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(Sqrt[a*x]*(Sqrt[b]*Sqrt[x]*Sqrt[1 - b*x]*(-1 + 2*b*x) + ArcSin[Sqrt[b]*Sqrt[x]]))/(4*b^(3/2)*Sqrt[x])


Integrate[Sqrt[a*x]*Sqrt[1 - b*x], x] ==
(Sqrt[a*x]*(Sqrt[b]*Sqrt[x]*Sqrt[1 - b*x]* (-1 + 2*b*x) + ArcSin[Sqrt[b]*Sqrt[x]]))/ (4*b^(3/2)*Sqrt[x])



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

(Sqrt[a x] (Sqrt[b] Sqrt[x] Sqrt[1 - b x] 
 
       (-1 + 2 b x) + ArcSin[Sqrt[b] Sqrt[x]])) / 
 
      3/2
  (4 b    Sqrt[x])

Time to compute: 0.09 second

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