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-((x^(1 - n)*Sqrt[(a + b*x)/a]*Hypergeometric2F1[1/2, 1 - n, 2 - n, -((b*x)/a)])/((-1 + n)*Sqrt[a + b*x]))


Integrate[1/(x^n*Sqrt[a + b*x]), x] ==
-((x^(1 - n)*Sqrt[(a + b*x)/a]*Hypergeometric2F1[1/2, 1 - n, 2 - n, -((b*x)/a)])/ ((-1 + n)*Sqrt[a + b*x]))



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

    1 - n      a + b x
-((x      Sqrt[-------] 
                  a
 
                        1                  b x
      Hypergeometric2F1[-, 1 - n, 2 - n, -(---)]) / 
                        2                   a
 
    ((-1 + n) Sqrt[a + b x]))

Time to compute: 0.05 second

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