Why both Wolfram Alpha and Wolfram Cloud don't simplify below BBP type sum, which I came up with?

Why both Wolfram Alpha and Wolfram Cloud don’t simplify below BBP type sum, which I came up with,

Sum((1/81)^(k+1)*(9/(2*k+1)+1/(2*k+2)),k = 0 .. infinity)

to

2*log(3) - 3*log(2)

Sum.nb (1.34 KB)

With Mathematica I get simplification with FullSimplify, but not with Simplify alone.

Thanks, Dear Gianluca!

doesn’t produce the final result either

And also ∑ Sum Calculator - Online Summation Σ Sigma from 1 to N Solver does not produce the desired final result