} b e -a \,b^{\frac {2}{3}} f +a^{\frac {5}{3}} h \right ) \arctan \! \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{3 b^{\frac {10}{3}}} \]
command
integrate(x**4*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ x^{4} \left (- \frac {a h}{4 b^{2}} + \frac {e}{4 b}\right ) + x^{3} \left (- \frac {a g}{3 b^{2}} + \frac {d}{3 b}\right ) + x^{2} \left (- \frac {a f}{2 b^{2}} + \frac {c}{2 b}\right ) + x \left (\frac {a^{2} h}{b^{3}} - \frac {a e}{b^{2}}\right ) + \operatorname {RootSum} {\left (27 t^{3} b^{10} + t^{2} \left (- 27 a^{2} b^{7} g + 27 a b^{8} d\right ) + t \left (- 9 a^{4} b^{4} f h + 9 a^{4} b^{4} g^{2} + 9 a^{3} b^{5} c h - 18 a^{3} b^{5} d g + 9 a^{3} b^{5} e f - 9 a^{2} b^{6} c e + 9 a^{2} b^{6} d^{2}\right ) + a^{7} h^{3} - 3 a^{6} b e h^{2} + 3 a^{6} b f g h - a^{6} b g^{3} - 3 a^{5} b^{2} c g h - 3 a^{5} b^{2} d f h + 3 a^{5} b^{2} d g^{2} + 3 a^{5} b^{2} e^{2} h - 3 a^{5} b^{2} e f g + a^{5} b^{2} f^{3} + 3 a^{4} b^{3} c d h + 3 a^{4} b^{3} c e g - 3 a^{4} b^{3} c f^{2} - 3 a^{4} b^{3} d^{2} g + 3 a^{4} b^{3} d e f - a^{4} b^{3} e^{3} + 3 a^{3} b^{4} c^{2} f - 3 a^{3} b^{4} c d e + a^{3} b^{4} d^{3} - a^{2} b^{5} c^{3}, \left ( t \mapsto t \log {\left (x + \frac {- 9 t^{2} a b^{7} f + 9 t^{2} b^{8} c - 3 t a^{4} b^{3} h^{2} + 6 t a^{3} b^{4} e h + 6 t a^{3} b^{4} f g - 6 t a^{2} b^{5} c g - 6 t a^{2} b^{5} d f - 3 t a^{2} b^{5} e^{2} + 6 t a b^{6} c d + a^{6} g h^{2} - a^{5} b d h^{2} - 2 a^{5} b e g h + 2 a^{5} b f^{2} h - a^{5} b f g^{2} - 4 a^{4} b^{2} c f h + a^{4} b^{2} c g^{2} + 2 a^{4} b^{2} d e h + 2 a^{4} b^{2} d f g + a^{4} b^{2} e^{2} g - 2 a^{4} b^{2} e f^{2} + 2 a^{3} b^{3} c^{2} h - 2 a^{3} b^{3} c d g + 4 a^{3} b^{3} c e f - a^{3} b^{3} d^{2} f - a^{3} b^{3} d e^{2} - 2 a^{2} b^{4} c^{2} e + a^{2} b^{4} c d^{2}}{a^{6} h^{3} - 3 a^{5} b e h^{2} + 3 a^{4} b^{2} e^{2} h - a^{4} b^{2} f^{3} + 3 a^{3} b^{3} c f^{2} - a^{3} b^{3} e^{3} - 3 a^{2} b^{4} c^{2} f + a b^{5} c^{3}} \right )} \right )\right )} + \frac {f x^{5}}{5 b} + \frac {g x^{6}}{6 b} + \frac {h x^{7}}{7 b} \]