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(15*Sqrt[-1 + x]*(-8 + 3*x) + Sqrt[15]*(2 + 3*x)^2*ArcTan[Sqrt[3/5]*Sqrt[-1 + x]])/(900*(2 + 3*x)^2)


Integrate[Sqrt[x - 1]/(2 + 3*x)^3, x] ==
(15*Sqrt[-1 + x]*(-8 + 3*x) + Sqrt[15]*(2 + 3*x)^2* ArcTan[Sqrt[3/5]*Sqrt[-1 + x]])/(900*(2 + 3*x)^2)



          Sqrt[x - 1]
Integrate[-----------, x] == 
                   3
          (2 + 3 x)

(15 Sqrt[-1 + x] (-8 + 3 x) + 
 
                      2             3
    Sqrt[15] (2 + 3 x)  ArcTan[Sqrt[-] Sqrt[-1 + x]])\
                                    5
 
                   2
   / (900 (2 + 3 x) )

Time to compute: 0.07 second

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