In certain tasks, I need to find all accurate positive (not just nonnegative) roots that exist of some multivariate polynomials like: 

nsd := 16*a*b*c*(9 + a^2 + b^2 + c^2)*(b*c + a*(b + c) + 3*(a + b + c)) - (3 + a + b + c)^2*(a*b + 3*c)*(3*b + a*c)*(3*a + b*c): # assume((a, b, c) >~ 0);

According to fsolve/details, for one general equation, the fsolve command only computes "a single real root", so it is inadequate to tackle this question. But if I use the solve command with floating-point values, the computation cannot finish in ten minutes instead! (Maybe a longer time will suffice, yet this is rather unacceptable.) 

Download solve_numerically.mws

Is there any workaround to obtain those (finitely many) positive solutions completely?

In practice, I need calculate the (principal) squareroot of some suitable large matrix exactly (so the desired result should not involve floating-point numbers, otherwise the decomposition will be of no theoretical value as a certificate …). But I find it difficult to do so in Maple. (What about the Efficient Computations - Maple Help (maplesoft.com)?)
Below are some matrices: (They are not contrived academic examples.) 

M__4 := <1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0|0,3,0,-6,0,3,0,0,0,0,0,0,0,0,0,0,-3,0,3,0,-3,0,0,0,0,0,0,0,0,6,0,0,0,0,-3,0|-2,0,7,0,-8,0,3,0,0,0,0,0,0,0,0,1,0,-7,0,2,0,0,0,0,0,0,4,0,3,0,0,0,0,-3,0,0|0,-6,0,13,0,-8,0,1,0,0,0,0,0,0,0,0,6,0,-4,0,4,0,0,0,0,0,0,-1,0,-11,0,0,0,0,6,0|1,0,-8,0,13,0,-6,0,0,0,0,0,0,0,0,4,0,8,0,-1,0,0,0,0,0,0,-11,0,-6,0,0,0,0,6,0,0|0,3,0,-8,0,7,0,-2,0,0,0,0,0,0,0,0,-3,0,-1,0,1,0,0,0,0,0,0,2,0,4,0,0,0,0,-3,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,0,0,1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,4,0,0,0,-2,0,0,0,0,0,0,0,0,-6,0,6,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,-2,0,-1,0,0,0,0,2,0,2,0,0,-1|0,0,0,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,-6,0,-6,0,0,0,0,0,0,6,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,-2,0,-6,0,4,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,0,0,-3,0,3,0,0,0,0,0,0,6,0,3,0,-6,0,0,0,0,-3,0,3,0,0,0|-2,0,1,0,4,0,-3,0,0,0,0,0,0,0,0,7,0,-1,0,2,0,0,0,0,0,0,-8,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|2,0,-7,0,8,0,-3,0,0,0,0,0,0,0,0,-1,0,7,0,-2,0,0,0,0,0,0,-4,0,-3,0,0,0,0,3,0,0|0,3,0,-4,0,-1,0,2,0,0,0,0,0,0,0,0,-3,0,7,0,-7,0,0,0,0,0,0,-2,0,8,0,0,0,0,-3,0|-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0|0,-3,0,4,0,1,0,-2,0,0,0,0,0,0,0,0,3,0,-7,0,7,0,0,0,0,0,0,2,0,-8,0,0,0,0,3,0|0,0,0,0,0,0,0,0,-6,0,-1,0,-6,0,6,0,0,0,0,0,0,13,0,8,0,-11,0,0,0,0,-8,0,4,0,0,1|0,0,0,0,0,0,0,0,0,-6,0,-6,0,6,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,-2,0,-3,0,3,0,0,0,0,0,0,8,0,7,0,-4,0,0,0,0,-7,0,-1,0,0,2|0,0,0,0,0,0,0,0,0,6,0,-6,0,0,0,0,0,0,0,0,0,0,-6,0,12,0,0,0,0,0,0,-6,0,0,0,0|0,0,0,0,0,0,0,0,6,0,-1,0,6,0,-6,0,0,0,0,0,0,-11,0,-4,0,13,0,0,0,0,4,0,-8,0,0,1|1,0,4,0,-11,0,6,0,0,0,0,0,0,0,0,-8,0,-4,0,-1,0,0,0,0,0,0,13,0,6,0,0,0,0,-6,0,0|0,0,0,-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,6,0,-11,0,4,0,1,0,0,0,0,0,0,0,0,-6,0,8,0,-8,0,0,0,0,0,0,-1,0,13,0,0,0,0,-6,0|0,0,0,0,0,0,0,0,3,0,2,0,3,0,-3,0,0,0,0,0,0,-8,0,-7,0,4,0,0,0,0,7,0,1,0,0,-2|0,0,0,0,0,0,0,0,0,-2,0,6,0,-2,0,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,4,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,2,0,-3,0,3,0,0,0,0,0,0,4,0,-1,0,-8,0,0,0,0,1,0,7,0,0,-2|0,0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,-6,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,2,0,1,0,0,0,0,-2,0,-2,0,0,1>:
M__3 := 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Unexpectedly, LinearAlgebra[MatrixPower] does not work well. For instance: 
 

Download performance_of_`sqrtm`.mw

The last six numerical experiments all fail to compute the square root. (Accordingly, here it is impossible to convert each numeric element to one of the "simplest" algebraic numbers that approximates it well.) As you can see, if I execute them directly (i.e., without converting each entry to the nearest floating-point value), the elapsed time required to run the procedure is still unacceptable! (Note that they can be evaluated symbolically in fact.) Is this a bug of Maple? And how do I get the desired results in Maple efficiently?

Let y＞1. It can be proved (maybe by hand) that the following four expressions are mathematically equivalent: 

assume(y > 1); # Assumption!
expr := [0, 0, 0, 0]: # Preallocation.
expr[1] := exp(y*LambertW(ln(y))):
expr[2] := (ln(y)/LambertW(ln(y)))^y:
expr[3] := eval(x^(x^x), x = exp(LambertW(ln(y)))):
expr[4] := eval(x^(x^x), x = ln(y)/LambertW(ln(y))):

But unfortunately, when I tried to simplify expr_i - expr_j (symbolically), I just got: 

seq(seq(ifelse(j <> i, [i, j, verify(expr[j], expr[i], equal)], NULL), j = 1 .. numelems(expr)), i = 1 .. numelems(expr)); # is(expr[j] = expr[i]) does not work as well.
 = 
   [1, 2, FAIL], [1, 3, FAIL], [1, 4, FAIL], [2, 1, FAIL], 

     [2, 3, FAIL], [2, 4, FAIL], [3, 1, FAIL], [3, 2, FAIL], 

     [3, 4, true], [4, 1, FAIL], [4, 2, FAIL], [4, 3, true]

In other words, Maple can only determine that expr[3] = expr[4].
One may check that, for example, 

MmaTranslator:-Mma:-Chop([seq](seq(evalhf(subs(y = log10(rand()), expr[i] - expr[j])), j = 1 .. numelems(expr)), i = 1 .. numelems(expr)), 2^(-26));
 = 
        [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

However, the desired approach is simplifying them symbolically. Is there a way to do so in Maple?

In accordance with this statement obtained by Чебышёв (1853), each of 

simplify(int(x^(1/2)*(x^2 + 1)^(-3/4), x), symbolic);
simplify(int((x^(1)*(1 - x^2))^(1/3), x), symnolic);
simplify(int(x^(-1)*(x^6 + 1)^(-1/6), x), symnolic);
simplify(int(x^(17/2)*(x^2 + 1)^(1/4), x), symnolic);

can be reduced to an integral of rational functions, which can be expressed in terms of elementary functions. But it appears that Maple 2023.0 is still unable to completely calculate them. For instance: 
 

Download Chebyshev_theorem_on_the_integration_of_binomial_differentials.mw

However, closed-form (and readable) solutions in elementary forms exist (cf. Regression reports for Computer Algebra Independent Integration Tests. Summer 2022 version (12000.org)); in fact, Mathematica returns: 

So, why can't Maple find these compact antiderivatives (expressed by elementary functions) directly here? In other words, is there a way to resolve them in Maple without applying some change of the variable to these indefinite integrals manually?

The issue arises from solving the following ODEs in Maple (where a is a suitable real parameter): 

ode__1 := a*(diff(y(x), x) + 1)^2 + (y(x) - x)^2*diff(y(x), x) = 0: # dsolve(ode__1);
ode__4 := a*(x*diff(y(x), x) + y(x))^2 - (y(x) + x)^2*diff(y(x), x) = 0: # dsolve(ode__4);

However, dsolve cannot give fully simplified solutions, so I have to compute these unevaluated integrals (i.e., expr₁) manually: (For the sake of completeness, I list some related ODEs below.) 
 

Download senseless_results_of_int.mw
 

Download senseless_results_of_int.mw

As you can see, the lengthy output of is nearly meaningless! (And if you want to simplify it, Maple will simply return: Error, (in simplify/recurse) indeterminate expression of the form 0/0.) So, how do I get the simplified results in Maple?
The integrals are: 

expr__1 := [Int(1/(z^2 + sqrt(z^4 + 4*a*z^2) + 4*a), z), Int(-1/(z^2 - sqrt(z^4 + 4*a*z^2) + 4*a), z)]: # (value(expr__1));
expr__4 := [Int((z^2 - 4*a*z + sqrt(-4*a*z^3 + z^4 - 8*a*z^2 + 4*z^3 - 4*a*z + 6*z^2 + 4*z + 1) + 2*z + 1)/(z*(-4*a*z + z^2 + 2*z + 1)), z), Int(-(z^2 - 4*a*z + 2*z + 1 - sqrt((-4*a*z + z^2 + 2*z + 1)*(z + 1)^2))/(z*(-4*a*z + z^2 + 2*z + 1)), z)]: # (value(expr__4)):

Note. By the way, Mma can solve the original ODEs directly and explicitly: 

In[1]:= DSolve[a*(y'[x]+1)^2+(y[x]-x)^2*y'[x]==0,y[x],x,IncludeSingularSolutions->Automatic]

                                   2                3                    2
                  a - x C[1] - C[1]             16 a  - 4 a x C[1] - C[1]
Out[1]= {{y[x] -> ------------------}, {y[x] -> --------------------------}}
                       x + C[1]                     4 a (4 a x + C[1])

In[2]:= DSolve[a*(x*y'[x]+y[x])^2-(y[x]+x)^2*y'[x]==0,y[x],x,IncludeSingularSolutions->Automatic]

                     2 a C[1]       2 a C[1]     2  2 a C[1]
                  a E         (-(a E        ) + a  E         + x)
Out[2]= {{y[x] -> -----------------------------------------------}, 
                                     2 a C[1]
                                  a E         - x
 
                2 a C[1]    2 a C[1]
               E         (-E         + 2 a x)
>    {y[x] -> --------------------------------}}
                    2 a C[1]              2
              2 a (E         - 2 a x + 2 a  x)

Unfortunately, Maple fails to do so.