Pxy[u_, ax_, bx_, cx_, dx_, ay_, by_, cy_, 
        dy_] :=  {ax*u^3 + bx*u^2 + cx*u  + dx,  
        ay*u^3 + by*u^2 + cy*u + dy}

Pxy[[1]]

Pxy[u, ax, bx, cx, dx, ay, by, cy, dy][[1]]

Clear[u]

Derivative[
        1][Pxy[u, ax, bx, cx, dx, ay, by, cy, dy][[1]]]

P0 = {2, 2};
P1  =  {3, 4};
P2  =  {4, 1};
P3  =  {5, 2};

v  =  
        Solve[
{Pxy[0, ax, bx,  cx, dx, ay, by, cy, 
                dy]  ==  
              P0, 
Pxy[1/3, ax, bx,  cx, dx, ay, by, cy, 
                dy] == P1, 
 
            Pxy[2/3, ax, bx,  cx, dx, ay, by, cy, dy] == 
              P2, 
Pxy[1, ax, bx,  cx, dx, ay, by, cy, 
                dy] == P3}, 
{ax, bx,  cx, dx, ay, 
            by, cy, dy}];

Derivative[1][Pxy[u, ax, bx, cx, dx, ay, by, cy, dy][[1]] /. 
        v[[1]]]

Derivative[1][Pxy[u, ax, bx, cx, dx, ay, by, cy, dy][[2]] /. 
        v[[1]]]

g[u_] := 
      Sqrt[(Derivative[
                  1][Pxy[u, ax, bx, cx, dx, ay, by, cy, dy][[1]] /. 
                  v[[1]]] )^2 +  (Derivative[
                  1][Pxy[u, ax, bx, cx, dx, ay, by, cy, dy][[2]] /. 
                  v[[1]]])^2]

Integrate[g, {u, 0, 1}]

N[Integrate[
        Sqrt[(Derivative[
                    1][Pxy[u, ax, bx, cx, dx, ay, by, cy, dy][[1]] /. 
                    v[[1]]] )^2 +  (Derivative[
                    1][Pxy[u, ax, bx, cx, dx, ay, by, cy, dy][[2]] /. 
                    v[[1]]])^2], {u, 0, 1}]]

g[0.5]