A side observation...
If one computes
theIntegral = Integrate[1/Sqrt[2 (1 + x)^3 + 1], x]
and then computes
theDerivative = D[theIntegral, x] // FullSimplify
which yields
-(((-1)^(5/6) Sqrt[3] Sqrt[
1 + 2 (1 + x)^3])/(Sqrt[-(-1)^(5/6) (1 + 2^(1/3) + 2^(1/3) x)]
Sqrt[(-1)^(5/6) (-1 + (-2)^(1/3) (1 + x))] Sqrt[
1 + (-2)^(1/3) (1 + x) + (-2)^(2/3) (1 + x)^2] Sqrt[
3 - (-1)^(1/6) Sqrt[3] + (1 + x) Root[108 + #1^6 &, 1]] Sqrt[
3 + (-1)^(5/6) Sqrt[3] + (1 + x) Root[108 + #1^6 &, 2]]))
and then attempts to Plot this using
Plot[-(((-1)^(5/6) Sqrt[3] Sqrt[
1 + 2 (1 + x)^3])/(Sqrt[-(-1)^(5/6) (1 + 2^(1/3) + 2^(1/3) x)]
Sqrt[(-1)^(5/6) (-1 + (-2)^(1/3) (1 + x))] Sqrt[
1 + (-2)^(1/3) (1 + x) + (-2)^(2/3) (1 + x)^2] Sqrt[
3 - (-1)^(1/6) Sqrt[3] + (1 + x) Root[108 + #1^6 &, 1]] Sqrt[
3 + (-1)^(5/6) Sqrt[3] + (1 + x) Root[108 + #1^6 &, 2]])) , {x, 0, 1}]
the result is that it crashes the Mathematica Kernel. (I will report this.)
In[1]:= SystemInformation["Small"]
Out[1]= {"Kernel" -> {"SystemID" -> "MacOSX-x86-64",
"ReleaseID" -> "10.0.0.0 (5098698, 5098537)",
"CreationDate" ->
DateObject[{2014, 6, 29}, TimeObject[{20, 38, 32}]]},
"FrontEnd" -> {"OperatingSystem" -> "MacOSX",
"ReleaseID" -> "10.0.0.0 (5098698, 2014062704)",
"CreationDate" ->
DateObject[{2014, 6, 27}, TimeObject[{23, 17, 40.}]]}}