Summer of 69!

Calculus Level 4

(x+x2+1)69dx \large \int { { (x+\sqrt { { x }^{ 2 }+1 } ) }^{ 69 } } dx

If the indefinite integral above is of the form:

ab[(x+x2+1)cc+(x+x2+1)dd]+α \large \frac { a }{ b } \left[ \frac { { (x+\sqrt { { x }^{ 2 }+1 } ) }^{ c } }{ c } +\frac { { (x+\sqrt { { x }^{ 2 }+1 } ) }^{ d } }{ d } \right] +\alpha

where α\alpha is constant of integration.

c>dc > d

Find a+b+cda+b+c-d.

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