Your truck should be able to handle this, but you need to be properly equipped.
I can't tell for sure from the pics, but it doesn't look like you have Class 4 trailer hitch. Here is list of items that are necessary to make this trip safely and legally.
Hitch - Minimum Class 4
Transmission cooler (if you don't already have one)
Trailer brake controller (unless the Uhaul has a built in trailer brake system)
Next thing is to determine the GVW and GCVWR of the truck (Should be a sticker on the drivers B-pillar with that info.) Then you need to know the actual weight of the trailer with the car on it.
GCVWR (Gross Combined Vehicle Weight Rating) is the maximum weight of the whole unit. This is truck + payload + loaded trailer.
GVW (Gross Vehicle Weight) is the maximum weight of the truck. This includes truck + fuel + humans + cargo + tongue weight of trailer.
When the trailer is properly loaded, about 10% of the gross trailer weight will be on the tongue, so if the trailer weights 2000lbs, and the car weighs 3500lbs, the Gross Trailer Weight will the 5500lbs, and the tongue will be about 550lbs. This 550 lbs is carried by the truck, so becomes part of the trucks 'payload'.
Something that is often overlooked is the rating of the ball mount. These are often rated at max 500lbs, so just be aware of this.
Depending on the rear suspension of your truck and the final tongue weight of the trailer, it may be necessary to use an equalizer hitch. This are generally recommended for trailer weights exceeding 4000lbs. I regularly pull 7000lbs of trailer and do not use an equalizer system, but I have taken the steps to ensure every part of my system is rated greater than the actual load I have on it.
If you have an RV dealer nearby, they can likely offer you some assistance in ensuring you are ready, safe and legal. Your Uhaul rep 'should' be able to do the same.
Not that I am some big expert on trailer towing, but if they give you information that conflicts with what I've given you here, I'd be cautious, then I'd come back here and call me on it. If I'm wrong on any point, I want that corrected so no one has an issue based on wrong information from me. If their stringing you along, there are a lot of members here that will point that out to you.
So by now your thinking 'I'm worried about my truck! What's all this crap about GVW....blah, blah.... I just want to know if my truck will make it without having to replace the engine, trans and differentials when I get there!'
I brought it up first because it 'IS' that important. If your truck has an engine failure, that would be a terrible and costly 'inconvenience', but chances are you would alive and well to complain about it. BUT on that note....
Make sure your truck is well serviced. Starting the trip with a fresh oil/filter change is recommended. If your transmission has not been recently serviced, I would recommend doing that now. You don't need to have it flushed, but a basic oil and filter change would be a great idea. Changing the oil in the diff's is also a smart thing to do, and check your U-joints, suspension components, brakes, wheel bearings, etc.
Depending on what gear ratio you have, will determine how heavy the load feels to your drivetrain. As you likely don't do this often, I will recommend that you just take your time on the trip, and make regular stops to allow the drivetrain to cool. Check the temp of the differential from time to time. After a run on the highway, if you touch it with your hand and get burned right away, you have a problem that needs addressing. Don't worry to much about your hand, it'll heal.
It is not uncommon for the diff to run hot enough that you can't hold your hand on it for any length of time, but you should not get an instant burn from it.
Also talk with the vehicle maintenance techs on base. These guys usually have an excellent understanding of these ratings, and can often help you in the task of checking over your truck to ensure the best chance of a successful trip.