Falcon 900 Captain reported Nose Landing Gear Failed to Retract after takeoff. The flight crew could not get the Nose Gear to retract using Alternate Procedures and requested priority handling. The flight crew diverted and made a precautionary landing with all Three Landing Gear Down.

Narrative

Departing ZZZZ; upon gear retraction we noticed the nose landing gear did not appear to be retracting. This was based upon the immediate feeling of a normal gear retraction; versus the feeling we felt of the gear continuing to hang in the wind. We reduced our acceleration rate to remain below normal gear retraction/extension speed; then selected the gear back to down. Upon selecting the gear back down; we received a normal indication of three green landing gear lights with no red gear door lights. ZZZZ is an extremely bumpy runway; so we believed it to be a momentary issue with a micro switch not quite making the contact it needed. After seeing a normal down indication; we attempted to raise the gear again. This led to another failure of the gear to raise. We then determined this to be a larger issue and selected the gear back down. This resulted in the nose gear not indicating down nor up. At this time we requested priority handling to begin our divert into ZZZ1.We followed the checklist to extend the gear with the secondary handle; bypassing the electrical system and using mechanical/hydraulic power; and that led to an immediate green indication with all three lights. This appeared to verify that it was a micro switch not quite making proper contact. We then elected to remain at a reduced altitude of 6;000 feet in order to burn off our excessive fuel loads to land under the limiting landing weight. The decision to request priority handling was based on an international divert; and having a prepared airport in ZZZ1 in case we had any further issues that we were not aware of. We chose ZZZ1 international for numerous reasons. We did not want to land back on the bumpiness of the runway; we knew ZZZ1 had excellent Airport Rescue and Firefighting (ARFF) services with a very long runway; and finally it was a major port of entry for customs. We believe our heavy weight and excessive runway roughness putting large loads on our nose strut contributed to the gear issue.