As a matter of fact, the 1/3 term is not an incorrect one, but an incomplete one. Besides the function "Integrate", you can also use "DSolve" to show that constant term which is
In:= DSolve[y'[x] == (8 x^2 + 1)^2 16 x, y[x], x]
Out = {{y[x] -> 8 x^2 + 64 x^4 + (512 x^6)/3 + C[1]}}
The C[1] is just the term missing in the solution by "Integrate".
