The integrals below involve `ax + b`, including forms where `ax + b` is raised to an exponent or in the denominator of a fraction.
1) `int 1/(ax+b) dx = 1/a ln(ax+b)`
2) `int x/(ax+b) dx = x/a - b/(a^2) ln(ax+b)`
3) `int x^2/(ax+b) dx = (ax+b)^2/(2a^3)-(2b(ax+b))/a^3+b^3/a^3 ln(ax+b)`
4) `int x^3/(ax+b) dx = (ax+b)^3/(3a^4)-(3b(ax+b)^2)/2a^4+(3b^2(ax+b))/a^4-b^3/a^4 ln(ax+b)`
5) `int 1/(x(ax+b)) dx = 1/b ln(x/(ax+b))`
6) `int 1/(x^2 (ax+b)) dx = -1/(bx)+a/b^2 ln((ax+b)/x)`
7) `int 1/(x^3(ax+b)) dx = (2ax-b)/(2b^2x^2)+a^2/b^3 ln(x/(ax+b))`
8) `int 1/(ax+b)^2 dx = -1/(a(ax+b))`
9) `int x/(ax+b)^2 dx = b/(a^2(ax+b))+1/a^2 ln(ax+b)`
10) `int x^2/(ax+b)^2 dx = (ax+b)/a^3-b^2/(a^3(ax+b))-(2b)/a^3 ln(ax+b)`
11) `int x^3/(ax+b)^2 dx = (ax+b)^2/(2a^4)-(3b(ax+b))/a^4+b^3/(a^4(ax+b))+(3b^2)/a^4 ln(ax+b)`
12) `int 1/(x(ax+b)^2) dx = 1/(b(ax+b))+1/b^2 ln(x/(ax+b))`
13) `int 1/(x^2(ax+b)^2) dx = (-a)/(b^2(ax+b))-1/(b^2x)+(2a)/b^3 ln((ax+b)/x)`
14) `int 1/(x^3(ax+b)^2) dx = -(ax+b)^2/(2b^4x^2)+(3a(ax+b))/(b^4x)-(a^3x)/(b^4(ax+b))-(3a^2)/b^4 ln((ax+b)/x)`
15) `int 1/(ax+b)^3 dx = (-1)/(2(ax+b)^2)`
16) `int x/(ax+b)^3 dx = (-1)/(a^2(ax+b))+b/(2a^2(ax+b^2))`
17) `int x^2/(ax+b)^3 = (2b)/(a^3(ax+b))-b^2/(2a^3(ax+b)^2)+1/a^3 ln(ax+b)`
18) `int x^3/(ax+b)^3 dx = x/a^3-(3b^2)/(a^4(ax+b))+b^3/(2a^4(ax+b)^2)-(3b)/a^4 ln(ax+b)`
19) `int 1/(x(ax+b)^3) dx = (a^2x^2)/(2b^3(ax+b)^2)-(2ax)/(b^3(ax+b))-1/b^3ln((ax+b)/x)`
20) `int 1/(x^2(ax+b)^3) dx = (-a)/(2b^2(ax+b)^2)-(2a)/(b^3(ax+b))-1/(b^3x)+(3a)/b^4 ln((ax+b)/x)`
21) `int 1/(x^3(ax+b)^3) dx = (a^4x^2)/(2b^5(ax+b)^2)-(4a^3x)/(b^5(ax+b)) - (ax+b)^2/(2b^5x^2)-(6a^2)/b^5 ln((ax+b)/x)`
22) `int (ax+b)^n dx = (ax+b)^(n+1)/((n+1)a), if n=-1, see (1)`
23) `int x(ax+b)^n dx = (ax+b)^(n+2)/((n+2)a^2)-(b(ax+b)^(n+1))/((n+1)a^2), n != -1, -2. If n=-1, -2, See (2), (9)`
24) `int x^2(ax+b)^n dx = (ax+b)^(n+3)/((n+3)a^3)-(2b(ax+b)^(n+2))/((n+2)a^3)+(b^2(ax+b)^(n+1))/((n+1)a^3)`
If n=-1, -2, -3, see (3), (10), (17)`
25) `int x^m(ax+b)^n dx = (x^(m+1)(ax+b^n))/(m+n+1)+(nb)/(m+n+1)int x^m(ax+b)^(n-1) dx`
or: `= (x^m(ax+b)^(n+1))/((m+n+1)a)-(mb)/((m+n+1)a)int x^(m-1)(ax+b)^n dx`
or: `=( -x^(m+1)(ax+b)^(n+1))/((n+1)b)+(m+n+2)/((n+1)b)int x^m(ax+b)^(n+1) dx`