B737-800
Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument
Time Pressure; Situational Awareness; Distraction
Deviation / Discrepancy - Procedural Published Material / Policy; Deviation / Discrepancy - Procedural Weight And Balance
Human Factors
We departed ZZZ without the flight release PIC signature.  The following summary is not offered as an excuse for this or to lessen the mistake that was made only to advise what happened and most importantly prevent it from happening again.  The weather for this flight was unusually good; only some forecast light chop at FL380 to FL410 for the first 200 miles of flight.  The FO (First Officer) and I spoke about this and decided to stop our climb at FL370 till past threat.  As we approached the gate I signed the release I thought; evidently the release did not go thru as expected.  The gate area was only partially full as our load was only 50 passengers. The aircraft was hot as the temperature was 106 degrees F and externally air was hooked to the aircraft. We dealt with that and unhooked the external air and started the APU to cool the aircraft.  During this time the FAs (Flight Attendants) boarded the aircraft and were briefed by me.  I was notified shortly after taken seat and starting the downloads to my EFB (Electronic Flight Bag) that the lights would not come on in the main cabin so we called Maintenance and they arrived quickly to fix the issue;  Quick fix and we continued; made the mask PA and we pushed early.  As we were pushing the FO noticed that the runway had changed to a west flow so runway change number 1; we completed the starting engines flows and set the parking brake and did not move.  We then got clearance to taxi and advise to set up for a south runway departure; change 2; we held position and as we completed this change we were advised we could depart on the north runway if we would be ready after the short taxi to Runway XX.  I called the FAs and asked if they would be ready for a short taxi; the response was yes. I accepted the short taxi; change 3.  We had set up for Runway XR before push; then after push Runway change XX; runway XYR and back to runway XX. Three runway changes after push along with checking and changing the route of flight as each runway has a different departure procedures and resetting the MCP 3 times.  Both the FO and myself have been on this aircraft for many years and both of us felt like we were in the green and had trapped all of the issues that had happened.  We never got an ACARS message saying the flight release had not been signed and during the distractions the final trap was missed as we briefed the takeoff data based on what we expected to happen.  The lesson which I learned on this flight is even when you feel like you're in the green; stop and run through your flows. This was the first time I ever took the runway without physically touching the closeout and it will be my last!  We were notified enroute that the release was not signed by Dispatch; signed release via ACARS and continued uneventfully to destination.  When you feel like your in the green; stop and run all the traps one final time before taking runway.
Air Carrier Flight Crew flying 737-800 aircraft reported not signing flight release prior to departure.