@Markiyan Hirnyk  Obviously the reason is in  rational  option. Examples show that in a solidly filled region the function takes complex values.
Look at these simple examples:

plots:-implicitplot(sqrt(-x^2-y^2), x=0..1,y=0..1);
plots:-implicitplot(sqrt(-x^2-y^2), x=0..1,y=0..1, rational);
plots:-implicitplot(sqrt(-x^2-y^2), x=0..1,y=0..1, gridrefine=2, rational);
 

@nm  The reason is probably in this:

whattype(1/sqrt(x));
op(1/sqrt(x));
                                                

A workaround:

expr:=1/sqrt(x)+sqrt(x);    
subs([x^(1/2)=abs(x)^(1/2), x^(-1/2)=(abs(x))^(-1/2)], expr);
                                           

 

@digerdiga  Yes. Here is an example:

simplify(sqrt((R-sqrt(R^2+z^2))^2));
simplify(sqrt((R-sqrt(R^2+z^2))^2))  assuming R-sqrt(R^2+z^2)<=0;
                              

@davtrs  This works. I added different colors to your lines.

Download Plot_problem_new.mw

@Carl Love I just thought  D(x)  was just a name, for example:

D(x):=5:
D(x);
                                     5

@vv 

In the original code I replaced  0.5  by  1/2 :
is(D(x)=DD(x));
                                        true

@whtitefall  Due to the parity of the integrand function, the answer to your question can be given without any computation  x=0 . Unfortunately, it seems that you have not understood my answer at all and again you have a lot of mistakes in your code.

@kuwait1  If we plot the expression in the denominator (without a power), it is clearly seen that it takes not only positive but also negative and zero values. Therefore, you should think about the adequacy of this model.

restart;
a := 2: Q := 5.48: d := 2.48: chi := (1/4)*Pi: phi := 0.: p := 5/2:
Expr:=(a*sin(X)*sin(phi)+d*cos(chi))^2+(a*cos(X)+d*sin(chi))^2+2*Q*((a*sin(X)*sin(phi)+d*cos(chi))^2+(a*cos(X)+d*sin(chi))^2)^(1/2)*cos(beta);
plot3d(Expr, beta=0..2*Pi, X=0..Pi);

@Markiyan Hirnyk  Where is your code to get this result?

@Markiyan Hirnyk  The point was  for  p=7/2  not for  p=1

@nm  It's just a way to programmatically code some symbols and combine them. Of course, in the above case, we can write down  ell  instead of  `&ell;` , but in other cases this method will be useful. Here are a few examples:

`&Delta;x`;
`&pm;`;
`&pm;3`;
30^`&deg;`;
                                              

@Mariusz Iwaniuk  convert(n, 'base', beta)  command returns the digits in reverse order.
                             

@mmcdara  In fact, these syntaxes are correct and they are documented. See the help on these commands.

@Carl Love  for the useful information on the binary search.

@Carl Love  I did not understand the meaning of your comment. Did you mean that the procedure should check that the given list already is sorted and produce an error message otherwise? Or something else?