What is the easiest or shortest way to show $A = \sum_{n=1}^{\infty} \frac{1}{n(n+6)(n+7)} = \frac{223}{5880} $

What is the easiest or shortest way to show

$$ A = \sum_{n=1}^{\infty} \frac{1}{n(n+6)(n+7)} = \frac{223}{5880} $$

?

Now A has many forms

$$A = \sum_{n=1}^{\infty} \frac{1}{n(n+6)(n+7)} = \sum_{n=1}^{\infty} \frac{1}{n^2 + 6n} - \frac{1}{n^2 + 7n} = \sum_{n=1}^{\infty} \frac{1}{7(n+7)} + \frac{1}{42 n} - \frac{1}{6(n+6)} = \sum_{n=1}^{\infty} \frac{1}{n^3 + 13 n^2 + 42n} = ... $$

and it relates to the digamma function.

Some forms are probably easier to work with than others.

I assume some telescoping method is in the top 4 ?

Can we avoid the digamma ?