I'm wondering whether anyone knows a reference or proof for finding the Fourier transform of $f(t):=(t+1)^{1/2}t_+^{1/2}$? (Here $t_+=\max (t,0)$.)

$$\tfrac{1}{2}\int_{0}^{\infty}e^{i\omega t}dt=\tfrac{1}{2}\pi\delta(\omega)+\tfrac{1}{2}i\omega^{1}$$ $$\int_{0}^{\infty}te^{i\omega t}dt=i\frac{d}{d\omega}\int_{0}^{\infty}e^{i\omega t}dt=i\pi\frac{d}{d\omega}\delta(\omega)\omega^{2}$$ $$\int_{0}^{\infty}[\sqrt{t(1+t)}t\tfrac{1}{2}]e^{i\omega t}dt=\omega^{2}\tfrac{1}{2}i\omega^{1}+\tfrac{1}{2}i\omega^{1}e^{i\omega/2}K_{1}(i\omega/2) $$ adding these three results gives the required Fourier transform $$ \int_{0}^{\infty}\sqrt{t(1+t)}e^{i\omega t}dt=\tfrac{1}{2}\pi\delta(\omega)i\pi\frac{d}{d\omega}\delta(\omega)+\tfrac{1}{2}i\omega^{1}e^{i\omega/2}K_{1}(i\omega/2) $$ with $K_1$ a modified Bessel function of the second kind; the derivative of the Dirac delta function should be understood in the context of an integral, $$\int_{\infty}^{\infty}f(\omega)\frac{d}{d\omega}\delta(\omega)d\omega=\lim_{\omega\rightarrow 0}\frac{d}{d\omega}f(\omega)$$ 

