I think the following identity is true, $$ \frac{4 (4 s+9)}{3 \Gamma \left(s+\frac{5}{2}\right) \Gamma \left(s+\frac{7}{2}\right)}-\frac{16 (s+2)}{3 \Gamma (s+3)^2}=\frac{\, _3F_2\left(2,s+\frac{5}{2},s+\frac{7}{2};s+4,s+5;1\right)}{\Gamma (s+4) \Gamma (s+5)} $$ But I haven't found anything useful in the literature.
This identity is inspired by this question.