without data my guess is, that the columns you are using are not numeric. Then it will be hard to calculate the rowsum. Make sure, that columns you use for summing (except 1:5) are indeed numeric, then the following code should work:
library(tidyverse)
df2 <- df1[,-c(1:5)] %>%
rowwise() %>%
mutate(rowsum = sum(c_across(everything()), na.rm = T))
df_result <- cbind(df1[,c(1:5)], df2$rowsum)
EDIT: I added na.rm = T (dont know if necessary). And you might want to rename the resulting "df2$rowsum" column of the resulting df_result dataframe this can be done using
df_result <- df_result %>% rename(rowsum_name = "df2$rowsum")