I want to evaluate the below integral $$ \int_{0}^{\infty} e^{-ax^{2}-bx} x^{\alpha -1}\,\mathrm{d}x, ~~~~~~ \operatorname{Re}(\alpha)>0 $$ where $a,b$ are real constants.
Somehow, I want to convert it in the form of the Gamma function but I don't know which substitution will work here. Any help will be appreciated. Please help me if there is any other method to evaluate this integral.