Question: How to find product

Maple does not find the product
product((k+a)*(k+b)/((k+c)*(k+d)), k = 1 .. infinity) assuming a > 0, b > 0, c > 0, d > 0, a+b = c+d;

                         GAMMA(c) GAMMA(d) infinity
                         ----------------------------------
                             GAMMA(a) GAMMA(b)    

in the general case, but Maple calculates it in the case of the concrete parameters. For example,

product((k+3-sqrt(2)+3^(1/3))*(k+5+sqrt(2)-3^(1/3))/((k+6-(1/2)*sqrt(5))*(k+2+(1/2)*sqrt(5))),
 k = 1 .. infinity);

- 55/512* (sin(Pi *(sqrt(2 ) - 3^(1/3))* (sqrt(2 ) - 3^(1/3))* (1633*sqrt( 5) - 2916))/
                                                                          
                      
(-621* 3^(2/3)      - 15132*sqrt(2)  + 13665*3^(1/3) + 297*sqrt(2)*3^(2/3) - 18166

          
   + 9941*sqrt(2)*3^(1/3))*sin(1/2* Pi*sqrt(5)))    
                     

Because of this reason the question arises:"How to find this product in the general case?"