Solving indefinite integral $\int^{\infty}_{0} x^{\alpha-1} e^{-px^{2} - qx} dx$

Question

I am working to solve an indefinite integral of the following form.

$\int^{\infty}_{0} x^{\alpha-1} e^{-px^{2} - qx} dx$

I ended up looking it up in V1 of Brychkov, Marichev, and Prudnikov's "Integrals and Series", where the solution is (Eq. 2.3.15.3)

I have gone through the chapter, the introduction, the appendix, and every table in the book I could find but I could not find the definition of $D$. Is $D$ just the differential operator? If so or if not, how do you solve for this integral? Apologies for such a basic question.

Answer 1

There is another formulation in terms of Kummer's confluent hypergeometric function $$I=\int^{\infty}_{0} x^{\alpha-1}\, e^{-px^{2} - qx}\, dx$$ $$2 p^{\frac{\alpha+1 }{2}}\,I=\sqrt{p}\, \Gamma \left(\frac{\alpha }{2}\right) \, _1F_1\left(\frac{\alpha }{2};\frac{1}{2};\frac{q^2}{4 p}\right)-q\,\Gamma \left(\frac{\alpha +1}{2}\right) \, _1F_1\left(\frac{\alpha +1}{2};\frac{3}{2};\frac{q^2}{4 p}\right)$$ provided that $\Re(q)>0\land \Re(\alpha )>0\land \Re(p)>0$.

Fot the particular case where $p=\frac{q^2}{4}$, this gives $$I=q^{-\alpha }\,\,\Gamma (\alpha )\,\, U\left(\frac{\alpha }{2},\frac{1}{2},1\right) $$ where appears Tricomi's confluent hypergeometric function.