Calculate the Ultraradical What is the Ultraradical?
The ultraradical, or the Bring radical, of a real number \$a\$ is defined as the only real root of the quintic equation \$x^5+x+a=0\$.
Here we use \$\text{UR}(\cdot)\$ to denote the ultraradical function. For example, \$\text{UR}(-100010)=10\$, since \$10^5+10-100010=0\$.
Challenge
Write a full program or a function, that takes a real number as input, and returns or outputs its ultraradical.
Requirements
No standard loopholes are allowed. The results for the test cases below must be accurate to at least 6 significant digits, but in general the program should calculate the corresponding values for any valid real number inputs.
Test Cases
9 decimal places rounded towards 0 are given for reference. Explanation is added for some of the test cases.
 a                         | UR(a)
---------------------------+---------------------
             0             |   0.000 000 000        # 0
             1             |  -0.754 877 (666)      # UR(a) &lt; 0 when a &gt; 0
            -1             |   0.754 877 (666)      # UR(a) &gt; 0 when a &lt; 0
             1.414 213 562 |  -0.881 616 (566)      # UR(sqrt(2))
            -2.718 281 828 |   1.100 93(2 665)      # UR(-e)
             3.141 592 653 |  -1.147 96(5 385)      # UR(pi)
            -9.515 716 566 |   1.515 71(6 566)      # 5th root of 8, fractional parts should match
            10             |  -1.533 01(2 798)
          -100             |   2.499 20(3 570)
         1 000             |  -3.977 89(9 393)
      -100 010             |  10.000 0(00 000)      # a = (-10)^5 + (-10)
 1 073 741 888             | -64.000 0(00 000)      # a = 64^5 + 64
    

Winning Criteria
The shortest valid submission in every language wins.
Wolfram Language (Mathematica), 20 bytes

Root[xx^5+x+#,1]&amp;


Try it online!

Still a built-in, but at least it isn't UltraRadical.

(the character  is displayed like |-&gt; in Mathematica, similar to =&gt; in JS)