Adding across columns ignoring NA

I am having difficulty finding a solution to devising an R code to do the
following:

I have 5 numerical variables and I would like to create a new variable that
is the sum of those 5 variables. However, there are many NA values
throughout these 5 variables and everytime I run the following code

new_variable=var1+var2+var3+var4+var5

I get NA as the sum whenever one of those 5 variables are NA. I cannot
figure out a way to have new_variable represent the sum for only those
values that are not NA.

As an example,
if var1=3
var2=3
var3=NA
var4=NA
var5=2

I would like new_variable to be 8 but I keep getting NA and I have
unsuccessfully tried different methods to do so. I feel there is a simple
method to solve my problem but I am unaware of such. I would appreciate any
guidance!