Integral involving exponential of exponential convergence

Question

Let $a$ and $b$ be two real numbers and $c>0$. I want to see (or compute if possible) if the following integral

$$ I = \int_0^{+\infty} y \exp(a y + b \sqrt{y} -c \exp(y)) \, dy$$

is convergent.

Answer 1

Prove that $ce^{y}-b\sqrt y -ay \geq M y$ for $y$ sufficiently large, say $y>y_0$ by showing that the ratio LHS/RHS $\to \infty$. The integral from $y_0$ to $\infty$ is therefore bounded by a constant times $\int_{y_0}^{\infty} ye^{-My} dy <\infty$.