Since the span is short that might be ok but you should use LVL's because they are 50% stronger. The more serious issue is the installation. The new joists must have adequate bearing at each end and there should be enough studs below to carry the additional load to the foundation.

Another advantage of LVL's is that the manufacturer will often design them and stamp the design for free through the lumber yard where they are bought.

When you sell the house you will want the stamped design drawing to be on file at the building department in case the prospective buyer uses a competent home inspector.

Leigh75:

You're calculations aren't finished. If your tub is a five-footer that's 33" x 60" = 1,980 sq.in. =13 s.q.feet divided into 990 lbs. which is 76 lbs. per square foot. This is easily within the live load of your floor.

No reinforcements are necessary. Install and enjoy.

The maximum allowed uniformly distributed live load for this floor is no greater than 50 PSF assuming the joists are of reasonable capacity, well supported and the loading is uniformly distributed.

But the true capacity of the joists is not known, the loading is obviously not uniform and the location on the span is not known so it would be unwise to leave them alone without the approval of an engineer who has seen the structure.

I have to agree with Renovator8..this project doesn't seem as straight forward as I first thought so i'll be bringing in an engineer.
I'll update.

I'm curious why the local building official would not require the stamp of an engineer or some other convincing evidence of structural soundness for this project.

Interesting that you posted your curiosity Renovator8.
I spoke to the inspector this morning to ask if he needed
an engineer's approval.
His answer was no..all he'll need is a diagram of the project for his yay or nay.
I'll still get an engineer's opinion.

The way this usually works is that common structural design issues are prescribed in a residential building code so that an engineer is not needed.

In the case of an unusual load on an existing structure the only way a building inspector could know if the design met the usual code standard of "accepted engineering practice" would be if the owner provided an engineer's design or some proof that it was acceptable from a manufacturer of engineered lumber like LVLs, etc.

In this case the inspector could be replaced by a receptionist with a rubber stamp. And so it goes.

I must be missing something, because this doesn't seem unusual to me. The only weight difference between a whirlpool and a regular bathtub is a motor and a few water jets.