I'm wondering if the following integral can be done analytically?
$$I(x) = \int_{x/4}^1 \text{d}y\,y^{3/2+a} \,\,(1-y)^{1/2}\,\, \exp \left(c\sqrt{\ln \left(\frac{4y}{x} \right)} \right),$$ where $a$ and $c$ are constants satisfying $3/2+a>0$ and $c>0$ with $x \ll 1$ (if this helps).