\int x^2 (x^3+1)^3\;dx = \frac{1}{3} \int 3x^2 (x^3+1)^3\;dx =\frac{1}{3}\int(x^3+1)^3 d(x^3+1)=\frac{1}{3}\frac{(x^3+1  )^4}{4}+c