I find it convenient to use cylindrical coordinates $(\phi,\rho,z)$ rather than spherical coordinates. I orient the $z$-axis along the vector $\vec{x}=x_0\hat{z}$. The desired integral is $$I=\int_{\mathbb{R}^{3}}\frac{e^{i|\vec{x}-\vec{r}|^2}}{1+|\vec{r}|}d^3 r=2\pi\int_{-\infty}^\infty dz\int_0^\infty \rho\,d\rho \,\frac{\exp[i(x_0-z)^2+i\rho^2]}{1+\sqrt{z^2+\rho^2}}.$$ I have not been able to evaluate this in closed form, but since the OP asks about the "existence" I consider instead an integral which decays even less rapidly, $$J=2\pi\int_{-\infty}^\infty dz\int_0^\infty \rho\,d\rho \,\frac{\exp[i(x_0-z)^2+i\rho^2]}{1+\rho},$$ i.e. in the denominator I replace $z^2+\rho^2$ by $\rho^2$. The integral $J$ is independent of $x_0$ and evaluates in terms of an exponential integral Ei and Fresnel sine and cosine integrals S,C: $$J=i\pi^2+ i \pi e^i \sqrt{\pi/2}\left[(1-i) \text{Ei}(-i)+2 \pi S\left(\sqrt{2/\pi}\right)+2\pi i C\left(\sqrt{2/\pi}\right)\right].$$

I find it convenient to use cylindrical coordinates $(\phi,\rho,z)$ rather than spherical coordinates. I orient the $z$-axis along the vector $\vec{x}=x_0\hat{z}$. The desired integral is $$I=\int_{\mathbb{R}^{3}}\frac{e^{i|\vec{x}-\vec{r}|^2}}{1+|\vec{r}|}d^3 r=2\pi\int_{-\infty}^\infty dz\int_0^\infty \rho\,d\rho \,\frac{\exp[i(x_0-z)^2+i\rho^2]}{1+\sqrt{z^2+\rho^2}}.$$ I have not been able to evaluate this in closed form, but since the OP asks about the "existence" I consider instead an integral which decays even less rapidly, $$J=2\pi\int_{-\infty}^\infty dz\int_0^\infty \rho\,d\rho \,\frac{\exp[i(x_0-z)^2+i\rho^2]}{1+\rho},$$ i.e. in the denominator I replace $z^2+\rho^2$ by $\rho^2$. The integral $J$ is independent of $x_0$ and evaluates in terms of an exponential integral Ei and Fresnel sine and cosine integrals S,C: $$J=i\pi^2+ i \pi e^i \sqrt{\pi/2}\left[(1-i) \text{Ei}(-i)+2 \pi S\left(\sqrt{2/\pi}\right)+2\pi i C\left(\sqrt{2/\pi}\right)\right].$$

The improper integral here is of the type $$\int_{-\infty}^\infty e^{iz^2}dz=(1+i)\sqrt{\pi/2},$$ discussed for example in this Physics.SE question.

I find it convenient to use cylindrical coordinates $(\phi,\rho,z)$ rather than spherical coordinates. I orient the $z$-axis along the vector $\vec{x}=x_0\hat{z}$. The desired integral is $$I=\int_{\mathbb{R}^{3}}\frac{e^{i|\vec{x}-\vec{r}|^2}}{1+|\vec{r}|}d^3 r=2\pi\int_{-\infty}^\infty dz\int_0^\infty \rho\,d\rho \,\frac{\exp[i(x_0-z)^2+i\rho^2]}{1+\sqrt{z^2+\rho^2}}.$$ I have not been able to evaluate this in closed form, but since the OP asks about the "existence" I consider instead an integral which decays even less rapidly, $$J=2\pi\int_{-\infty}^\infty dz\int_0^\infty \rho\,d\rho \,\frac{\exp[i(x_0-z)^2+i\rho^2]}{1+\rho},$$ i.e. in the denominator I replace $z^2+\rho^2$ by $\rho^2$. The integral $J$ is independent of $x_0$ and evaluates in terms of an exponential integral Ei and Fresnel sine and cosine integrals S,C: $$J=i\pi^2+ 2^{-1/2}i \pi ^{3/2}e^i \left[(1-i) \text{Ei}(-i)+2 \pi S\left(\sqrt{2/\pi}\right)+2\pi i C\left(\sqrt{2/\pi}\right)\right].$$

I find it convenient to use cylindrical coordinates $(\phi,\rho,z)$ rather than spherical coordinates. I orient the $z$-axis along the vector $\vec{x}=x_0\hat{z}$. The desired integral is $$I=\int_{\mathbb{R}^{3}}\frac{e^{i|\vec{x}-\vec{r}|^2}}{1+|\vec{r}|}d^3 r=2\pi\int_{-\infty}^\infty dz\int_0^\infty \rho\,d\rho \,\frac{\exp[i(x_0-z)^2+i\rho^2]}{1+\sqrt{z^2+\rho^2}}.$$ I have not been able to evaluate this in closed form, but since the OP asks about the "existence" I consider instead an integral which decays even less rapidly, $$J=2\pi\int_{-\infty}^\infty dz\int_0^\infty \rho\,d\rho \,\frac{\exp[i(x_0-z)^2+i\rho^2]}{1+\rho},$$ i.e. in the denominator I replace $z^2+\rho^2$ by $\rho^2$. The integral $J$ is independent of $x_0$ and evaluates in terms of an exponential integral Ei and Fresnel sine and cosine integrals S,C: $$J=i\pi^2+ i \pi e^i \sqrt{\pi/2}\left[(1-i) \text{Ei}(-i)+2 \pi S\left(\sqrt{2/\pi}\right)+2\pi i C\left(\sqrt{2/\pi}\right)\right].$$