As it appears today

it is partially mistaken by WolframAlpha:

In[1]:= Remove[fI, f1, f3, a]
fI[L_, l_, 
  u_, \[Alpha]_, \[Nu]_] := ((L + 
       l) Cos[\[Alpha]]) l (((L + l) Sin[\[Alpha]]) u - ((L + 
           l) Cos[\[Alpha]]) \[Nu])/(((L + l) Sin[\[Alpha]])^2 + ((L +
             l) Cos[\[Alpha]])^2)^(3/2) + \[Nu]
f1[L_, l_, u_, \[Alpha]_, \[Nu]_] := 
 l Cos[\[Alpha]] Sqrt[(l + 
       L)^2] (u Sin[\[Alpha]] - \[Nu] Cos[\[Alpha]]) + \[Nu]
f3[L_, l_, 
  u_, \[Alpha]_, \[Nu]_] := (2 \[Nu] Sqrt[(l + L)^2] + 
    l u Sin[2 \[Alpha]] - 
    2 l \[Nu] Cos[\[Alpha]]^2)/(2 Sqrt[(l + L)^2])


In[11]:= f3[L, l, u, \[Alpha], \[Nu]] - 
  f1[L, l, u, \[Alpha], \[Nu]] // FullSimplify

Out[11]= (l (-1 + l + L) (1 + l + 
   L) Cos[\[Alpha]] (\[Nu] Cos[\[Alpha]] - 
   u Sin[\[Alpha]]))/Sqrt[(l + L)^2]

In[19]:= f3[L, l, u, \[Alpha], \[Nu]] - 
  fI[L, l, u, \[Alpha], \[Nu]] // FullSimplify

Out[19]= 0

In[23]:=    {fI[Sequence @@ #], f3[Sequence @@ #], 
   f1[Sequence @@ #]} & /@ RandomReal[{0, 8}, {10, 5}]

Out[23]= {{6.60003, 6.60003, 20.5665}, {1.89881, 
  1.89881, -70.8277}, {1.21599, 1.21599, 18.0041}, {2.1538, 2.1538, 
  2.14085}, {3.30499, 3.30499, 2.29929}, {5.12323, 5.12323, 
  2.08673}, {0.586871, 0.586871, -26.8678}, {1.97635, 1.97635, 
  17.4329}, {5.57199, 5.57199, -226.114}, {2.49095, 2.49095, 13.5429}}

f3 and the input form fI agree with each other, but f1 and f2 (where f2 is a mere rewriting of f1) do not agree with fI.