You can use the tee
command to redirect output:
/usr/bin/mysqldump -u user -pupasswd my-database | \
tee >(gzip -9 -c > /home/user/backup/mydatabase-backup-`date +\%m\%d_\%Y`.sql.gz) | \
gzip> /home/user/backup2/mydatabase-backup-`date +\%m\%d_\%Y`.sql.gz 2>&1
see documentation here