If 2x - 1 x^6 [5] 2x¹⁰ + 3x^8 - x^7 dx = - A B (2 + 3x^C - x^D)^ E/F + K, x > 0, where K is the constant of integration, A, B, E, F are positive integers with (A,B)=1 and (E,F)=1 , and C, D are integers, then the value of A + B + |C| + |D| + E + F is equal to ________ .

Step-by-step solution

Given integral is I = 2x - 1 x^6 [5] 2x¹⁰ + 3x^8 - x^7 dx Extract x¹⁰ from the fifth root: I = 2x - 1 x^6 x^2 [5] 2 + 3x⁻² - x⁻³ dx I = ( 2x - 1 x^4 ) (2 + 3x⁻² - x⁻³ )^ 1/5 dx I = (2x⁻³ - x⁻⁴) (2 + 3x⁻² - x⁻³)^ 1/5 dx Let t = 2 + 3x⁻² - x⁻³ . Differentiating both sides with respect to x : dt = (-6x⁻³ + 3x⁻⁴) dx = -3(2x⁻³ - x⁻⁴) dx (2x⁻³ - x⁻⁴) dx = - dt 3 Substituting this into the integral: I = - 1 3 t^ 1/5 dt I = - 1 3 ( t^ 6/5 6/5 ) + K = - 5 18 t^ 6/5 + K Substituting back t = 2 + 3x⁻² - x⁻³ : I = - 5 18 (2 +