An Integration Problem

$\begin{aligned} \int_0^\infty \frac{x^5}{(x+1)^7} \ln^2(x+1)\,dx & = \; \frac{d^2}{da^2} \int_0^\infty \frac{x^5}{(x+1)^{6+a}}\,dx \Big|_{a=1} \; = \; \frac{d^2}{da^2} B(6,a)\Big|_{a=1} \\ & = \; \frac{\Gamma(6)\Gamma(a)}{\Gamma(6+a)}\big[(\psi(a) - \psi(6+a))^2 + (\psi'(a) - \psi'(6+a))\big] \Big|_{a=1} \; = \; \frac{1}{6}\big[(\psi1)-\psi(7))^2 + (\psi'(1) - \psi'(7))\big] \\ & = \; \frac{13489}{10800} \end{aligned}$ making the answer $13489 - 10800 = \boxed{2689}$ .