Infinite dimensions, not quite infinite volume

Calculus Level 5

\[\lim _{ x\rightarrow \infty }{ { \left( 1+6\int _{ 0 }^{ B }{ \frac { y{ \left( x+{ y }^{ 2 } \right) }^{ 2 } }{ { x }^{ 3 } } dy } \right) }^{ x } } ={ e }^{ 2016 }\] The real number \(B\) satisfies the equation above and can be expressed in the form \(\alpha \sqrt { \beta }\), with \(\alpha\) and \(\beta\) positive, root-free integers with a greatest common divisor of 2. Find \(\alpha+\beta\).

If you believe that the following limit goes to infinity for all real \(B\), submit your answer as 2016.

Details and Assumptions

\(e\) denotes Euler's number, the base of the natural logarithm.


This problem is original. The picture of the graph was produced from Desmos.
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