Hello Dear Wolfram Community, I am trying to compute the following definite integral in Mathematica:

Sqrt[u]/(e + u)

over du from 0 to infinity where e>0 The expected result is

\Pi / Sqrt[e] 

I have tried the following Mathematica inputs: 1.

Assuming[e > 0, Integrate[Sqrt[u] / (e + u), {u, 0, Infinity}, GenerateConditions -> False]]

This input does not produce a convergence error but returns the incorrect result:

-Sqrt[e] \[Pi]

where the incorrect scaling with Sqrt[e] is the main issue. I'm not worried about the sign.

Other approaches like:

Integrate[Sqrt[u] / (e + u), {u, 0, Infinity}, Assumptions -> {e > 0}, PrincipalValue -> True]

or

Integrate[Sqrt[u] / (e + u), {u, 0, Infinity}, Assumptions -> {e > 0}]

also fails to give the expected result, suggesting that Mathematica struggles with interpreting the integral's convergence or symmetry. (Both give as Output, that the Integral doesn't converge)

How can I adjust my approach to compute this integral correctly in Mathematica? Any advice would be greatly appreciated!