B737 Captain reported the flight departed without a signed final DG summary.

Narrative

Unsigned Dangerous Goods (DG). We pushed back and departed without acknowledging the final DG summary by mistake. After a rough night of sleep with loud air conditioner I was moving a bit slow this flight. Our van departed late which added to a small stress level of feeling slightly rushed on top of gate agents looking to push us out as quickly as possible. In the middle of this when I got the DG; I did something out of my normal habit pattern by putting it to my side rather than clipping it to my yoke clip like I normally would until signed. This ultimately led to me completely forgetting about it. The warning message on ACARS must have come at the same time as our finally weights during pushback and was somehow cleared without that fail; save being seen by either of us. It wasn't till halfway though the flight that I realized I had never signed the final DG; though the final DG summary was on board. I looked through received messages and did see we got a warning message but again neither of us had seen this when it did come through. I sent a message to Dispatch to confirm all was good and that we had the final DG to at least try and do something rather than ignore it. Unfortunately it was a snowball effect of issues that led to this and several safety catches failed to highlight our mistake till later in the flight. This highlights the importance of keeping with normal habit patterns!