Another incorrect Maxima integral

[kcrisman]

This is already reported upstream at ​this bug tracker ticket.

The original report from a summer PREP user:

integral((2/3)*x^(5/2)*(x+1)^.5, x,0,1)

I got the answer -0.888888889*sqrt(2)

integral((2/3)*x^(5/2)*(x+1)^(1/2), x,0,1)

And the answer is


-5/192*I*pi + 61/288*sqrt(2) + 5/192*log(-sqrt(2) + 1) -
5/192*log(sqrt(2) + 1)

[kcrisman]

#11493 may be related.

[mforets]

This prompts me to see what happens with alternative integration interfaces.

Try with SymPy?:

sage: sage: integral((2/3)*x^(5/2)*(x+1)^.5, x,0,1, algorithm='sympy')   
# no output at all after ~10mins

Try with giac:

sage: giac('integrate((2/3)*x^(5/2)*sqrt(x+1), x,0,1)')   # correct and almost instantaneous
1/288*(sqrt(2)*61+15*ln(sqrt(2)-1))
sage: (1/288*(sqrt(2)*61+15*ln(sqrt(2)-1))).n()
0.253633414928700
sage: (-5/192*I*pi + 61/288*sqrt(2) + 5/192*log(-sqrt(2) + 1) -
....: 5/192*log(sqrt(2) + 1)).n()    # compare with formula given by OP
0.253633414928700

[mforets]

fixed at Sage v8.0.beta3:

sage: integrate((2/3)*x^(5/2)*sqrt(x+1), x,0,1, algorithm='maxima')
-5/192*I*pi + 61/288*sqrt(2) - 5/192*log(sqrt(2) + 1) + 5/192*log(-sqrt(2) + 1)
sage: _.n()
0.253633414928700

[mforets]

not really.. the problem is with the 0.5:

sage: integral((2/3)*x^(5/2)*(x+1)^.5, x,0,1, algorithm='maxima') # wrong
-0.8888888888888888*sqrt(2)
sage: integral((2/3)*x^(5/2)*(x+1)^.5, x,0,1, algorithm='giac') # ok
0.253633414929
sage: integral((2/3)*x^(5/2)*(x+1)^.5, x,0,1, algorithm='sympy') # timeout

[rws]

That is user error. No one can expect to get correct integrals from expressions containing floating point numbers. It is not supported and maybe Sage should throw an exception in these cases anyway. If you agree, Karl-Dieter, I'll open a ticket for just this.

[rws]

Together with #22894 and ex._convert(QQ) one could even provide a default option that automatically converts 0.5 to 1/2 before integration.