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Integrate[Log[1 + Sqrt[b + a*x]]/x, x] ==
Log[1 + Sqrt[b + a*x]]* (Log[1 + (1 + Sqrt[b + a*x])/(-1 + Sqrt[b])] + Log[1 - (1 + Sqrt[b + a*x])/(1 + Sqrt[b])]) + PolyLog[2, -((1 + Sqrt[b + a*x])/(-1 + Sqrt[b]))] + PolyLog[2, (1 + Sqrt[b + a*x])/(1 + Sqrt[b])]



          Log[1 + Sqrt[b + a x]]
Integrate[----------------------, x] == 
                    x

Log[1 + Sqrt[b + a x]] 
 
            1 + Sqrt[b + a x]
   (Log[1 + -----------------] + 
              -1 + Sqrt[b]
 
             1 + Sqrt[b + a x]
     Log[1 - -----------------]) + 
                1 + Sqrt[b]
 
               1 + Sqrt[b + a x]
  PolyLog[2, -(-----------------)] + 
                 -1 + Sqrt[b]
 
             1 + Sqrt[b + a x]
  PolyLog[2, -----------------]
                1 + Sqrt[b]

Time to compute: 0.27 second

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