Question: (r-1)*y1=a mess

HI all,

I have this polynom

> y1 := c[0]*x^r+c[0]*x^(r+1)/(r*(r-2))+c[0]*x^(r+2)/(r*(r-2)*(r+1)*(r-1))+c[0]*x^(r+3)/(r^2*(r-2)*(r+1)*(r-1)*(r+2))+c[0]*x^(r+4)/(r^2*(r-2)*(r+1)^2*(r-1)*(r+2)*(r+3))+c[0]*x^(r+5)/(r^2*(r-2)*(r+1)^2*(r-1)*(r+2)^2*(r+3)*(r+4));

                                (r + 1)               (r + 2)      
         
                                (r + 1)               (r + 2)      
                      r   c[0] x                c[0] x             
          y1 := c[0] x  + ------------- + -------------------------
                            r (r - 2)     r (r - 2) (r + 1) (r - 1)

                               (r + 3)           
                         c[0] x                  
             + ----------------------------------
                2                                
               r  (r - 2) (r + 1) (r - 1) (r + 2)

                                    (r + 4)               
                              c[0] x                      
             + -------------------------------------------
                2                2                        
               r  (r - 2) (r + 1)  (r - 1) (r + 2) (r + 3)

                                        (r + 5)                    
                                  c[0] x                           
             + ----------------------------------------------------
                2                2                2                
               r  (r - 2) (r + 1)  (r - 1) (r + 2)  (r + 3) (r + 4)

As you cans see, there is a singularity at r=1.  So I want to multiply y1 by (x-1) without disturbing the rest of the multiplication so it will be easy for me to find the n-th coeffcicients (n in this  cas range from 0 to 5)

Expand is not the way, it create more problem than solving it.

Any idea will be appreciate

Mario