The chain rule in reverse is substitution; the product rule in reverse is integration by parts. Both will probably be useful. Have you tried either?
You can't just integrate each factor and multiply them together; so why do you think you need to do this? To integrate by parts, you generally choose a "part" that you can integrate.
Why would you integrate that 3rd degree polynomial? Do you plan on using using by parts. Remember that \(\displaystyle \int f(x)g(x)\neq\int f(x)dx\int g(x)dx.\)