# Definite integral including the ratio and power functions of a single variable

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..

• If $a$ is an integer, you should be able to do it by using partial fractions. – Raskolnikov Apr 24 '13 at 7:45
• Since you're interested in an approximation, what are the relative sizes of $m,s,a,R$? – Antonio Vargas Apr 24 '13 at 15:12

$$F(x)=\int \frac{m\cdot x}{m+s\cdot x^a} \,dx=x^2*\,_2F_1[1, 2/a, 1 + 2/a, -sx^a/m]/2$$
Result: $$\int_0^R \frac{m\cdot x}{m+s\cdot x^a} \,dx=F(R)-F(0)=R^2*\,_2F_1[1, 2/a, 1 + 2/a, -sR^a/m]/2$$