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I have this simple integral but I don't know why Mathematica can't solve it?!

FullSimplify[Integrate[x/(-x^2 + a x + b)^(1/2), {x, Sqrt[b], Sqrt[R^2 + b]},Assumptions -> {a > 0, b > 0}]]

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Clear["Global`*"]

Include the assumption that R is real.

Assuming[{a > 0, b > 0, Element[R, Reals]},
 Integrate[x/(-x^2 + a x + b)^(1/2), {x, Sqrt[b], Sqrt[R^2 + b]}] // 
  FullSimplify]

enter image description here

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We can test the result by uses Newton-Leibniz formula.

f = Integrate[x/(-x^2 + a x + b)^(1/2), x];
expr2 = (f /. x -> Sqrt[R^2 + b]) - (f /. x -> Sqrt[b])
expr1 = Integrate[
  x/(-x^2 + a x + b)^(1/2), {x, Sqrt[b], Sqrt[R^2 + b]}, 
  Assumptions -> {a > 0, b > 0, R > 0}]
FullSimplify[expr1 == expr2, Assumptions -> {a > 0, b > 0, R > 0}]

True

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