Assume that $a_1<1$, $a_3<1,$ $a_1+a_2+a_5>1$, $a_3+a_4+a_5>1,$ $a_1+a_2+a_3+a_4+a_5>2,$ and $b_1, b_2>0$. For $x,y>0,$ define a fucntion $$H(x,y)=\frac{x^{\frac{1}{2}}\int_0^{\infty}\int_0^{\infty}\frac{1}{t^{a_1}~ (1+t)^{a_2+1}~ s^{a_3}~(1+s)^{a_4}~ (1+t+s)^{a_5}}\exp\big\{-\frac{x}{1+t}-\frac{b_1 y}{1+t+s}\big\}dt ds}{\int_0^{\infty}\int_0^{\infty}\frac{1}{t^{a_1}~ (1+t)^{a_2}~ s^{a_3}~(1+s)^{a_4}~(1+t+s)^{a_5}}\exp\big\{-\frac{x}{1+t}-\frac{b_2 y}{1+t+s}\big\}dt ds}.$$ Then $H(x,y)$ is uniformly bounded over $x,y$, i.e. there is a constant C, such that $H(x,y)\le C.$

How to prove it ??? Fedor Petrov gave the answer [Uniformly Bounded (updating) for the case when $b_1=b_2.$ I still didn't know how to prove the general case. I need your help.

Maybe, the following result is useful:

If $a_1<1$ and $a_1+a_2>1,$ then $$f(x)\equiv\int_0^{\infty}\frac{1}{t^{a_1}~ (1+t)^{a_2}}\exp\big\{-\frac{x}{1+t}\big\}dt\approx C_1\min\{C_2,x^{1-a_1-a_2}\}$$ for some positive constants $C_1$ and $C_2$.