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See the documentation of integrate:

To prevent automatic evaluation use the hold argument.

Indeed,

sage: var('alpha beta x')
sage: denom = integral(x**(alpha-1) * (1-x)**(beta-1), x, hold=True)
sage: diff(denom,x)
x^(alpha - 1)*(-x + 1)^(beta - 1)

See the documentation of integrateintegral:

To prevent automatic evaluation use the hold argument.

Indeed,

sage: var('alpha beta x')
sage: denom = integral(x**(alpha-1) * (1-x)**(beta-1), x, hold=True)
sage: diff(denom,x)
x^(alpha - 1)*(-x + 1)^(beta - 1)

See the documentation of integral:

To prevent automatic evaluation use the hold argument.

Indeed,

sage: var('alpha beta x')
sage: denom = integral(x**(alpha-1) * (1-x)**(beta-1), x, hold=True)
hold=True); denom
integrate(x^(alpha - 1)*(-x + 1)^(beta - 1), x)
sage: diff(denom,x)
x^(alpha - 1)*(-x + 1)^(beta - 1)