You either pay for it to be cleaned or live with it

Damp dog will be ingrained in the carpet the only way is to deep clean. It may take more than one attempt..

Sprinkle baking soda onto the carpet
Wait a few hours and vacuum

Get a dog

As mentioned baking soda is an excellent odor absorber. In addition I would put out small bowls of vinegar. Vinegar also absorbs odors.

I'm with @PrettyFlamingos - cleaning it properly is the only option besides living with the smell

Do you rent? Have you mentioned it to the landlord?

Nothing will ever get the dog smell out. An ozone generator might help, tho.