Hi,
I took Wellbutrin SR for about two years many years ago (from 2003 to 2005, if my memory is serving me correctly), and I didn't notice any changes in my weight in the time I spent taking the medication. The Wellbutrin SR did seem to reduce my appetite some, despite the fact that I didn't notice any changes in my weight one way or another, but after the psychiatrist that I was seeing at the time prescribed me lithium to take along with the Wellbutrin SR at least a few months into my Wellbutrin trial, my appetite seemed to return to a level that was normal for me. Still though, my weight didn't seem to change in any noticeable way in the time that I spent taking Wellbutrin, whether or not I was taking lithium along with the Wellbutrin. I'm not saying that I can be sure that you won't gain weight while taking Wellbutrin simply because I didn't gain weight on the medication, but I did take the medication for two years without any noticeable weight changes, and my understanding about Wellbutrin's potential to affect weight is consistent with your doctor's idea that taking Wellbutrin is less likely to lead to weight gain than other antidepressants. In fact, there's evidence from a scientific review article (Li et al., 2005) that Wellbutrin can promote modest weight loss. The article, which included an analysis of three studies that examined the effectiveness of Wellbutrin in the treatment of obesity among individuals who had major depressive disorder or depressive symptoms found that those who took Wellbutrin lost an average of 2.77 kg (6.1 pounds) more than those who took a placebo over a period of 6 to 12 months. So, I'm not saying that it's impossible to gain weight on Wellbutrin, but losing weight on Wellbutrin seems to be more likely than gaining weight on the medication, judging from what I've read.
I wish you luck with taking Wellbutrin, assuming that you're already taking it or will go ahead with taking it soon.
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REFERENCE
Li, Z., Maglione, M., Tu, W., Mojica, W., Arterburn, D., Shugarman, L.R., et al. (2005). Meta-analysis: Pharmacologic treatment of Obesity.
Annals of Internal Medicine, 142, 532-546. Abstract:
https://www.ncbi.nlm.nih.gov/pubmed/15809465