calculus and analysis - Symbolic integration error

fixed in 10.1 (windows)

I'm running Mathematica 10.0.0 and encountered a disturbing error in the symbolic integration of a rather simple function

Integrate[(1 - x)*(1 + 2*x)^6/Sqrt[1 - x^2], {x, -1, 1}]/Pi

The correct value for this integral is 15 (and NIntegrate gives that correctly) but Mathematica evaluates it symbolically as 1/π+29/2. I tried Wolfram Alpha, and it also gives the wrong answer. Any idea what is going on?

incorrect Wolfram Alpha symbolic evaluation of this integral

the correct answer is 15π=47.1239

splitting the integrand into two parts does give the correct answer 15,

Integrate[(1 + 2*x)^6/Sqrt[1 - x^2], {x, -1, 1}]/Pi - 

 Integrate[x*(1 + 2*x)^6/Sqrt[1 - x^2], {x, -1, 1}]/Pi

somehow Mathematica has difficulty with square root singularities in the integrand?

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