I am trying to solve the following integral where $x$ is a random variable from a sub-Gaussian distribution while $c\geq 2$ and $\beta>0$:
$$\int_0^\infty x^{c-1} \exp{\left(-\frac{x^c}{\beta^c}-\frac{x^2}{2 \beta^2}\right)} dx $$
I know the solution to the following known integral:
$$\int_0^{\infty} x^n \exp\,(-a x^b)\,dx = \frac{1}{b}\, a^{-\frac{n+1}{b}} \Gamma \left(\frac{n+1}{b} \right) $$
However, the exponential function in my problem has a polynomial power. Can anyone please help me through this?