For available “quadrature” routines see:
See also the “MathCore” and “MathMore” documentation.
In principle, ROOT internally often uses:
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for “special functions”: GSL → Special Functions
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for “quadrature”: GSL → Numerical Integration and GSL → Monte Carlo Integration
If you find some appropriate routine in GSL (GNU Scientific Library), you should be able to use it in ROOT.
BTW. The integrals over “dc” can be calculated analytically. In principle, one should use clean “analytical formulas” whenever possible.