Integral that should have closed form is not evaluating

[jacluff1]

I have an integral of the form:
t / sy.sqrt( At**2 + Bt + C ),

where:
t = sy.symbols('t', real=True, positive=True)
A,B,C = sy.symbols('A B C', real=True)

when I try to integrate in sympy, it does not evaluate; when I integrate in wolfram alpha, it does. I am writing code in Jupyter Notebook at http://localhost:8888/notebooks/meeting3.ipynb

[Dirivian]

1. This doesn't evaluate for integrals even when I replace A,B and C with 1.

2. You linked the local path of your notebook (accessible only to you). If the notebook is important, put it up on github and link it,

[oscarbenjamin]

I get this on master for that general form:

In [3]: A, B, C = symbols('A B C')                                                                                                

In [4]: integrate(t / (A*t**2 + B*t + C), t)                                                                                      
Out[4]: 
                                ⎛            ⎛       _____________      ⎞      ⎛       _____________      ⎞      ⎞              
                                ⎜            ⎜      ╱           2       ⎟      ⎜      ╱           2       ⎟      ⎟              
                                ⎜            ⎜  B⋅╲╱  -4⋅A⋅C + B      1 ⎟    2 ⎜  B⋅╲╱  -4⋅A⋅C + B      1 ⎟      ⎟              
⎛       _____________      ⎞    ⎜    - 4⋅A⋅C⋅⎜- ────────────────── + ───⎟ + B ⋅⎜- ────────────────── + ───⎟ + 2⋅C⎟   ⎛     _____
⎜      ╱           2       ⎟    ⎜            ⎜       ⎛         2⎞    2⋅A⎟      ⎜       ⎛         2⎞    2⋅A⎟      ⎟   ⎜    ╱     
⎜  B⋅╲╱  -4⋅A⋅C + B      1 ⎟    ⎜            ⎝   2⋅A⋅⎝4⋅A⋅C - B ⎠       ⎠      ⎝   2⋅A⋅⎝4⋅A⋅C - B ⎠       ⎠      ⎟   ⎜B⋅╲╱  -4⋅A
⎜- ────────────────── + ───⎟⋅log⎜t + ────────────────────────────────────────────────────────────────────────────⎟ + ⎜──────────
⎜       ⎛         2⎞    2⋅A⎟    ⎝                                         B                                      ⎠   ⎜     ⎛    
⎝   2⋅A⋅⎝4⋅A⋅C - B ⎠       ⎠                                                                                         ⎝ 2⋅A⋅⎝4⋅A⋅

                   ⎛            ⎛     _____________      ⎞      ⎛     _____________      ⎞      ⎞
                   ⎜            ⎜    ╱           2       ⎟      ⎜    ╱           2       ⎟      ⎟
                   ⎜            ⎜B⋅╲╱  -4⋅A⋅C + B      1 ⎟    2 ⎜B⋅╲╱  -4⋅A⋅C + B      1 ⎟      ⎟
________      ⎞    ⎜    - 4⋅A⋅C⋅⎜────────────────── + ───⎟ + B ⋅⎜────────────────── + ───⎟ + 2⋅C⎟
      2       ⎟    ⎜            ⎜     ⎛         2⎞    2⋅A⎟      ⎜     ⎛         2⎞    2⋅A⎟      ⎟
⋅C + B      1 ⎟    ⎜            ⎝ 2⋅A⋅⎝4⋅A⋅C - B ⎠       ⎠      ⎝ 2⋅A⋅⎝4⋅A⋅C - B ⎠       ⎠      ⎟
──────── + ───⎟⋅log⎜t + ────────────────────────────────────────────────────────────────────────⎟
     2⎞    2⋅A⎟    ⎝                                       B                                    ⎠
C - B ⎠       ⎠                                                                                  

Probably a simpler output is possible if you say what A, B and C should be.

Is this still an issue or should I close it?

[asmeurer]

I get the same. And simplify(integrate(t / (A*t**2 + B*t + C), t).diff(t)) produces the original expression. I think it can be closed, unless we want to add a test for it.

[oscarbenjamin]

It's possible that @jacluff1 had a particular combination of coefficients that caused trouble. We should allow time to respond.

I'm going to create a label "could-close" for this situation - after some time if there is no response it can be closed.

[jksuom]

I think that there is currently no support for completing the square in sqrt(A*t**2 + B*t + C). With B = 0, manualintegrate will find an antiderivative.

[oscarbenjamin]

Good point @jksuom :)

I missed off the square root:

In [31]: A, B, C = symbols('A B C')                                                                                               

In [32]: integrate(t / sqrt(A*t**2 + B*t + C), t)                                                                                 
Out[32]: 
⌠                       
⎮          t            
⎮ ─────────────────── dt
⎮    ________________   
⎮   ╱    2              
⎮ ╲╱  A⋅t  + B⋅t + C    
⌡  

With B=0 it does work:

In [33]: integrate(t / sqrt(A*t**2 + C), t)                                                                                       
Out[33]: 
⎧   __________           
⎪  ╱    2                
⎪╲╱  A⋅t  + C            
⎪─────────────  for A ≠ 0
⎪      A                 
⎨                        
⎪      2                 
⎪     t                  
⎪    ────       otherwise
⎪    2⋅√C                
⎩   

So I guess the issue is adding support for completing the square inside the square root.