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

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

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

(Sqrt[a x] (Sqrt[a] Sqrt[x] Sqrt[b - a x] 
 
                       2        Sqrt[a] Sqrt[x]
       (-b + 2 a x) + b  ArcTan[---------------])) / 
                                 Sqrt[b - a x]
 
      3/2
  (4 a    Sqrt[x])
Time to compute: 0.12 second
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