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



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

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

Time to compute: 0.22 second

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