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Sqrt[-b + a*x^2] + Sqrt[b]*ArcTan[Sqrt[b]/Sqrt[-b + a*x^2]]

Integrate[Sqrt[-b + a*x^2]/x, x] ==
Sqrt[-b + a*x^2] + Sqrt[b]* ArcTan[Sqrt[b]/Sqrt[-b + a*x^2]] 

                       2
          Sqrt[-b + a x ]
Integrate[---------------, x] == 
                 x

             2                       Sqrt[b]
Sqrt[-b + a x ] + Sqrt[b] ArcTan[---------------]
                                              2
                                 Sqrt[-b + a x ]
Time to compute: 0.05 second
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