#int(-2x^3 + 4x)/(-2x^2 + x + 7)dx#
Multiply by 1 in the form #(-1)/-1#:
#int(2x^3 - 4x)/(2x^2 - x - 7)dx#
The power of the numeration is greater than the power of the denominator, therefore, we do the implied division:
#........................x + 1/2#
#2x^2 - x - 7|2x^3 + 0x^2 - 4x + 0#
#..................-2x^2 + x^2 + 7x#
#...............................x^2 + 3x#
#..............................-x^2 +1/2x + 7/2#
#........................................(7/2)x + 7/2#
#int(2x^3 - 4x)/(2x^2 - x - 7)dx = intxdx + 1/2intdx + 7/2int(x + 1)/(2x^2 - x -7)dx#
#intxdx = x^2/2#:
#int(2x^3 - 4x)/(2x^2 - x - 7)dx = x^2/2 + 1/2intdx + 7/2int(x + 1)/(2x^2 - x -7)dx#
#1/2intdx = x/2#:
#int(2x^3 - 4x)/(2x^2 - x - 7)dx = x^2/2 + x/2 + 7/2int(x + 1)/(2x^2 - x -7)dx#
Integration of the last integral by wolframalpha
#int(2x^3 - 4x)/(2x^2 - x - 7)dx = x^2/2 + x/2 + 7/456 ((57+5 sqrt(57)) ln(-4 x+sqrt(57)+1)+(57-5 sqrt(57)) ln(4 x+sqrt(57)-1))+ C#
