What is the value of this Integration?

Question

What is the value of this : $$\int_{0}^{1}x^{{(\frac{1}{\sum_{a=0}^{\infty}\sum_{b=0}^{\infty}\sum_{c=0}^{\infty}\frac{1}{(a+b+c)!}})}x^{(\frac{1}{\sum_{a=0}^{\infty}\sum_{b=0}^{\infty}\sum_{c=0}^{\infty}\frac{1}{(a+b+c)!}})}}dx$$
My work: we have
(1)..$$\sum_{a=0}^{\infty}\sum_{b=0}^{\infty}\sum_{c=0}^{\infty}\frac{1}{(a+b+c)!}=\frac{7e}{2}$$ and : $$\int_{0}^{1}x^{\alpha x^{\beta}}dx=\sum_{n=0}^{\infty}\frac{\alpha^n(-1)^n}{{(\beta n+1)}^{n+1}}$$
So $$\int_{0}^{1}x^{{(\frac{1}{\sum_{a=0}^{\infty}\sum_{b=0}^{\infty}\sum_{c=0}^{\infty}\frac{1}{(a+b+c)!}})}x^{(\frac{1}{\sum_{a=0}^{\infty}\sum_{b=0}^{\infty}\sum_{c=0}^{\infty}\frac{1}{(a+b+c)!}})}}dx=\int_{0}^{1}x^{(\frac{2}{7e}) x^{(\frac{2}{7e})}}dx=\sum_{n=0}^{\infty}\frac{(\frac{2}{7e})^n(-1)^n}{{((\frac{2}{7e}) n+1)}^{n+1}}$$
But$$\sum_{n=0}^{\infty}\frac{(\frac{2}{7e})^n(-1)^n}{{((\frac{2}{7e}) n+1)}^{n+1}}=?$$
You can help calculate the series or find an integral value in another way