Help with an Incomplete Gamma function-like integral $\int_x^\infty t^{m-1}e^{-0.5 t^2 \pm pt} dt$

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I was working on some mathematical derivations where I was struck with integrals of the form given below:

$$\int_x^\infty t^{m-1}e^{-0.5 t^2 \pm pt} dt$$

The integral seems very close to the incomplete gamma function integral. Note here that m and p are positive real numbers. I tried different forms of substitutions but none did work for me.

Any helpful notes, hints, references will be very much appreciated.