#[Pyth](https://pyth.readthedocs.io), <s>20</s> 19 bytes

My current prefix approach (will hopefully golf further).

    +*tlQsQssM+_M.__Q._

**[Test Suite](http://pyth.herokuapp.com/?code=%2B%2atlQsQssM%2B_M.__Q._&input=%22123%22&test_suite=1&test_suite_input=%2212%22%0A%22101%22%0A%22123%22%0A%221234567%22&debug=0)** or try **[an alternative approach with the same byte count](http://pyth.herokuapp.com/?code=%2B%2alQsQssM%2B_MP.__Q._&input=%22123%22&test_suite=1&test_suite_input=%2212%22%0A%22101%22%0A%22123%22%0A%221234567%22&debug=0).**

#Explanation

    +*tlQsQssM+_M.__Q._  - Full program that reads a String from STDIN, with implicit input.

      tlQ                - Length of the input - 1.
         sQ              - The input converted to an integer.
     *                   - Product of the above two elements. We will call this P.
                     ._  - Prefixes of the input.
              +          - Concatenated with:
               _M.__Q    - The prefixes of the reversed input, reversed.
            sM           - Convert each to an integer.
           s             - Sum.
    +                    - Addition of the product P and the sum above.

To understand the concept better, we shall take an example, say `"123"`.

- We first get the prefixes of the input. Those are `['1', '12', '123']`.

- Then, we get the prefixes of the reversed input, i.e: `['3', '32', '321']` and reverse each, hence we get `['3', '23', '123']`.

- We concatenate the two lists and convert each element to an integer, so we obtain `[3, 23, 123, 1, 12, 123]`. 

- By summing the list, the result is `285`.

- The product `P` is the length of the input - 1 (i.e `2`) multiplied by the integer representation of it (`2 * 123 = 246`).

- In the end, we sum the two results: `285 + 246`, hence we obtain `531`, which is the correct result.

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#[Pyth](https://pyth.readthedocs.io), 20 bytes

    +*hlQsQsm+s>Qds<QdtU

**[Test Suite.](http://pyth.herokuapp.com/?code=%2B%2ahlQsQsm%2Bs%3EQds%3CQdtU&input=%22123%22&test_suite=1&test_suite_input=%221234567%22%0A%22123%22%0A%22101%22%0A%2212%22&debug=0)**

#Explanation

<s>Explanation to come after further golfing.</s> I didn't succeed to golf this further for now (I have ideas though).

    +*hlQsQsm+s>Qds<QdtUQ  - Full program. Reads from STDIN. Q means input, and is implicit at the end.

      hlQ                  - Length of the input + 1.
         sQ                - The input converted to an integer.
     *                     - Multiply the above. We'll call the result P.
            m         tUQ  - Map over [1...length of the input)
              s>Qd         - input[currentItem:] casted to an integer.
                  s<Qd     - input[:currentItem] casted to an integer.
             +             - Sum the above.
           s               - Sum the list.
    +                      - Add the sum of the list and P.