I find trouble in calculating the following integral:
$$ \int_0^R \frac{m\cdot x}{m+s\cdot x^a} \,dx $$
Mathematica does not provide an output for this function, however, there seems to be an output in the online http://integrals.wolfram.com/ tool for an upper limit to infinity. The result given in this tool is as follows: online integration
Any suggestions on the computation of this integral? Perhaps any suggestions for approximating the integral under specific assumptions for the values of m, s, a or R..
I am thinking that since the indefinite integral of this function equals to
$$ F(x)=\int \frac{m\cdot x}{m+s\cdot x^a} \,dx=x^2*\,_2F_1[1, 2/a, 1 + 2/a, -sx^a/m]/2 $$
Then, by taking F(R)-F(0) we get the result of the definite integral. Any remarks?
Result: $$ \int_0^R \frac{m\cdot x}{m+s\cdot x^a} \,dx=F(R)-F(0)=R^2*\,_2F_1[1, 2/a, 1 + 2/a, -sR^a/m]/2 $$