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

Benchmark:

    > library(mvtnorm)
    > library(OwenQ)
    > library(microbenchmark)
    > 
    > a <- 2
    > b <- 3
    > w <- 1
    > 
    > microbenchmark(
    +   integral = integrate(function(x) dnorm(x)*pnorm(a+b*x), lower=-Inf, upper=w),
    +   mvtnorm = {rho <- -b/sqrt(1+b^2); pmvnorm(upper=c(a/sqrt(1+b^2), w), sigma=cbind(c(1,rho),c(rho,1)))},
    +   OwenT = 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))
    + )
    Unit: microseconds
         expr     min       lq      mean   median       uq      max neval cld
     integral  80.677  83.5860 116.97275  90.0240  93.0625 2878.062   100  b 
      mvtnorm 320.550 327.0625 339.22625 330.3975 336.0315  595.829   100   c
        OwenT  22.682  24.6360  28.89006  29.2685  31.9955   51.015   100 a  

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