Try this way:
u=(x^(p-1))/(x+1) and dv=ln(x)dx. Find du and v, then you will get I(p)=u*v-Integrate[(v*du)]
seems useless to me.
So I tried substituting
which apparently gave me
I am not sure if this is really true. Moreover, I don't seem to be able to solve them (I actually expect to get an Euler's gamma function somehow).
Answer is supposed to be
which leaves me almost certain this can be achieved by making Euler's beta function
Edit: It took some hours of pure thinking, but satisfaction is well worth it.
Integral is already a form of Euler's beta function and known manipulations can be done: