# Integrate and ConditionalExpression Output

I am integrating some difficult functions and using the conditional output to place bounds on parameters to ensure integrability, but it seems that not all of the conditions are being returned. The following simple example illustrates the issue.

When I perform the 1D integration

Integrate[Exp[b y], {y, 0, Infinity}]


ConditionalExpression[-(1/b), Re[b] < 0]


but when I perform the 2D integration

Integrate[Exp[a x + b y], {x, 0, Infinity}, {y, 0, Infinity}]


ConditionalExpression[1/(a b), Re[a] < 0].


That is, the condition on the parameter b ensuring integrability is not returned. In this case does Mathematica only return the condition associated with the last integration step?

Thanks...

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It seems you have to set GenerateConditions explicitly
Integrate[Exp[a x + b y], {x, 0, Infinity}, {y, 0, Infinity},