A weighted quadrature formula is called of Chebyshev type if it has equal coefficients and real (but not necessarily distinct) nodes. Among such quadrature rules we construct an optimal one, i. e., one which has maximum algebraic degree of accuracy and minimum error when applied to the first power not exactly integrated. Optimal quadrature rules, typically, have multiple nodes. Their construction requires the complete solution of systems of algebraic equations involving generalized power sums. Numerical results are presented for the case of constant weight function on a finite interval, as well as for weight functions of the Hermite and Laguerre type on infinite intervals.