\[ \int \frac {1}{x \left (a+b x^n+c x^{2 n}\right )} \, dx \]
Optimal antiderivative \[ \frac {\ln \left (x \right )}{a}-\frac {\ln \left (a +b \,x^{n}+c \,x^{2 n}\right )}{2 a n}+\frac {b \arctanh \left (\frac {b +2 c \,x^{n}}{\sqrt {-4 a c +b^{2}}}\right )}{a n \sqrt {-4 a c +b^{2}}} \]
command
integrate(1/x/(a+b*x**n+c*x**(2*n)),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {4 b c n \log {\left (x \right )}}{b^{3} n + 2 b^{2} c n x^{n}} - \frac {4 b c \log {\left (\frac {b}{2 c} + x^{n} \right )}}{b^{3} n + 2 b^{2} c n x^{n}} + \frac {4 b c}{b^{3} n + 2 b^{2} c n x^{n}} + \frac {8 c^{2} n x^{n} \log {\left (x \right )}}{b^{3} n + 2 b^{2} c n x^{n}} - \frac {8 c^{2} x^{n} \log {\left (\frac {b}{2 c} + x^{n} \right )}}{b^{3} n + 2 b^{2} c n x^{n}} & \text {for}\: a = \frac {b^{2}}{4 c} \\- \frac {x^{- n}}{b n} - \frac {c \log {\left (x^{n} \right )}}{b^{2} n} + \frac {c \log {\left (\frac {b}{c} + x^{n} \right )}}{b^{2} n} & \text {for}\: a = 0 \\\frac {\log {\left (x \right )}}{a + b + c} & \text {for}\: n = 0 \\\frac {\log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x^{n} \right )}}{a n} & \text {for}\: c = 0 \\- \frac {b \log {\left (\frac {b}{2 c} + x^{n} - \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )}}{2 a n \sqrt {- 4 a c + b^{2}}} + \frac {b \log {\left (\frac {b}{2 c} + x^{n} + \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )}}{2 a n \sqrt {- 4 a c + b^{2}}} + \frac {\log {\left (x \right )}}{a} - \frac {\log {\left (\frac {b}{2 c} + x^{n} - \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )}}{2 a n} - \frac {\log {\left (\frac {b}{2 c} + x^{n} + \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )}}{2 a n} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________