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 $\gamma>\max(0,-a/2,-b/2)$; then Mathematica gives the result in terms of the <A HREF="https://en.wikipedia.org/wiki/Meijer_G-function">Meijer G-function</A>, see also Bateman and Erdélyi, Higher Transcendental Functions, Vol. I, paragraph 5.3.1:

$$F\left ( x \right )=\frac{1}{2i\pi}\int_{\gamma-i\infty}^{\gamma+i\infty} \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}\,ds=\frac{1}{\pi}2^{a+b-2} G_{0,5}^{5,0}\left(\frac{x}{16}|
\begin{array}{c}
 0,\frac{a}{2},\frac{a+1}{2},\frac{b}{2},\frac{b+1}{2} \\
\end{array}
\right).$$