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Composing left and right derived functors

Hi,

probably this question is obvious. I apologize for this.

Given functors $F$ and $G$ left exact, with as good properties as you want, on the "correct" category, we have a spectral sequence $R^p F\circ R^q G$ abutting to $R^{p+q}F\circ G$. I am looking for an analogous for a "mixed version" in the following case: $F$ left exact and $G$ right exact. What appens to $L^pG\circ R^q F$?

Thanks