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


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



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

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

Time to compute: 0.17 second

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