The integrals below involve `sqrt(ax^2+bx+c)`
If `b^2=4ac`, then `sqrt(ax^2+bx+c)=sqrta(x+b/(2a))`; use the results in the tables for Integrals Involving `ax+b` or `sqrt(ax+b)`.
If `b=0`, use the results in the tables for Integrals Involving `sqrt(x^2+a^2), sqrt(x^2-a^2), or sqrt(a^2-x^2`.
If `a=0` or `c=0`, use the results in the tables for Integrals Involving `ax+b` or `sqrt(ax+b)`.
1) `int 1/sqrt(ax^2+bx+c) dx = 1/sqrta ln (2sqrtasqrt(ax^2+bx+c)+2ax+b)`
OR `= -1/sqrt(-a) sin^-1((2ax+b)/sqrt(b^2-4ac))`
OR `= 1/sqrtasinh^-1((2ax+b)/sqrt(4ac-b^2))`
2) `int x/sqrt(ax^2+bx+c) dx = sqrt(ax^2+bx+c)/a-b/(2a) int 1/sqrt(ax^2+bx+c) dx`
3) `int x^2/sqrt(ax^2+bx+c) dx = (2ax-3b)/(4a^2)sqrt(ax^2+bx+c)+(3b^2-4ac)/(8a^2) int 1/sqrt(ax^2+bx+c) dx`
4) `int 1/(xsqrt(ax^2+bx+c))dx = -1/sqrtc ln((2sqrtcsqrt(ax^2+bx+c)+bx+2c)/x)`
OR `= 1/sqrt(-c) sin^-1((bx+2c)/(absxsqrt(b^2-4ac)))`
OR `= -1/sqrtc sinh^-1((bx+2c)/(absxsqrt(4ac-b^2)))`
5) `int 1/(x^2sqrt(ax^2+bx+c)) dx = -sqrt(ax^2+bx+c)/(cx)-b/(2c) int 1/(xsqrt(ax^2+bx+c)) dx`
6) `int sqrt(ax^2+bx+c) dx = ((2ax+b)sqrt(ax^2+bx+c))/(4a)+(4ac-b^2)/(8a) int 1/sqrt(ax^2+bx+c) dx`
7) `int xsqrt(ax^2+bx+c) dx = (ax^2+bx+c)^(3/2)/(3a)-(b(2ax+b))/(8a^2)sqrt(ax^2+bx+c)-(b(4ac-b^2))/(16a^2) int 1/sqrt(ax^2+bx+c) dx`
8) `int x^2sqrt(ax^2+bx+c) dx = (6ax-5b)/(24a^2)(ax^2+bx+c)^(3/2)+(5b^2-4ac)/(16a^2) int sqrt(ax^2+bx+c) dx`
9) `int sqrt(ax^2+bx+c)/x dx = sqrt(ax^2+bx+c)+b/2 int 1/sqrt(ax^2+bx+c) dx+c int 1/(xsqrt(ax^2+bx+c)) dx`
10) `int sqrt(ax^2+bx+c)/x^2 dx = -sqrt(ax^2+bx+c)/x+a int 1/sqrt(ax^2+bx+c) dx+b/2 int 1/(xsqrt(ax^2+bx+c)) dx`
11) `int 1/(ax^2+bx+c)^(3/2) dx = (2(2ax+b))/((4ac-b^2)sqrt(ax^2+bx+c))`
12) `int x/(ax^2+bx+c)^(3/2) dx = (2(bx+2c))/((b^2-4ac)sqrt(ax^2+bx+c))`
13) `int x^2/(ax^2+bx+c)^(3/2) dx = ((2b^2-4ac)x+2bc)/(a(4ac-b^2)sqrt(ax^2+bx+c))+1/a int 1/sqrt(ax^2+bx+c) dx`
14) `int 1/(x(ax^2+bx+c)^(3/2)) dx = 1/(csqrt(ax^2+bx+c))+1/c int 1/(xsqrt(ax^2+bx+c)) dx-b/(2c) int 1/(ax^2+bx+c)^(3/2) dx`
15) `int 1/(x^2(ax^2+bx+c)^(3/2)) dx = -(ax^2+2bx+c)/(c^2xsqrt(ax^2+bx+c))+(b^2-2ac)/(2c^2) int 1/(ax^2+bx+c)^(3/2) dx-(3b)/(2c^2) int 1/(xsqrt(ax^2+bx+c)) dx`
16) `int (ax^2+bx+c)^(n+1/2) dx = ((2ax+b)(ax^2+bx+c)^(n+1/2))/(4a(n+1))+((2n+1)(4ac-b^2))/(8a(n+1)) int (ax^2+bx+c)^(n-1/2) dx`
17) `int x(ax^2+bx+c)^(n+1/2) dx = (ax^2+bx+c)^(n+3/2)/(a(2n+3))-b/(2a) int (ax^2+bx+c)^(n+1/2) dx`
18) `int 1/(ax^2+bx+c)^(n+1/2) dx = (2(ax+b))/((2n-1)(4ac-b^2)(ax^2+bx+c)^(n-1/2))+(8a(n-1))/((2n-1)(4ac-b^2)) int 1/(ax^2+bx+c)^(n-1/2) dx`
19) `int 1/(x(ax^2+bx+c)^(n+1/2)) dx = 1/((2n-1)c(ax^2+bx+c)^(n-1/2))+1/c int 1/(x(ax^2+bx+c)^(n-1/2)) dx-b/(2c) int 1/(ax^2+bx+c)^(n+1/2) dx`