Integral $\int_{-\infty}^{at}\dfrac{|x|^u t^v}{(|x|+t+1)^{1+u+v+\varepsilon}}dx$

Question

I want to study the following integral: $F(t)=\int_{-\infty}^{at}\dfrac{|x|^u t^v}{(|x|+t+1)^{1+u+v+\varepsilon}}dx$ where $a,u,v,\varepsilon>0$.

I wish to know what is $A=\sup_{t>0}F(t)$? Does there exist a constant $C=C(a,u,v)>0$ such that $A\ge C. \dfrac1{\varepsilon}$ ??

I indeed have no Idea.