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(-(x*Sqrt[a^2 + x^2]) + 2*x^2*Log[x + Sqrt[a^2 + x^2]] + a^2*Log[2*(x + Sqrt[a^2 + x^2])])/4


Integrate[x*Log[x + Sqrt[a^2 + x^2]], x] ==
(-(x*Sqrt[a^2 + x^2]) + 2*x^2*Log[x + Sqrt[a^2 + x^2]] + a^2*Log[2*(x + Sqrt[a^2 + x^2])])/4



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

           2    2        2               2    2
(-(x Sqrt[a  + x ]) + 2 x  Log[x + Sqrt[a  + x ]] + 
 
     2                  2    2
    a  Log[2 (x + Sqrt[a  + x ])]) / 4

Time to compute: 0.07 second

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