Tribute to Ramanujan on his upcoming birthday!

Calculus Level 5

\[\large \large \large \large \large \large \large \large \large \large \int _{ 0 }^{ \infty }{ \left[ \frac { { \left( \sqrt { 1729x+11957764+x\sqrt { 1729\left( x+1729 \right) +11957764+\left( x+1729 \right) \sqrt { \ldots } } } -3458 \right) }^{ \frac { -1 }{ 2 } }\left( \gamma x+ln\Gamma \left( 1+x \right) \right) }{ { x }^{ 2 } } \right] } dx\]

Evaluate the integral above. Take \(\zeta \left( \frac { 3 }{ 2 } \right) =2.612\).

Extra Credits:

  • Find a general solution to the problem.

  • The first person who posts the solution explaining all the steps wins my respect.

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