Altho' it is true that side effects are often the first thing we notice when trying an antidepressant, I think 5 weeks is time enough to know if a medicine is right for you.
There are many antidepressants on the market, as gofio mentions. It is a matter of trial and error until the one that works is found.
Call your doctor. Often a combo is the answer....depending on what the core problem is.
Keep us posted. Most of us have been where you are. I actually tried Prozac at least 3 times during the years. As anxiety is my primary problem...it gave me a jittery energy and insomnia:).
Best wishes....don't give up...stay in close communication with your doctor.
5 weeks is not really long enough to have given it a fair trial. The downside of meds clicks in much quicker than the 'good sides'. Try to be patient, give a few more weeks, if there's no increase in your mood then discuss other options with your Doc. There's a plethora of things you can try if prozac fails. Good Luck, George