Does this telescope?

Calculus Level 5

For all \( n \) let \(\ell_n \) denote \( \displaystyle\int_{0}^{1} {\frac{x^n}{\sqrt{x^3+1}}} \). Suppose that \( m = \underbrace{2014...2014}_{\text{2014 times}} \), evaluate to 3 decimal digits the value of \( (2m-1)\ell_m + 2(m-2) \ell_{m-3} \).

\( \tiny \text{This is not original but I cannot find the source. If you know its source, please tag me in the comments. Thank you!} \)

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