Is there closed formula to calculate the probability of beta distribution?

In general, we need a special function for $P(Xle x)$. However, if $a$ and/or $b$ is a positive integer, we can obtain a closed-form solution (indeed, if both are positive integers it's a polynomial). Suppose $a$ is an integer:$$begin{align}int_0^x(1-t)^{b-1}dt&=frac{1-(1-x)^b}{b},\int_0^xt^{c+1}(1-t)^{b-1}dt&=int_0^xt^c(1-t)^{b-1}dt-int_0^xt^c(1-t)^bdt.end{align}$$Further, the normalization constant that ensures probabilities sum to $1$ will still be a value of the Beta function. Suppose $a$ is an integer:$$operatorname{B}(1,,b)=frac1b,,frac{operatorname{B}(c+1,,b)}{operatorname{B}(c,,b)}=frac{c}{b+c}.$$