This integral can be found in *D. B. Owen (1980) A table of normal integrals, Communications in Statistics - Simulation and Computation, 9:4, 389-419*:
[![enter image description here][1]][1]

`BvN` denotes the bivariate normal probability function. 

Check in R:

    > a <- 2
    > b <- 3
    > w <- 5
    > f <- function(x) dnorm(x)*pnorm(a+b*x)
    > integrate(f, lower=-Inf, upper=w)
    0.7364551 with absolute error < 1.3e-06
    > 
    > rho <- -b/sqrt(1+b^2)
    > Sigma <- cbind(c(1,rho),c(rho,1))
    > mvtnorm::pmvnorm(upper=c(a/sqrt(1+b^2), w), sigma=Sigma)
    [1] 0.7364551
    attr(,"error")
    [1] 1e-15
    attr(,"msg")
    [1] "Normal Completion"

Alternatively, you can express this integral with the Owen $T$-function:

    > library(OwenQ)
    > 1/2*(pnorm(a/sqrt(1+b^2))  + pnorm(w) - 2*OwenT(w, (b*w+a)/w) - 2*OwenT(-a/sqrt(1+b^2), (a*b+w*(1+b^2))/a) - (a <= 0))
    [1] 0.7364551

  [1]: https://i.sstatic.net/fIOSQ.png