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



                     1
Integrate[------------------------, x] == 
                                 m
          Sqrt[a + b x] (c + d x)

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

Time to compute: 0.12 second

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