This bash script will compile a static Ffmpeg build with NVENC and VAAPI hardware-accelerated support on Ubuntu in your home directory. You can modify the script to customize the build options as you see fit.

compile-ffmpeg-nvenc.sh

#!/bin/bash

# Changed to work with Ubuntu 19.04 from  Brainiarc7 script

#install required things from apt
installLibs(){
echo "Installing prerequisites"
sudo apt-get update
sudo apt-get -y --force-yes install autoconf automake build-essential libass-dev libfreetype6-dev libgpac-dev \
  libsdl1.2-dev libtheora-dev libtool libva-dev libvdpau-dev libvorbis-dev libxcb1-dev libxcb-shm0-dev \
  libxcb-xfixes0-dev pkg-config texi2html zlib1g-dev
}

#install CUDA SDK
InstallCUDASDK(){
echo "Installing CUDA and the latest driver repositories from repositories"
cd ~/ffmpeg_sources
wget -c -v -nc https://developer.download.nvidia.com/compute/cuda/repos/ubuntu1604/x86_64/cuda-repo-ubuntu1604_9.2.88-1_amd64.deb
sudo dpkg -i cuda-repo-ubuntu1604_9.2.88-1_amd64.deb
sudo apt-key adv --fetch-keys http://developer.download.nvidia.com/compute/cuda/repos/ubuntu1604/x86_64/7fa2af80.pub
sudo apt-get -y update
sudo apt-get -y install cuda
sudo add-apt-repository ppa:graphics-drivers/ppa
sudo apt-get update && sudo apt-get -y upgrade
}

#Install nvidia SDK
installSDK(){
echo "Installing the nVidia NVENC SDK."
cd ~/ffmpeg_sources
cd ~/ffmpeg_sources
git clone https://git.videolan.org/git/ffmpeg/nv-codec-headers.git
cd nv-codec-headers
make
sudo make install
}

#Compile nasm
compileNasm(){
echo "Compiling nasm"
cd ~/ffmpeg_sources
wget http://www.nasm.us/pub/nasm/releasebuilds/2.14rc0/nasm-2.14rc0.tar.gz
tar xzvf nasm-2.14rc0.tar.gz
cd nasm-2.14rc0
./configure --prefix="$HOME/ffmpeg_build" --bindir="$HOME/bin"
make -j$(nproc)
make -j$(nproc) install
make -j$(nproc) distclean
}

#Compile libx264
compileLibX264(){
echo "Compiling libx264"
cd ~/ffmpeg_sources
wget http://download.videolan.org/pub/x264/snapshots/last_x264.tar.bz2
tar xjvf last_x264.tar.bz2
cd x264-snapshot*
PATH="$HOME/bin:$PATH" ./configure --prefix="$HOME/ffmpeg_build" --bindir="$HOME/bin" --enable-static
PATH="$HOME/bin:$PATH" make -j$(nproc)
make -j$(nproc) install
make -j$(nproc) distclean
}

#Compile libfdk-acc
compileLibfdkcc(){
echo "Compiling libfdk-cc"
sudo apt-get install unzip
cd ~/ffmpeg_sources
wget -O fdk-aac.zip https://github.com/mstorsjo/fdk-aac/zipball/master
unzip fdk-aac.zip
cd mstorsjo-fdk-aac*
autoreconf -fiv
./configure --prefix="$HOME/ffmpeg_build" --disable-shared
make -j$(nproc)
make -j$(nproc) install
make -j$(nproc) distclean
}

#Compile libmp3lame
compileLibMP3Lame(){
echo "Compiling libmp3lame"
sudo apt-get install nasm
cd ~/ffmpeg_sources
wget http://downloads.sourceforge.net/project/lame/lame/3.99/lame-3.99.5.tar.gz
tar xzvf lame-3.99.5.tar.gz
cd lame-3.99.5
./configure --prefix="$HOME/ffmpeg_build" --enable-nasm --disable-shared
make -j$(nproc)
make -j$(nproc) install
make -j$(nproc) distclean
}

#Compile libopus
compileLibOpus(){
echo "Compiling libopus"
cd ~/ffmpeg_sources
wget http://downloads.xiph.org/releases/opus/opus-1.2.1.tar.gz
tar xzvf opus-1.2.1.tar.gz
cd opus-1.2.1
./configure --prefix="$HOME/ffmpeg_build" --disable-shared
make -j$(nproc)
make -j$(nproc) install
make -j$(nproc) distclean
}

#Compile libvpx
compileLibPvx(){
echo "Compiling libvpx"
cd ~/ffmpeg_sources
git clone https://chromium.googlesource.com/webm/libvpx
cd libvpx
PATH="$HOME/bin:$PATH" ./configure --prefix="$HOME/ffmpeg_build" --disable-examples --enable-runtime-cpu-detect --enable-vp9 --enable-vp8 \
--enable-postproc --enable-vp9-postproc --enable-multi-res-encoding --enable-webm-io --enable-better-hw-compatibility --enable-vp9-highbitdepth --enable-onthefly-bitpacking --enable-realtime-only \
--cpu=native --as=nasm
PATH="$HOME/bin:$PATH" make -j$(nproc)
make -j$(nproc) install
make -j$(nproc) clean
}

#Compile ffmpeg
compileFfmpeg(){
cd ~/ffmpeg_sources && \
wget -O ffmpeg-snapshot.tar.bz2 https://ffmpeg.org/releases/ffmpeg-snapshot.tar.bz2 && \
tar xjvf ffmpeg-snapshot.tar.bz2 && \
cd ffmpeg && \
PATH="$HOME/bin:$PATH" PKG_CONFIG_PATH="$HOME/ffmpeg_build/lib/pkgconfig" ./configure \
  --prefix="$HOME/ffmpeg_build" \
  --pkg-config-flags="--static" \
  --extra-cflags="-I$HOME/ffmpeg_build/include" \
  --extra-ldflags="-L$HOME/ffmpeg_build/lib" \
  --bindir="$HOME/bin" \
  --enable-cuda-sdk \
  --enable-cuvid \
  --enable-libnpp \
  --extra-cflags="-I/usr/local/cuda/include/" \
  --extra-ldflags=-L/usr/local/cuda/lib64/ \
  --enable-gpl \
  --enable-libass \
  --enable-libfdk-aac \
  --enable-vaapi \
  --enable-libfreetype \
  --enable-libmp3lame \
  --enable-libopus \
  --enable-libtheora \
  --enable-libvorbis \
  --enable-libvpx \
  --enable-libx264 \
  --enable-nonfree \
  --enable-nvenc
PATH="$HOME/bin:$PATH" make -j$(nproc)
make -j$(nproc) install
make -j$(nproc) distclean
hash -r
}

#The process
cd ~
mkdir ffmpeg_sources
installLibs
InstallCUDASDK
installSDK
compileNasm
compileLibX264
compileLibfdkcc
compileLibMP3Lame
compileLibOpus
compileLibPvx
compileFfmpeg
echo "Complete!"

Tried to use your script on stock Xubuntu 16.04.3 ("stock" meaning I never run sudo apt-get upgrade or dist upgrade, only sudo apt-get update)
Doesn't work, up until the last steps, which are for building the ffmpeg executable itself.
Here are the last output lines of the terminal, can you perhaps guess where it is failing? (I don't have experience with compiling)

(edited out as seen below...)

It seems to be stuck at the last part (build ffmpeg) so I cut out all the rest from the script, and even tried to run this command line here manually, but strangely enough even ./configure refuses to work, as seen below:

(edited out as seen below...)

unbelievable... it was only a matter of including "sudo" before each command inside the script... it is running ./configure now, waiting to see if it will succeed

stopped with error right after creating files in "Docs" directory (at the last steps I suppose)
Tried 3 different automated scripts until now, have all the dependencies installed for all of them (including the ones referred at the official ffmpeg site) and can't make this thing work. "It's easy" they said. "Linux is cool" they said. XDDD