Is something like this what you are after?
clear vars *;
% Define the function - Note I simplify the denominator in the last term
% Double check the numbers are typed correctly!
F =@(x) 2.906663106 .* x .* ( 1 ./(1 + 1.38 .* x.^4)).^0.4311594203...
- ( (3.458929096 .* x.^5) ./ (1 + 1.38 .* x.^4).^1.4311594203 );
% define teh Y values and the steps
Y = 0:0.01:20;
nSteps = length(Y);
% preallocate the integral values for speed
Ival = NaN(nSteps-1, 1);
% Do the loop
for ii = 1:nSteps-1
Ival(ii) = integral(F, Y(ii), Y(ii+1));
end
% Check it all adds up...
% The whole integral
integral(F, 0, 20)
% The sum of the parts
sum(Ival)
% Then add some plotting function here
