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


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



                             b
Integrate[(Sqrt[2] + Sqrt[x]) , x] == 

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

Time to compute: 0.11 second

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