Question: Approximating a function near a point

I realise this is probably a really obvious question...

I have this function:

(2^(1/3)*Pi*AiryAi(x))/(b*(a*AiryAi(x) + b*AiryBi(x)))

And I want to approximate it near to the root of the equation

a*AiryAi(x) + b*AiryBi(x)=0 (say the solution is x=x0)

The first term - I know - is a term for 1/(x-x0) for which the coefficient is

AiryAi(RootOf(AiryAi(_Z)*a + b*AiryBi(_Z)))*2^(1/3)*Pi/(b*a*AiryAi(1, RootOf(AiryAi(_Z)*a + b*AiryBi(_Z))) + b^2*AiryBi(1, RootOf(AiryAi(_Z)*a + b*AiryBi(_Z)))).

However the next term, which should be a constant, comes out as a really large term which I can't make sense of (I've included it at the bottom of this post but it is very large).

Am I doing something wrong? Because I feel that this should be easier than I'm making it.

The determined expansion was:

    (1/3)                                       2
-3 2      Pi RootOf(AiryAi(_Z) a + b AiryBi(_Z))  

                                             3               
  AiryBi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  AiryAi(RootOf(

                                 3      (1/3)    
  AiryAi(_Z) a + b AiryBi(_Z))) b  - 9 2      Pi

                                     2
  RootOf(AiryAi(_Z) a + b AiryBi(_Z))  

                                             2
  AiryBi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  

                                             2    2      (1/3)    
  AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  a b  - 9 2      Pi

                                     2                           
  RootOf(AiryAi(_Z) a + b AiryBi(_Z))  AiryBi(RootOf(AiryAi(_Z) a

                                                               3
   + b AiryBi(_Z))) AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  

   2        (1/3)                                       2
  a  b - 3 2      Pi RootOf(AiryAi(_Z) a + b AiryBi(_Z))  

                                             4  3      (1/3)    
  AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  a  - 4 2      Pi

  RootOf(AiryAi(_Z) a + b AiryBi(_Z)) AiryBi(RootOf(AiryAi(_Z) a

   + b AiryBi(_Z)))

                                                2               
  AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  AiryAi(RootOf(

                                 3      (1/3)                       
  AiryAi(_Z) a + b AiryBi(_Z))) b  - 8 2      Pi RootOf(AiryAi(_Z) a

   + b AiryBi(_Z)) AiryBi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

  AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(1,

  RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(RootOf(AiryAi(_Z) a

                       2      (1/3)                       
   + b AiryBi(_Z))) a b  - 4 2      Pi RootOf(AiryAi(_Z) a

   + b AiryBi(_Z)) AiryBi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

                                                2               
  AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  AiryAi(RootOf(

                                 2        (1/3)                  
  AiryAi(_Z) a + b AiryBi(_Z))) a  b - 4 2      Pi RootOf(AiryAi(

  _Z) a + b AiryBi(_Z))

                                                2
  AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  

                                             2    2      (1/3)    
  AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  a b  - 8 2      Pi

  RootOf(AiryAi(_Z) a + b AiryBi(_Z)) AiryBi(1,

  RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(1,

  RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

                                             2  2        (1/3)    
  AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  a  b - 4 2      Pi

  RootOf(AiryAi(_Z) a + b AiryBi(_Z))

                                                2
  AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  

                                             2  3      (1/3)    
  AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  a  + 2 2      Pi

                                             2           
  AiryBi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  AiryBi(1,

  RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(RootOf(AiryAi(_Z) a

                     3      (1/3)    
   + b AiryBi(_Z))) b  + 2 2      Pi

                                             2           
  AiryBi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  AiryAi(1,

  RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(RootOf(AiryAi(_Z) a

                       2      (1/3)                              
   + b AiryBi(_Z))) a b  + 4 2      Pi AiryBi(RootOf(AiryAi(_Z) a

   + b AiryBi(_Z))) AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

                                             2    2      (1/3)    
  AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  a b  + 4 2      Pi

  AiryBi(RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(1,

  RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

                                             2  2         (1/3)    
  AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  a  b + 12 2      Pi

                                                3           
  AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  AiryAi(1,

                                        3       (1/3)    
  RootOf(AiryAi(_Z) a + b AiryBi(_Z))) b  + 36 2      Pi

                                                2
  AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  

                                                2    2      
  AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  a b  + 36

   (1/3)                                                   
  2      Pi AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

                                                3  2        (1/3)    
  AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  a  b + 2 2      Pi

  AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

                                             3  2  
  AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  a  b

         (1/3)                                                  4
   + 12 2      Pi AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  

   3      (1/3)                                                   
  a  + 2 2      Pi AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

                                             3  3\//   
  AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  a / \12

                                                4  4        
  AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  a  b + 48

                                                3           
  AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  AiryBi(1,

                                        3  2      
  RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a  b  + 72

                                                2
  AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  

                                                2  2  3      
  AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  a  b  + 48

  AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

                                                3    4
  AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  a b

                                                      4  5\
   + 12 AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))  b /