\[ \int \frac {x^8}{\left (a x+b x^3+c x^5\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {b x}{2 c \left (-4 a c +b^{2}\right )}+\frac {x^{3} \left (b \,x^{2}+2 a \right )}{2 \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {\arctan \! \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \left (b^{2}-6 a c -\frac {b \left (-8 a c +b^{2}\right )}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{4 c^{\frac {3}{2}} \left (-4 a c +b^{2}\right ) \sqrt {b -\sqrt {-4 a c +b^{2}}}}+\frac {\arctan \! \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}}\right ) \left (b^{2}-6 a c +\frac {b \left (-8 a c +b^{2}\right )}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{4 c^{\frac {3}{2}} \left (-4 a c +b^{2}\right ) \sqrt {b +\sqrt {-4 a c +b^{2}}}} \]
command
integrate(x**8/(c*x**5+b*x**3+a*x)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {a b x + x^{3} \left (- 2 a c + b^{2}\right )}{8 a^{2} c^{2} - 2 a b^{2} c + x^{4} \left (8 a c^{3} - 2 b^{2} c^{2}\right ) + x^{2} \left (8 a b c^{2} - 2 b^{3} c\right )} + \operatorname {RootSum} {\left (t^{4} \left (1048576 a^{6} c^{9} - 1572864 a^{5} b^{2} c^{8} + 983040 a^{4} b^{4} c^{7} - 327680 a^{3} b^{6} c^{6} + 61440 a^{2} b^{8} c^{5} - 6144 a b^{10} c^{4} + 256 b^{12} c^{3}\right ) + t^{2} \left (- 61440 a^{5} b c^{5} + 61440 a^{4} b^{3} c^{4} - 24064 a^{3} b^{5} c^{3} + 4608 a^{2} b^{7} c^{2} - 432 a b^{9} c + 16 b^{11}\right ) + 1296 a^{5} c^{2} - 360 a^{4} b^{2} c + 25 a^{3} b^{4}, \left ( t \mapsto t \log {\left (x + \frac {49152 t^{3} a^{4} c^{7} - 40960 t^{3} a^{3} b^{2} c^{6} + 12288 t^{3} a^{2} b^{4} c^{5} - 1536 t^{3} a b^{6} c^{4} + 64 t^{3} b^{8} c^{3} - 1728 t a^{3} b c^{3} + 656 t a^{2} b^{3} c^{2} - 88 t a b^{5} c + 4 t b^{7}}{324 a^{3} c^{2} - 81 a^{2} b^{2} c + 5 a b^{4}} \right )} \right )\right )} \]