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Integrate[x^m*Log[x + Sqrt[-a^2 + x^2]], x] ==
(x^(1 + m)*((x*Sqrt[1 - x^2/a^2]* (a^2 - x*(x + Sqrt[-a^2 + x^2]))* Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2, x^2/a^2])/((2 + m)*(-a^2 + x^2)* (x + Sqrt[-a^2 + x^2])) + Log[x + Sqrt[-a^2 + x^2]]))/(1 + m)



           m                2    2
Integrate[x  Log[x + Sqrt[-a  + x ]], x] == 

                      2
  1 + m              x
(x      ((x Sqrt[1 - --] 
                      2
                     a
 
           2                 2    2
         (a  - x (x + Sqrt[-a  + x ])) 
 
                                             2
                           1      m      m  x
         Hypergeometric2F1[-, 1 + -, 2 + -, --]) / 
                           2      2      2   2
                                            a
 
                   2    2              2    2
       ((2 + m) (-a  + x ) (x + Sqrt[-a  + x ])) + 
 
                     2    2
      Log[x + Sqrt[-a  + x ]])) / (1 + m)

Time to compute: 0.62 second

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