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



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

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

Time to compute: 0.14 second

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