Proofs of McIntosh's conjecture on Franel integrals and two generalizations

We provide a proof of conjecture made by Richard McIntosh in 1996 on the values Franel integrals, $$\int_0^1((ax))((bx))((cx))((ex))\,dx,$$ where $((x))$ is first periodic Bernoulli function. Secondly, we extend our ideas to prove similar theorem for $$\int_0^1((a_1x))((a_2x))\cdots ((a_{n}x))\,dx.$$ Lastly, further generalization which replaced any particular function with odd index.