\[ \int \frac {x^8 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx \]
Optimal antiderivative \[ \frac {\left (-b e +c d \right ) x^{3}}{3 c^{2}}+\frac {e \,x^{6}}{6 c}-\frac {\left (a c e -b^{2} e +b c d \right ) \ln \! \left (c \,x^{6}+b \,x^{3}+a \right )}{6 c^{3}}-\frac {\left (3 a b c e -2 a \,c^{2} d -b^{3} e +b^{2} c d \right ) \arctanh \! \left (\frac {2 c \,x^{3}+b}{\sqrt {-4 a c +b^{2}}}\right )}{3 c^{3} \sqrt {-4 a c +b^{2}}} \]
command
integrate(x**8*(e*x**3+d)/(c*x**6+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
\[ x^{3} \left (- \frac {b e}{3 c^{2}} + \frac {d}{3 c}\right ) + \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac {a c e - b^{2} e + b c d}{6 c^{3}}\right ) \log {\left (x^{3} + \frac {2 a^{2} c e - a b^{2} e + a b c d + 12 a c^{3} \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac {a c e - b^{2} e + b c d}{6 c^{3}}\right ) - 3 b^{2} c^{2} \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac {a c e - b^{2} e + b c d}{6 c^{3}}\right )}{3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d} \right )} + \left (\frac {\sqrt {- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac {a c e - b^{2} e + b c d}{6 c^{3}}\right ) \log {\left (x^{3} + \frac {2 a^{2} c e - a b^{2} e + a b c d + 12 a c^{3} \left (\frac {\sqrt {- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac {a c e - b^{2} e + b c d}{6 c^{3}}\right ) - 3 b^{2} c^{2} \left (\frac {\sqrt {- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac {a c e - b^{2} e + b c d}{6 c^{3}}\right )}{3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d} \right )} + \frac {e x^{6}}{6 c} \]