Comment #3 on issue 2477 by [email protected]: Sympy can't do
elementary integration of type quadratic * sqrt(quadratic)
http://code.google.com/p/sympy/issues/detail?id=2477
My code *should* be able to do P(x)*(x**a + b)**c out of the box [P(x)
being a polynomial]. Indeed
In [1]: integrate((x**2 + x + 3)*sqrt(x**2 + 1), x, meijerg=True)
Out[1]:
⎛
ⅈ⋅π⎞
⎽⎽⎽⎽⎽⎽⎽⎽
⎽⎽⎽⎽⎽⎽⎽⎽ ⎽⎽⎽⎽⎽⎽⎽⎽ ⎜ ───⎟
5 3 2 ╱ 2 ╱
2 ╱ 2 ⎜ 2 ⎟
x 3⋅x x ⋅╲╱ x + 1 3⋅x⋅╲╱ x +
1 x ╲╱ x + 1 ⅈ⋅asin⎝x⋅ℯ ⎠ 3
──────────────────── + ──────────────────── + ────────────── +
─────────────── + ──────────────────── + ─────────── + ────────────── + ─
⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ 3
2 ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ 3 8
╱ 2 ⅈ⋅π ╱ 2
ⅈ⋅π ╱ 2 ⅈ⋅π
4⋅╲╱ - x ⋅ℯ + 1 8⋅╲╱ - x ⋅ℯ +
1 8⋅╲╱ - x ⋅ℯ + 1
⋅asinh(x)
─────────
2
[but I did not verify this]. Note the somewhat peculiar factors of
exp_polar. For indefinite integration they can safely be replaced by exp.
The more general quadratics can be done by substitution and completing the
square, there is not currently any infrastructure for this in my code.
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