to avoid all three poles in the Mellin inversion formula you want to integrate along the line $\int_{\gamma-i\infty}^{\gamma+i\infty}ds$ where $\gamma>\max(0,-a/2,-b/2)$; the inverse Mellin transform is a modified Bessel function of the second kind,

$$F\left ( s \right )=\frac{1}{2i\pi}\int_{\gamma-i\infty}^{\gamma+i\infty} \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}\,ds=x^{(a+b)/4}K_{|a-b|}(2x^{1/4}).$$