While working with a generating function for the Catalan numbers, I came across the integral representation $$\frac1{1+\sqrt{1-4x}}=\frac1{2\pi}\int_0^{\infty}\frac{\sqrt{t}}{(t+\frac14)(t-x+\frac14)}\,dt.$$ But, now, I wish to ask:

QUESTION. Is there a similar (real) integral formulation for $$\frac1{1+2x+\sqrt{1-4x}}$$ with "simple" (hopefully linear) factors in the integrand?