[integrals] Intent to work on manualintegrate TODO: handle n < -1 case

[AB]

Hi everyone,

I would like to work on a TODO found in sympy/integrals/manualintegrate.py inside the sqrt_quadratic_rule function:

elif n == -1:

        generic_step = sqrt_quadratic_denom_rule(f_poly, integrand)

    else:

        return  # todo: handle n < -1 case

    return _add_degenerate_step(generic_cond, generic_step, degenerate_step)

This TODO handles integrals involving odd negative powers of a quadratic, such as (a + bx + cx**2) ** {-3/2}, (a + bx + cx**2) ** {-5/2}, etc.

Proposed Plan:

I plan to implement a recursive reduction step using the formula from Gradshteyn & Ryzhik (2.263.3).

This formula rewrites integrals of the form integral(dx / sqrt(R ** (2k+1))) in terms of an algebraic term and a simpler integral integral(dx / sqrt(R ** (2k-1))). By applying this recursively, any n < -1 case can be reduced until it reaches the n = -1 base case, which is already correctly handled by the existing sqrt_quadratic_rule.

Questions:

1. Is anyone currently working on this?

2. Are there any known blockers for this implementation?

If there are no objections, I would like to proceed with the implementation and a PR.

Thanks,

Ayush Bothra

[Oscar Benjamin]

Sounds reasonable to me. Go ahead 

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