Laplace Transforms · Superchallenge problem

Let $\operatorname{erf}(x) = \dfrac{2}{\sqrt\pi}\displaystyle\int_0^x e^{-w^2}dw$. Using the derivative rule and the shift rule for Laplace transforms, evaluate $\displaystyle\int_0^\infty e^{-t}\operatorname{erf}\!\left(\sqrt{t}\right)dt$.