How can i integrate?

Question

it is
sqrt[ ((x-a)^2 + b)^2 +cx -d ]

Answer 1

Sorry, this integral is usually not elementary.
That means it cannot be expressed as a finite
combination of radicals, trig functions, exponentials
and logarithms.
In fact, the integral of the square root of a cubic
or quartic polynomial is usually an elliptic integral.
Even if elliptic functions are allowed, the integral is
enormously complicated. Try running it through
integrals.wolfram.com!

Answer 2

int(sqrt[ ((x-a)^2 + b)^2 +cx -d ]) when int is integral
not to look to bad:
(e + f)^2 = e^2 + 2ef + f^2 ... so:
(x - a)^2 = x^2 - 2ax + a^2 and
(x^2 - 2ax + a^2)^2 = x^4 - 4a*x^3+6a^2*x^2-4a^3*x+a^4
(() + b)^2 = ()^2 + 2b*() + b^2 put it together:
(x^2 - 2ax + a^2)^2 + 2b*(x^2 - 2ax + a^2) + b^2 simplify
(x^2 - 2ax + a^2)^2 + 2b*x^2 - 4a*b*x + 2b*a^2 + b^2 substitute
x^4-4a*x^3+6a^2*x^2-4a^3*x+a^4+2b*x^2-4a*b*x+2b*a^2+b^2
at to it the cx and subtract d.
the you can use this rule:
int(sqrt(u(x))) = 1/(2*sqrt(u(x))) * (du/dx)
where u is the really long expression above :D
so du/dx = 4x^3-12a*x^2+12a^2*x-4a^3+4b*x-4a*b+c
I think that might be helpful, otherwise... I don't know.