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Integrate[x^3/Sqrt[a + b*x + c*x^2], x] ==
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(15*b^2 - 10*b*c*x + 8*c*(-2*a + c*x^2)) + (-15*b^3 + 36*a*b*c)* Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/ (48*c^(7/2))



                    3
                   x
Integrate[--------------------, x] == 
                            2
          Sqrt[a + b x + c x ]

(2 Sqrt[c] Sqrt[a + x (b + c x)] 
 
          2                             2
     (15 b  - 10 b c x + 8 c (-2 a + c x )) + 
 
          3
    (-15 b  + 36 a b c) 
 
     Log[b + 2 c x + 2 Sqrt[c] Sqrt[a + x (b + c x)]])
 
           7/2
    / (48 c   )

Time to compute: 0.40 second

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