NIntegrate[] vs Integrate[]//N
​​UsingWolfram14.0youget:
In[]:=
Integrate
t
(-1+t)(-1+5t)
(-3+5t)
,{t,1,7/5}//FullSimplify
Out[]=
2
25
-5
2
+5EllipticK[5]-5EllipticPi
5
3
,5+
5
-EllipticFArcSin[
7
],
1
5
+EllipticPi
1
3
,ArcSin[
7
],
1
5

In[]:=
N[%]
Out[]=
0.245649-0.565685
Regardlessoftheterm
2
25
-5
2
,itresults:
In[]:=
2
25
5EllipticK[5]-5EllipticPi
5
3
,5+
5
-EllipticFArcSin[
7
],
1
5
+EllipticPi
1
3
,ArcSin[
7
],
1
5
//N//Chop
Out[]=
0.245649
In[]:=
NIntegrate
t
(-1+t)(-1+5t)
(-3+5t)
,{t,1,7/5}
Out[]=
0.245649
​