There is such an equation, but it's not easy to obtain. Because the magnus effect depends on the ball's current velocity, it changes when external forces are applied. You cannot calculate the position of the ball by linearly summing the contributions of different forces. If you were hoping you could simply add a few terms to \$x(t)=\frac{1}{2}at^2+vt\$, you're out of luck.
Numerical integration is a valid way of predicting the balls trajectory and doesn't require you to actually move the ball if you use a different set of variables. If you want to use a direct approach, you might want to use a tool that can solve a system of differential equations. I used Mathematica to obtain the following equations for a ball starting off with a set velocity and being subject to a gravitational acceleration -g in the z-direction and a magnus force \$F_{mag}=c\omega\times v\$, for brevity condensed into: \$a_{mag}=r\times v\$.
system = {
vx'[t]==ry vz[t]-rz vy[t],
vy'[t]==rz vx[t]-rx vz[t],
vz'[t]==rx vy[t]-ry vx[t]-g,
vx[0]==vx0,vy[0]==vy0,vz[0]==vz0
};
DSolve[system,{vx,vy,vz},t]
Integrate[%,t]
With a few tweaks, assumptions and some rewriting here and there, the result still looks dreadful, but is computationally not all that expensive.
Direct equations according to Mathematica:
$$\begin{align}
x(t) =&\frac{1}{r^3}(-\frac{1}{2}t(grtr_xr_z+2grr_y-2rr_xr_yv_y(0)-2rr_xr_zv_z(0)-2rr^2_xv_x(0))+\\
&sin(rt)(gr_y+r^2_yv_x(0)-r_xr_yv_y(0)+r^2_zv_x(0)-r_xr_zv_z(0))-\\
&\small{\frac{cos(rt)(gr_xr_z+r^2_xr_yv_z(0)-r^2_xr_zv_y(0)+r^3_yv_z(0)-r^2_yr_zv_y(0)+r_yr^2_zv_z(0)-r^3_zv_y(0))}{r})}\\
y(t)=&\frac{1}{r^3}(-\frac{1}{2}t(grtr_yr_z-2grr_x-2rr_xr_yv_x(0)-2rr_yr_zv_z(0)-2rr^2_yv_y(0))-\\
&sin(rt)(gr_x+r^2_x(-v_y(0))+r_xr_yv_x(0)-r^2_zv_y(0)+r_yr_zv_z(0))-\\
&\small{\frac{cos(rt)(gr_yr_z-r_xr^2_yv_z(0)+r^2_yr_zv_x(0)+r^3_x(-v_z(0))+r^2_xr_zv_x(0)-r_xr^2_zv_z(0)+r^3_zv_x(0))}{r})}\\
z(t)=&\frac{1}{r^3}(-\frac{1}{2}tr_z(grtr_z-2rr_xv_x(0)-2rr_yv_y(0)-2rr_zv_z(0))+\\
&\small{\frac{cos(rt)(gr^2_x+gr^2_y-r_xr^2_zv_y(0)+r_yr^2_zv_x(0)+
r^3_x(-v_y(0))+r^2_xr_yv_x(0)-r_xr^2_yv_y(0)+r^3_yv_x(0))}{r}}-\\
&sin(rt)(r^2_x(-v_z(0))+r_xr_zv_x(0)+r_yr_zv_y(0)-r^2_yv_z(0)))\end{align}$$
If you didn't think these equations were getting out of hand already, you most likely will if you were to add more forces to the model. Unless you feel the need for the accuracy and speed these equations can provide, I recommend sticking with code that fellow mortals can read and maintain.