Travelling Chef Problem

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When Chef is travelling  that's the problem. No one knows when he will change his velocity. But mathematicians are certain that Chef's velocity depends on the current time, and can be described as a cubic polynomial. A cubic polynomial can be defined as:
v(t) = A×t^{3} + B×t^{2} + C×t + D, where A, B, C, D are rational numbers (A is non zero), and t is the current time.
As for us, all we know is: Chef's velocity at 4 time instances. And we are interested in finding the total distance covered by Chef at time T. We can assume that Chef has started his travel at zero time moment.
Input
The first line contains an integer N denoting the number of test cases. The first line of each testcase contains an integer T, the time up to which we want to know the distance covered by Chef from time 0. Then 4 lines follow, each containing two integers: ct and cv. Here ct is the time instant and cv is the velocity at time ct.
Please note, that we pick ct in such a way that cv comes out to be integer. In general the velocity of Chef is real number.
Output
For each test case output a single line containing a real number – the distance covered by the Chef in time interval 0 to T. Your answer will be considered as correct if it has relative error less than 10^{10}.
Constraints
 1 ≤ N ≤ 1000
 1 ≤ T ≤ 3600
 1 ≤ ct ≤ 3600
 1000000 ≤ cv ≤ 1000000
 For each testcase values of ct are distinct.
Example
Input: 2 5 2 11 3 31 4 69 5 131 8 12 480 13 693 14 960 15 1287 Output: 173.750000 80.000000
Author:  vinayak garg 
Tester:  gerald 
Editorial  http://discuss.codechef.com/problems/TCP 
Tags  cook45, integrals, interpolation, medium, vinayak garg 
Date Added:  15072013 
Time Limit:  1 sec 
Source Limit:  50000 Bytes 
Languages:  C, CPP14, JAVA, PYTH, PYTH 3.5, CS2, PAS fpc, PAS gpc, RUBY, PHP, GO, NODEJS, HASK, SCALA, D, PERL, FORT, WSPC, ADA, CAML, ICK, BF, ASM, CLPS, PRLG, ICON, SCM qobi, PIKE, ST, NICE, LUA, BASH, NEM, LISP sbcl, LISP clisp, SCM guile, JS, ERL, TCL, PERL6, TEXT, CLOJ, FS 
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