Suppose that $a, b, c\_1$ and $c\_2$ are real constant. 


Is there the **necessary and sufficient** conditions of $a ,b, c\_1,c\_2 $ for the following integration is integrable? i.e.
$$\int\_1^{\infty}\int\_1^{\infty}\int\_1^{\infty}\frac{1}{~x^{a}~y^{b}~(x+y)^{c\_1}~(x+y+t)^{c\_2}}~t^{-\frac{1}{2}}e^{-\frac{1}{t}} dx dy dt < \infty.$$


#


# I know the necessary and sufficient conditions of $a ,b, c\_1$ for 


$$\int\_1^{\infty}\int\_1^{\infty}\frac{1}{~x^{a}~y^{b}~(x+y)^{c\_1}}dx dy 
 < \infty$$


is $a+c\_1>1, b+c\_1>1$ and $a+b+c\_1>2.$



Integrable function
The factor $e^{-1/t}$ is $\asymp1$ for $t>1$, and so, it may be dropped. So, the integral in question is finite iff $I\_j<\infty$ for all $j=1,\dots,6$, where 
\begin{equation}
 I\_j:=\iiint\limits\_{R\_j}\frac{dx\,dy\,dt}{x^a\,y^b\,(x+y)^{c\_1}\,(x+y+t)^{c\_2}\,t^{1/2}},
\end{equation}
\begin{align}
 R\_1&:=\{(x,y,t)\colon 1<x<y<t\}, \\ 
 R\_2&:=\{(x,y,t)\colon 1<y<x<t\}, \\ 
R\_3&:=\{(x,y,t)\colon 1<x<t<y\}, \\ 
R\_4&:=\{(x,y,t)\colon 1<y<t<x\}, \\ 
R\_5&:=\{(x,y,t)\colon 1<t<x<y\}, \\ 
R\_6&:=\{(x,y,t)\colon 1<t<y<x\}. 
\end{align}


We have 
\begin{equation}
 I\_1\asymp\int\_1^\infty\frac{dx}{x^a}\int\_x^\infty\frac{dy}{y^{b+c\_1}}
 \int\_y^\infty\frac{dt}{t^{c\_2+1/2}}, 
\end{equation}
so that $I\_1<\infty$ iff 
\begin{equation}
 c\_2+1/2>1,\quad b+c\_1+c\_2>3/2,\quad a+b+c\_1+c\_2>5/2. 
\end{equation}
Similarly, $I\_2<\infty$ iff 
\begin{equation}
 c\_2+1/2>1,\quad a+c\_1+c\_2>3/2,\quad a+b+c\_1+c\_2>5/2. 
\end{equation}


The integrals $I\_3,\dots,I\_6$ are treated similarly. As the result, the integral in question is finite iff 
\begin{equation}
 c\_2+1/2>1,\quad a+c\_1+c\_2>3/2,\quad b+c\_1+c\_2>3/2,\quad a+b+c\_1+c\_2>5/2. 
\end{equation}


