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