Region of validity for integral of gamma functions

Question

In Gradshteyn's "Table of Integrals, Series and Products", page 651, formula 6.414 (4), the following integral is evaluated as $$ \int^{+\infty}_{-\infty} {\Gamma(\gamma+x)\Gamma(\delta+x) \over \Gamma(\alpha + x) \Gamma(\beta + x)} \, dx = {2\pi^2 i \, \Gamma(\alpha + \beta - \gamma - \delta -1) \over \sin[\pi(\gamma-\delta)] \Gamma(\alpha - \gamma) \Gamma(\alpha-\delta)\Gamma(\beta-\gamma)\Gamma(\beta-\delta)} $$ if $\textrm{Re}(\alpha + \beta - \gamma -\delta ) > 1$, $\textrm{Im}(\gamma)<0$ and $\textrm{Im}(\delta)<0$ but it's zero if $\textrm{Im}(\delta)>0$. My question is, what should be the integral if all the imaginary parts are $0$? I have looked in all the standard references I can think of but this case is never mentioned.