What you have done is not right because you cannot have "x" in the "du" integral- since x is not a constant, you cannot treat that remaining "2+ 2x" as a constant. It's not so much a matter of the "chain rule" as it is the "quotient rule". If you were to differentiate you cannot just differentiate the " " and cancel the "2+2x" in the denominator. . That, of course, is NOT

What youcando is "complete the square": write as = so that . Now, and you can use the linear substitution u= x+ 1 so that du= dx. Now the integral is . Of course that integral cannot be done in any simple form, but now you can write it in terms of the "error function" Erf(x).