I need your kind help in the computation of the small-$q$ expansion of the following definite integral
$\displaystyle\int_0^\infty\frac{dp}{p \left(p^2+q^2\right) \Big[(a/p)^{\eta }+1\Big]^{1/2} \Big[a^\eta/(p^2+q^2)^{\eta/2}+1\Big]^{1/2}}$,
where $2>\eta>0$, $a>0$, and $q\ll a$.
I can show that for $\eta=2$ the integral behaves as $\displaystyle \frac{2}{a^2}\log\left(\frac{a}{q}\right)$. However, I am not able to find its small-$q$ asymptote for other values of $\eta$.
The main contribution to the integral comes from the vicinity of $p=0$. To get the leading term, it's sufficient to replace the integrand with a simpler one that has the same behavior for small $p$ and small $q$: $$\int_0^\infty \frac 1 {p \left( p^2+q^2 \right)} \left( \left( a/p \right) ^ \eta + 1 \right) ^ {-1/2} \left( a ^ \eta / \left( p^2+q^2 \right) ^ {\eta/2} + 1 \right) ^ {-1/2} dp \sim\\ \int_0^\infty \frac 1 {p \left( p^2+q^2 \right)} \left( \left( a/p \right) ^ \eta \right) ^ {-1/2} \left( a ^ \eta / \left( p^2+q^2 \right) ^ {\eta/2} \right) ^ {-1/2} dp =\\ \frac {\Gamma \left( \frac \eta 4 \right) \Gamma \left( \frac 1 2 - \frac \eta 4 \right)} {2 ^ {\eta/2 + 1} \sqrt \pi a ^ \eta q ^ {2 - \eta}}.$$
The leading term for $\eta=2$ can be obtained in the same manner; it'll be $-\ln(q)/a^2$ (you have an extra factor of 2 in your result).