How can I integrate x^2 dx/ sqrt(x^2+21) without using the sqrt (a^2+u^2) form formulas?

? x^2 / ?(x^2+21)  dx = by parts
 ? x  d (?(x^2+21)) =
 x?(x^2+21) -  ? ?(x^2+21) dx =
 x?(x^2+21) -  21? ?((x/?21)^2+1) d(x / ?21) =
I continue for
? ?((x/?21)^2+1) d(x / ?21)
using x / ?21 = sinh ( u)
1/?21 dx = cosh(u) du
? ?(sinh^2(u) +1) cosh(u) du = ? cosh^2 (u) du = (1/2)? (1+ cosh(2u)) du = (1/2)u + (1/4)sinh(2u) + C
back to x and you get the answer

x = sqrt(21) * tan(t)
dx = sqrt(21) * sec(t)^2 * dt

x^2 * dx / sqrt(21 + x^2) =>
21 * tan(t)^2 * sqrt(21) * sec(t)^2 * dt / sqrt(21 + 21tan(t)^2) =>
21 * tan(t)^2 * sqrt(21) * sec(t)^2 * dt / (sqrt(21) * sqrt(1 + tan(t)^2)) =>
21 * tan(t)^2 * sec(t)^2 * dt / sqrt(sec(t)^2) =>
21 * tan(t)^2 * sec(t)^2 * dt / sec(t) =>
21 * tan(t)^2 * sec(t) * dt

To solve this, we'll need to integrate by parts:

u = tan(t)
du = sec(t)^2 * dt
dv = sec(t) * tan(t) * dt
v = sec(t)

int(u * dv) =>
u * v - int(v * du) =>
tan(t) * sec(t) - int(sec(t)^3 * dt) =>
tan(t) * sec(t) - int(sec(t) * (1 + tan(t)^2) * dt) =>
tan(t) * sec(t) - int(sec(t) * dt) - int(sec(t) * tan(t)^2 * dt)

int(tan(t)^2 * sec(t) * dt) = tan(t) * sec(t) - int(sec(t) * dt) - int(sec(t) * tan(t)^2 * dt)
2 * int(tan(t)^2 * sec(t) * dt) = tan(t) * sec(t) - int(sec(t) * dt)
int(tan(t)^2 * sec(t) * dt) = (1/2) * (tan(t) * sec(t) - int(sec(t) * dt))
int(tan(t)^2 * sec(t) * dt) = (1/2) * (tan(t) * sec(t) - ln|sec(t) + tan(t)|) + C


21 * tan(t)^2 * sec(t) * dt =>
21 * (1/2) * (tan(t) * sec(t) - ln|sec(t) + tan(t)|) + C =>
(21/2) * (tan(t) * sec(t) - ln|sec(t) + tan(t)|) + C


x = sqrt(21) * tan(t)
x / sqrt(21) = tan(t)
x^2 / 21 = tan(t)^2
x^2 / 21 = sec(t)^2 - 1
(21 + x^2) / 21 = sec(t)^2
sec(t) = sqrt((21 + x^2) / 21)

(21/2) * (tan(t) * sec(t) - ln|sec(t) + tan(t)|) + C
(21/2) * ((x/sqrt(21)) * sqrt((21 + x^2) / 21)  -  ln|sqrt(21 + x^2) / sqrt(21)  +  x / sqrt(21)|) + C
(21/2) * (x * sqrt(21 + x^2) / 21  -  ln|x + sqrt(21 + x^2)| - ln(sqrt(21)) + C

We can bring out ln(sqrt(21)) and multiply it with 21/2.  This will give us some real constant value which we can add to the constant of integration C to create some new constant C

(21/2) * ((1/21) * x * sqrt(21 + x^2) - ln|x + sqrt(21 + x^2)|) + C
(1/2) * x * sqrt(21 + x^2) - (21/2) * ln|x + sqrt(21 + x^2)| + C

x^2 dx/ sqrt(x^2+21)
(x^2+21 -21)/sqrt(x^2+21)
= sqrt(x^2+21)- 21/sqrt(x^2+21).........Now use bothe formulas