close formula for hypergeometric ${}_3 F_2$

Question

I search a close form for the general hypergeometric ${}_3 F_2(1, 1, 1/2+A/2; 3/2, 2; 1 )$ where $A$ is exclusive a negative integer even.

It seems that the close form is ( rational number ) + $(2\ln 2)/(-A+1)$.

So i need help for the formula about the rational number. Thanks.

Answer 1

Wolfram Alpha gives the answer which can simplify as $$\, _3F_2\left(1,1,A+\frac{1}{2};\frac{3}{2},2;1\right)=\frac{\psi (1-A)+\gamma +2\log (2)}{1-2 A}=\frac{H_{-A}+2\log (2)}{1-2 A}$$