I have been using Mathematica 10 for a while now and noticed that the time required for multiple integrals seems to be excessive.
In[2]:= Timing[
Integrate[(x^2 + y^2)/a, {y, 0, a / Sqrt[2]}, {x, y,
Sqrt[a^2 - y^2]}] // PowerExpand // Simplify]
Out[2]= {21.903075, (a^3 \[Pi])/16}
In[3]:= Timing[
Integrate[
Integrate[(x^2 + y^2)/a, {x, y, Sqrt[a^2 - y^2]}], {y, 0,
a / Sqrt[2]}] // PowerExpand // Simplify]
Out[3]= {1.542915, (a^3 \[Pi])/16}
In[4]:= Timing[
g[y_] = Integrate[Integrate[(x^2 + y^2)/a, {x, y, Sqrt[a^2 - y^2]}],
y]; (g[a/Sqrt[2]] - g[0]) // PowerExpand // Simplify]
Out[4]= {0.048080, (a^3 \[Pi])/16}
Apparently, using one call to integrate a double integral takes 14 times as long as calling integrate twice, and integrating to a function and subsequently evaluating that function at the limits is 32 times faster than giving the limits to Integrate.
Seems like there is a bug there.