title: Use API Gateway as http proxy with uri (request path+variable query params) passthrough
body: <p>I'm trying to do some vary basic routing with API Gateway.
  I need to achieve the following scenario:</p>
  
  <ul>
  <li><p>user makes request xxxx-execute-api.eu-west-1.amazonaws.com/prod/<strong>api1</strong>/a/b/../n?param1=val1&amp;parma2=val2...&amp;paramn=valn request should go to <strong>api1</strong>.back.end/a/b/../n?param1=val1&amp;parma2=val2...&amp;paramn=valn</p></li>
  <li><p>user makes request xxxx-execute-api.eu-west-1.amazonaws.com/prod/<strong>api2</strong>/a/b/../n?param1=val1&amp;parma2=val2...&amp;paramn=valn request should go to <strong>api2</strong>.back.end/a/b/../n?param1=val1&amp;parma2=val2...&amp;paramn=valn</p></li>
  <li><p>user makes request xxxx-execute-api.eu-west-1.amazonaws.com/prod/*****/a/b/../n?param1=val1&amp;parma2=val2...&amp;paramn=valn request should go to <strong>api3</strong>.back.end/a/b/../n?param1=val1&amp;parma2=val2...&amp;paramn=valn</p></li>
  </ul>
  
  <p>The routing should be done based on first path index after stage, and everything else after that should be passed to the http backend (like a transparent proxy).</p>
  
  <p>In other words, if path index 1 is <strong>api1</strong>, forward request to <strong>api1</strong>.back.end with full URI after path index 1;   if path index 1 is <strong>api2</strong>, forward request to <strong>api2</strong>.back.end with full URI after path index 1; if path index 1 is <strong>anything else</strong> than the explicit values api1 or api2, forward request to <strong>api3</strong>.back.end with full URI after path index 1;</p>
  
  <p>How would I achieve this, without adding any extra layers (lambda, cloudfront, ec2, etc.) ?</p>
  
  <p>Thank you!</p>