How come the antiderivative of e^(2x+x^2) isn't e^(2x+x^2)/(x+2x)??

I know the anti derivative of e^(x^2) cannot be defined using elem. functions, so i guess e^(2x+x^2) is kinda like that, but it seemed to make sense to me when I antiderived it using substitution like this :

S e^u = e^u + C

let u= 2x+x^2

du = 2+2x dx

dx= du/2+2x

Then:

S e^(2x+x^2) dx= S e^u * du/2+2x = e^(2x+x^2)/2+2x

Doesn't that make sense? what am I doing wrong? or is that the antiderivative of e^(2x+x^2) altho I'm pretty sure it isn't?? does this have something to do with chain rule??

thanks!