Polynomial -> Integrate Given a polynomial in one variable with rational coefficients, output an equivalent expression containing only 1, variables, and definite integrals. For example, -x2 can be expressed as ∫x∫111dtxdu.

E := 1 | var | ∫EEEdvar

Any reasonable input/output method is allowed.

Examples:



Your score will be the your code length multiplied by the number of ∫ symbols  used on the test cases. You should be able to score your program. Lowest score wins.

Test cases:

4/381*x^2+49/8*x^3-17/6
311/59*x^2-92/9*x^3-7/15*x
333/29*x^3+475/96*x^8



  Golfing is going to be difficult, because I can't golf just the code or just the output, and I therefore do not know if a change will help or hurt my score until I try it, which sucks in my opinion.


Don't let the score restrict your creation. You are welcomed to answer with mainly one part of the score well optimized, even if the other one badly left.
Python 2, 315 bytes * 5113 = 1610595 score
I'm still working on golfing the score. Golfing is going to be difficult, because I can't golf just the code or just the output, and I therefore do not know if a change will help or hurt my score until I try it, which sucks in my opinion.
Despite the annoyance of golfing this, I did enjoy the calculus.

t='t'
Z=lambda n:N(Z(-n))if n&lt;0else[1,t,N(1),Z(n-1)]if n&gt;1else[[1,t,1,1],1][n]
N=lambda a:[1,t,a,Z(0)]
x=lambda n:n&gt;1and[x(n-1),t,Z(0),'x']or'x'
M=lambda a,b:[b,t,Z(0),a]
print reduce(lambda a,b:[1,t,N(a),b],[M((lambda a,b:M(Z(a),[x(b-1)if b&gt;1else 1,'x',Z(0),1]))(*c),x(i)if i else 1)for i,c in enumerate(input())])

Try it online!
Run all test cases - to score, count all [ in the output.
The input polynomial is taken as a list of (numerator, denominator) coefficient pairs in order from lowest to highest power of x. (0, 1) (zero) is used for missing powers.
Output is given with each integral represented by a list [f,t,a,b] to represent ∫abf dt
Verification
Here is a slightly less golfed version that outputs valid Mathematica syntax for integration, which may be tested in an online notebook. Unfortunately, decently-sized programs will not complete in a free notebook.
Go here, scroll to bottom, &quot;Create New Notebook&quot;, paste (Wolfram Language Input), and evaluate (Shift+Enter) (Note that using num-pad Enter doesn't work).
Explanation
Uses these equations:






Links

Tool that converts the output to Mathematica nested functions

Tool that converts the output to Mathematica, avoiding recursion

Run the Mathematica output on TIO

Equation images made with this tool.