Upper bound of polynomial integral

In my recent research, I encountered the following integral inequality$$\int_0^1kx^{k-1}(1+x)^{2k+1}\mathrm{d}x<2^{2k},$$where $k$ is a positive integer. This inequality can be transformed to binomial coefficient form, but the induction method doesn't work.

The following picture gives the value of $RHS-LHS$ for $k\leq500$. So the difference is increasing at speed $10^k$.

I also want to know where can I find some messages about the integral form $\int_0^1x^m(1+x)^n\mathrm{d}x$ ?