It comes from the change of variables when doing the integration with respect to x. This, of course, could be easily done by hand. Note:
Collect[D[w1, x], x]
gives
w110 + w111 y1 + w112 y1^2 + x (2 w120 + 2 w121 y1 + 2 w122 y1^2)
I am not sure why you have an Expand in your computation --and also why you have a 0.5 rather than a 1/2. But here is the integral more exposed
In[36]:= Integrate[1/2 (D[w1, x])^2, x]
Out[36]= (w110 + 2 w120 x +
y1 (w111 + 2 w121 x + w112 y1 + 2 w122 x y1))^3/(12 (w120 +
y1 (w121 + w122 y1)))