\[ \int \frac {1}{x^2 \left (a x+b x^3+c x^5\right )} \, dx \]
Optimal antiderivative \[ -\frac {1}{2 a \,x^{2}}-\frac {b \ln \! \left (x \right )}{a^{2}}+\frac {b \ln \! \left (c \,x^{4}+b \,x^{2}+a \right )}{4 a^{2}}-\frac {\left (-2 a c +b^{2}\right ) \arctanh \! \left (\frac {2 c \,x^{2}+b}{\sqrt {-4 a c +b^{2}}}\right )}{2 a^{2} \sqrt {-4 a c +b^{2}}} \]
command
integrate(1/x**2/(c*x**5+b*x**3+a*x),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \left (\frac {b}{4 a^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{2} + \frac {- 8 a^{3} c \left (\frac {b}{4 a^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} \left (\frac {b}{4 a^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 3 a b c - b^{3}}{2 a c^{2} - b^{2} c} \right )} + \left (\frac {b}{4 a^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{2} + \frac {- 8 a^{3} c \left (\frac {b}{4 a^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} \left (\frac {b}{4 a^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 3 a b c - b^{3}}{2 a c^{2} - b^{2} c} \right )} - \frac {1}{2 a x^{2}} - \frac {b \log {\left (x \right )}}{a^{2}} \]