Wolfram Mathematica Online Integrator
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Integrate[(a + b*x + c*x^2)^(-2^(-1) - n), x] ==
-((2^(-1/2 - n)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*
((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/
Sqrt[b^2 - 4*a*c])^(1/2 + n)*
(a + x*(b + c*x))^(-1/2 - n)*Hypergeometric2F1[
1/2 - n, 1/2 + n, 3/2 - n,
(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/
(2*Sqrt[b^2 - 4*a*c])])/(c*(-1 + 2*n)))
2 -(1/2) - n Integrate[(a + b x + c x ) , x] ==
-(1/2) - n 2
-((2 (b - Sqrt[b - 4 a c] + 2 c x)
2
b + Sqrt[b - 4 a c] + 2 c x 1/2 + n
(----------------------------)
2
Sqrt[b - 4 a c]
-(1/2) - n
(a + x (b + c x))
1 1 3
Hypergeometric2F1[- - n, - + n, - - n,
2 2 2
2
-b + Sqrt[b - 4 a c] - 2 c x
-----------------------------]) /
2
2 Sqrt[b - 4 a c]
(c (-1 + 2 n)))
Time to compute: 0.49 second
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