MATLAB: Symbolic integration inside numerical integration

Question

I would like to solve the following integral numerically:

As far as I can tell, the built-in function integral2 is not applicable here due to the nature of the expression. Instead, from Evaluating Double Integrals, I arrive at the following solution which takes forever to compute due to the third line:

syms x r
firstint=int(1./(1+x.^2.5),x,r,Inf)
answer=int(r.*exp(-r.^2).*firstint,r,0,Inf),
double(answer)

The last line is just to get a numerical answer out of the symbolic integration. To speed up things, I am inclined to replace the third line with the built-in function integral, but it requires me to make firstint into a handle first, for example by using another built-in function called matlabFunction. Unfortunately, I get an error when doing the following:

syms x r
firstint=int(1./(1+x.^2.5),x,r,Inf),
firstintHandle = matlabFunction(firstint),
answer=integral(r.*exp(-r.^2).*firstintHandle,0,Inf),

Any ideas on what to do? Please be very specific. Your help is greatly appreciated.

Answer 1

See if this does what you want:

firstint = @(r) integral(@(x) 1./(1+x.^2.5), r, Inf, 'ArrayValued',1);
secondint = @(r) r.*exp(-r.^2).*exp(-firstint(r));
answer = integral(secondint, 0, Inf, 'ArrayValued',1)
answer =
   0.274214226644371

The code appears to me to be correct. I defer to you to be certain it is.