Assume that $a_1<1$, $a_3<1,$ $a_1+a_2+a_5>1$, $a_3+a_4+a_5>1,$ $a_1+a_2+a_3+a_4+a_5>2.$ For $x,y>0,$ define a fucntion $$H(u,v)=\frac{u^{\frac{1}{2}}\int_0^{\infty}\int_0^{\infty}\frac{1}{x^{a_1}~ (1+x)^{a_2+1}~ y^{a_3}~(1+y)^{a_4}~ (1+x+y)^{a_5}}\exp\big\{-\frac{u}{1+x}-\frac{v}{1+x+y}\big\}dx dy}{\int_0^{\infty}\int_0^{\infty}\frac{1}{x^{a_1}~ (1+x)^{a_2}~ y^{a_3}~(1+y)^{a_4}~(1+x+y)^{a_5}}\exp\big\{-\frac{2u}{1+x}-\frac{2v}{1+x+y}\big\}dx dy}.$$ Then $H(u,v)$ is uniformly bounded over positive constant $u,v$, i.e. there is a constant C, such that $H(u,v)\le C.$

How to prove it? Fedor Petrov gave the answer [Uniformly Bounded (updating) for a similar case. I need your help.

Maybe, the following result is useful:

If $a_1<1$ and $a_1+a_2>1,$ then $$f(u)\equiv\int_0^{\infty}\frac{1}{x^{a_1}~ (1+x)^{a_2}}\exp\big\{-\frac{u}{1+x}\big\}dt\approx \min\{1,u^{1-a_1-a_2}\}.$$