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

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

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

  1 + m
(x      Sqrt[a + b x] 
 
                        1                   b x
    Hypergeometric2F1[-(-), 1 + m, 2 + m, -(---)]) / 
                        2                    a
 
                a + b x
  ((1 + m) Sqrt[-------])
                   a
Time to compute: 0.05 second
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