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


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



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

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

Time to compute: 0.06 second

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