Air Carrier Captain reported encountering severe turbulence during climb while on a SID. The weather caused a speed and altitude deviation. The Captain stated; at one point; the pilots were along for the ride. Reports to ATC; Dispatch and Maintenance were made. A Logbook entry was completed.

Narrative

We departed ZZZ from Runway XXR on the ZZZZZ 3 Departure. ZZZZZ was the first fix on the LEGS page with LNAV/VNAV engaged. Sometime from setting takeoff trust to 1000 ft.; there became a discontinuity between the runway and ZZZZZ. After flap retraction I closed the discontinuity and re-verified the route. It was very turbulent with 20+ kts. of airspeed fluctuation. The airspeed started increasing rapidly and the aircraft climbing about 3000 FPM. My FO (First Officer) was flying; he was at flight idle and pitched down. We finally leveled off at about 8600 ft. MSL and about 305 kts. For about 1000 ft.-1200 ft. of climb we were along for the ride and the aircraft was not responding to control inputs. The crossing restriction and airspeed restriction at ZZZZZ2 is 230 kts. max and 8000 ft. or below. We crossed ZZZZZ2 at about 8600 ft. and 300 kts. After we got back down to 8000 ft. and slowed to 230 kts. it was very rough. After climbing to smoother air; I reported severe turbulence to ATC between ZZZZZ and ZZZZZ2 from 7500 ft.-10;000 ft. ATC did not mention our altitude or airspeed deviation. I reported the turbulence to Dispatch via ACARS. We continued onto ZZZ1 and landed without incident. I had a conference call with Dispatch; the Superintendent of Dispatch; Maintenance Control; and the Chief Pilot on Call. I made an entry in the logbook about the severe turbulence and doing 300 kts. at 8600 ft.