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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: 0.51 second

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