If I rewrite this as:
x^2 sqrt((x^2)^2 + (sqrt5)^2) prop u sqrt(u^2 + a^2
then you can do the trig substitution method of letting:
x^2 = sqrt5tantheta
sqrt(x^4 + 5) = sqrt5sec^2theta
x = 5^(1/4)sqrttantheta
dx = 5^(1/4)*1/(2sqrttantheta)sec^2thetad theta = 5^(1/4)/2sec^2theta/(sqrttantheta)d theta
= int sqrt5tantheta sqrt5sec^2theta 5^(1/4)/2sec^2theta/(sqrttantheta)d theta
= 5^(5/4)/2int (tantheta)^(1/2) sec^4thetad theta
= 5^(5/4)/2int (tantheta)^(1/2) (tan^2theta + 1)sec^2thetad theta
since 1+tan^2theta = sec^2theta.
Then just do some u-substitution with u = tantheta and du = sec^2thetad theta and you'll just have some polynomials to work with, integrate that, substitute back in previous variables, add +C.
I think you can do it from there (I'm in a hurry. If someone wants to finish this, go ahead).
