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



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

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

Time to compute: 0.21 second

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