At the start of battle it will almost always use Plasma Cannon, which inflicts moderate to severe damage on unprotected characters. They can deal around 2,000 to even 4,500 damage to characters in range.
The key to beating the Aquila Velocycle is to switch to Evened Odds (MED/SYN/SAB) at the start of the battle, and wait until the Saboteur has inflicted Imperil, Slow and Deprotect on him, while the Medic keeps the party healthy. The Synergist should have also buffed the party with at least Haste and Faith.
Then, use Delta Attack (COM/RAV/SEN) until the Aquila Velocycle is at 40% of its Chain Gauge, then use Mystic Tower (RAV/RAV/SEN) to stagger it rapidly. Once staggered, the player should stick to Mystic Tower until its Chain Bonus is at a reasonable bonus. Lightning's Army of One ability is a great way to send the chain bonus skyrocketing. Using this strategy, one will be able to drastically cut down the battle duration, so long as the player changes to a more offensive Paradigm once the chain bonus reaches a reasonable amount.
Another way of fighting this resistant enemy is by starting with the Recuperation paradigm (SYN/SYN/MED). Cast Protect and Haste onto all characters quickly. Then, when the enemy charges up for its devastating Plasma Cannon, quickly switch from the augment paradigm to one with at least a single Sentinel at hand. After it blasts the party, quickly switch to the Salvation or Combat Clinic paradigms. Heal as quickly as possible before it charges up again.
Switch back to Recuperation to finish up any and all augments and buffs along with Enfire or Enwater. Have one or two party members debuff and debilitate the Aquila Velocycle while the other one or two focus on defending, enhancing, or healing the party. When the Velocycle has several status ailments, that's the chance to stagger the enemy. Once it is staggered, use Ruin or charge up the chain gauge and use the Full ATB Skills of Lightning, Sazh, or Hope.
Aquila is the Italian and Latin word for "eagle", as well as the name of a constellation.