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