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Integrate[Sqrt[(a + b*x)*(c + d*x)], x] ==
(Sqrt[(a + b*x)*(c + d*x)]*((2*a)/b + (2*c)/d + 4*x - ((b*c - a*d)^2*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]* Sqrt[c + d*x]])/(b^(3/2)*d^(3/2)* Sqrt[a + b*x]*Sqrt[c + d*x])))/8



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

(Sqrt[(a + b x) (c + d x)] 
 
     2 a   2 c
    (--- + --- + 4 x - 
      b     d
 
                  2
      ((b c - a d)  Log[b c + a d + 2 b d x + 
 
           2 Sqrt[b] Sqrt[d] Sqrt[a + b x] 
 
            Sqrt[c + d x]]) / 
 
         3/2  3/2
       (b    d    Sqrt[a + b x] Sqrt[c + d x]))) / 8

Time to compute: 0.21 second

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