A simple general expression for a definite integral Let
$$I_k = \int_0^{\frac{\pi}{2}} \sin x \cos^{2k} x \sqrt{1 + \sin x} \, dx,$$
where $k = 0,1,2,\ldots$. I wish to find a simple general expression for $I_k$ in terms of $k$. Simple here is the operative word.
Making a tangent half-angle substitution of $t = \tan \frac{x}{2}$ produces
$$I_k = 4\int_0^1 \frac{t(1 + t)^{2k + 1} (1 - t)^{2k}}{(1 + t^2)^{2k + \frac{5}{2}}} \, dt.$$
Trying to guess its general form one notices:
\begin{align*}
k = 0 : \quad I_0 &= 4\int_0^1 \frac{t(1 + t)}{(1 + t^2)^{\frac{5}{2}}} \, dt = 4 \left [\frac{(t - 1)(t^2 + t + 1)}{3(t^2 + 1)^{\frac{3}{2}}} \right ]_0^1 = \frac{4}{3}\\
k = 1 : \quad I_1 &= 4\int_0^1 \frac{t(1 + t)^3(1 - t)^2}{(1 + t^2)^{\frac{9}{2}}} \, dt = 4 \left [\frac{(t - 1)^3(11t^4 + 33t^3 + 52t^2 + 33t + 11)}{105(t^2 + 1)^{\frac{7}{2}}} \right ]_0^1 = \frac{44}{105}\\
k = 2 : \quad I_2 &= 4 \int_0^1 \frac{t(1 + t)^5(1 - t)^4}{(1 + t^2)^{\frac{13}{2}}} \, dt = 4\left [\frac{(1 - t)^5(211t^6 + 1055t^5 + 2593 t^4 + 3370t^2 + 1055t + 211)}{3465(t^2 + 1)^{\frac{11}{2}}} \right ]_0^1 = \frac{844}{3465}
\end{align*}
So it appears
$$I_k = 4 \left [\frac{(t - 1)^{2k + 1}p_k(t)}{a_k(t^2 + 1)^{2k + \frac{3}{2}}} \right ]_0^1,$$
where $a_k$ is a positive integer and $p_k(t)$ is a symmetric polynomial of degree $(2k + 2)$. But what is $a_k$ and $p_k(t)$?
Alternatively, a closed-form expression in terms of the hypergeometric function can be found. It is
$$I_k = \frac{2}{3\sqrt{2}} {}_2 F_1 \left (\frac{3}{2},-2k;\frac{5}{2};1 \right ) + \frac{1}{2^{2k + \frac{3}{2}}(2k+1)} {}_2F_1 \left (2k + 1,2k + \frac{5}{2};2k + 2; \frac{1}{2} \right ),$$
but this is hardly simple since ultimately $I_k$ is just a positive rational number. Perhaps this expression can be simplified in some way.
So my question is, can a simple expression for $I_k$ be found?

I think that we could simplify the problem letting
$$\sqrt{1+\sin(x)}=t \implies x=-\sin ^{-1}\left(1-t^2\right)\implies dx=\frac{2 t}{\sqrt{1-\left(1-t^2\right)^2}}\,dt$$ This makes
$$I_k=2\int_1^{\sqrt 2}\left(t^2-1\right) t^{2 k+1} \left(2-t^2\right)^{k-\frac{1}{2}}\,dt$$
$$\color{blue}{I_k=\frac{\sqrt{\pi }\, \Gamma (2 k+1)-4^k \Gamma \left(2 k+\frac{3}{2}\right) \left((2
   k-1) B_{\frac{1}{2}}\left(k+1,k-\frac{1}{2}\right)-4 k
   B_{\frac{1}{2}}\left(k+1,k+\frac{1}{2}\right)\right)}{\sqrt{2}\, \Gamma \left(2
   k+\frac{5}{2}\right)}}$$ which is valid $\forall k\geq 0$.
For integer $k$, this generates the sequence
$$\left\{\frac{4}{3},\frac{44}{105},\frac{844}{3465},\frac{7676}{45045},\frac{271372}
   {2078505},\frac{785244}{7436429},\frac{40395868}{456326325},\frac{343225244}{450
   8102925},\frac{3309796628}{49589132175},\cdots\right\}$$
These sequences are not found in $OEIS$.
