(Reference request) Asymptotic of generalized exponential integral

I'm interested in asymptotics of the integral$$f_a(x)=\int_1^{x} t^a e^t\,dt, \ a\in \mathbb R,$$as $x\to+\infty$.

Mathematica gives $f_a(x)\sim x^a e^{x}$, $x\to \infty$.

I didnt' find it in Abramiovits, Stegun, only integer values of $a$ are considered there. DLMF introduces an integral $E_p(z)$ generally, starts with complex arguments:$$E_{p}\left(z\right)=z^{p-1}\Gamma(1-p,z)=z^{p-1}\int_{z}^{\infty}\frac{e^{-t}}{t^{p}}\,{d}t,\quad z,p\in \mathbb{C},$$not giving a reference for the last equality, and defines a principal value for the rhs.

Is there a reference for asymptotics for the special case of $f_a$ where the function in integral is positive without need to define $t^p$ for negative $t$ etc as the general formula requires?