It seems you do an indefinite integral:
In[7]:= Integrate[(8 x^2 + 1)^2 16 x, x]
Out[7]= 8 x^2 + 64 x^4 + (512 x^6)/3
you can add any constant and differentiation will give you your original equation back
In[8]:= D[% + c, x]
Out[8]= 16 x + 256 x^3 + 1024 x^5
on the other hand a definite integral from a to be is found like this:
In[9]:= Integrate[(8 x^2 + 1)^2 16 x, { x, a, b}]
Out[9]= -8 a^2 - 64 a^4 - (512 a^6)/3 + 8 b^2 + 64 b^4 + (512 b^6)/3
