Series representation for this integral

[ConfusedCat]

I am trying to find a series representation for the following expression
$$\int_{i=0}^\infty {x^{\frac{2n-1}{2}}(b+x)^{-n}}e^{\left(-{\frac{x^2}{2m}}+\frac{x}{p}\right)} dx$$ ; b,m,n,p are constant.

Is there a name for this function?

I found a series representation for $$\int_{i=0}^\infty {x^{\frac{2n-1}{2}}}e^{\left(-{\frac{x^2}{2m}}+\frac{x}{p}\right)} dx$$ in Table of Integrals, Series and Products by Gradshteyn and Ryzhik, involving parabolic cylinder function, but nothing that fits the first expression.

Any help would be appreciated.

 

[HallsofIvy]

I would try writing power series representations for the exponential and $$(b+ x)^{-n}$$, multiply the series, then integrate term by term.