Integral [x^3 + 1]^2 dx = ? I found that it was [(x^3 + 1)^3] / 9x^2, but that is wrong? Details inside.

So I solved this question by going through these steps:

First, I used the power rule of integration. So, it then looked like this:

[(x^3 + 1)^3] / 3

So then I realized that when you derive that, the chain rule applies to the portion in the numerator. To compensate, I added the value of the derivative of the part in parenthesis to which the chain rule applies, 3x^2, into the denominator. I got my answer, then, which was:

[(x^3 + 1)^3] / 9x^2

This is the incorrect answer, apparently. Is there somewhere where I took a wrong turn? It must be somewhere in my recognition of the power rule. Do you have any help for me?

Thank you!