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Default sumif returns blank if all cells blank

Hi all, I have rows in which columns b-f may be blank or have numerical
values in them, and in column e of the same row I need to be able to sum
these numbers if there are any, but remain blank if ALL of columns b-f are
blank. The problem I'm having with:

{=IF(ISBLANK(B2:F2),"",SUM(B2:F2))} in cell e2

is that it returns "" if ANY of the cells in b2:f2 are blank. In other
words, I need a formula in e2 that sums b2:f2 if any of those cells are not
blank, but returns blank if all cells in b2:f2 are blank. If all cells in
b2:f2 contain zeroes, then e2 should contain a zero, but if all cells in
b2:f2 are blank, e2 should be blank.

Thanks in advance for your help.
 
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