help solving "unsolvable" definite integral describing aerodynamic property I need some urgent help with this particular integral:
\begin{equation}
\int_{0}^{b} \frac{r^3 \sqrt{1-\frac{r}{b}}}{\sqrt{1 + k^2 r^2}} dr
\end{equation}
where k and b are a constant. 
This is an integral that helps describing the forces along an helicopter blade, and it would be amazing if I could find a solution to this integral, as it would solve my system of equations and allow me to size all parameters accordingly. 
I tried to solve it first with my graphical calculator, then moved on to wolfram alpha, but none were able to write me a solution. I tried to solve it by hand, but it never ended, and I tried to prepare a numerical way to solve it, but had the problem that I couldnt pull out the k constant outside of the integral. 
Any help or suggestions would be greatly appreciated!
Cheers!
 A: The first thing to do is reduce the problem from one with two parameters ($b$ and $k$) to one with a single parameter, say $a=bk$, by the substitution $x=r/b$. Thus: $$\int_{0}^{b} \frac{r^3 \sqrt{1-r/b}}{\sqrt{1 + k^2 r^2}} \,\mathrm dr=b^4\!\!\int_{0}^{1} \frac{x^3 \sqrt{1-x}}{\sqrt{1-a^2x^2}}\,\mathrm dx.$$An alternative, slightly simpler-looking, form of this integral can be got by substituting $x=a^{-1}\sin\theta$, which yields $$\frac1{k^4\surd a}\int_0^{\arcsin a}\sqrt{a-\sin\theta}\,\mathrm d\theta.$$Any practical evaluation of such an integral requires numerical approximation, and the simplest way to do it is by a standard numerical integration method applied to the (single-parameter) integral for various appropriate numerical choices of the parameter $a$. 
A: I am not sure what is going on (apparently you have had a typo in the post now corrected), but W|A actually does find a primitive:

Since you integrate from $0$ to $b$, just substitute $r=b$ and you are done.
A: I really do not know what you could be doing with this monster : using 
dbanet's answer, using integration bounds and simplifying you have $$
\int_{0}^{b} \frac{r^3 \sqrt{1-\frac{r}{b}}}{\sqrt{1 + k^2 r^2}} dr
=\frac{32}{315} b^4 \,
   _3F_2\left(\frac{1}{2},2,\frac{5}{2};\frac{11}{4},\frac{13}{4};-b^2 k^2\right)$$ where appears the generalized hypergeometric function.
If $kb$ is small, we can develop some expansions $$\, _3F_2\left(\frac{1}{2},2,\frac{5}{2};\frac{11}{4},\frac{13}{4};-x\right)=1-\frac{40 x}{143}+\frac{336 x^2}{2431}-\frac{3840 x^3}{46189}+\frac{5376
   x^4}{96577}-\frac{43008 x^5}{1077205}+O\left(x^6\right)$$
I wish you a lot of fun !
Back to serious, as John Bentin answered, I think that numerical methods would be the most appropriate approach.
