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The code shown below is based on code from Alfred Gray's _Differential
Geometry of Curves and Surfaces with Mathematica_ 2nd Edition.

\!\(scp[t_] := {t\^2, t\^3}\[IndentingNewLine]
  \(alenp[alpha_]\)[t_] := 
    Sqrt[Simplify[D[alpha[tt], tt] . D[alpha[tt], tt]]] /. 
      tt -> t\[IndentingNewLine]
  \(alen[a_, b_]\)[alpha_] := 
    Integrate[\(alenp[alpha]\)[u], {u, a, b}]\[IndentingNewLine]
  Simplify[
    PowerExpand[\[IndentingNewLine]\(alen[a, b]\)[scp] /. 
        Sqrt[x_] :> Sqrt[Factor[x]]\[IndentingNewLine]]]\)

(-a + b)*If[Re[a/(a - b)] >= 1 || Re[a/(-a + b)] >= 
    0 || Im[a/(-a + b)] != 0, 
  ((a^((2*3)/2)*(4 + 9*a^2)^(3/2))*b^3 - 
    a^3*(b^((2*3)/2)*(4 + 9*b^2)^(3/2)))/
   (27*a^3*(a - b)*b^3), Integrate[
   Sqrt[Factor[(a - a*u + b*u)^2*
      (4 + 9*(a - a*u + b*u)^2)]], {u, 0, 1}, 
   Assumptions -> Im[a/(-a + b)] == 0 && 
     Re[a/(a - b)] < 1 && Re[a/(-a + b)] < 0]]

The result Gray shows is only the one satisfying the first of the above
conditions, and the constraints are not shown at all.  That suggests to me
that something has changed in Mathematica since he wrote the book.  I
believe the result from the newer version is more correct.  Nonetheless, I
would like to know how to get the old behavior if possible.  Is there a way
to do this?
-- 
The Mathematica Wiki: http://www.mathematica-users.org/
Math for Comp Sci http://www.ifi.unizh.ch/math/bmwcs/master.html
Math for the WWW: http://www.w3.org/Math/


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