First, rationalize the denominator:
#color(white)=int 1/(sqrt(x+1)-sqrtx)dx#
#=int 1/(sqrt(x+1)-sqrtx)color(red)(*(sqrt(x+1)+sqrtx)/(sqrt(x+1)+sqrtx))dx#
#=int (sqrt(x+1)+sqrtx)/((sqrt(x+1)-sqrtx)(sqrt(x+1)+sqrtx))dx#
#=int (sqrt(x+1)+sqrtx)/((sqrt(x+1))^2-(sqrtx)^2)dx#
#=int (sqrt(x+1)+sqrtx)/(x+1-x)dx#
#=int (sqrt(x+1)+sqrtx)/(color(red)cancelcolor(black)x+1color(red)cancelcolor(black)(color(black)-x))dx#
#=int (sqrt(x+1)+sqrtx)/1dx#
#=int (sqrt(x+1)+sqrtx) dx#
#=intsqrt(x+1)# #dx+intsqrtx# #dx#
#=int(x+1)^(1/2)# #dx+intx^(1/2)# #dx#
Power rule:
#=((x+1)^(1/2+1))/(1/2+1)+(x^(1/2+1))/(1/2+1)#
#=((x+1)^(3/2))/(3/2)+(x^(3/2))/(3/2)#
#=2/3(x+1)^(3/2)+2/3x^(3/2)#
You can factor out the #2/3#, and don't forget to add #C#:
#=2/3((x+1)^(3/2)+x^(3/2))+C#
That's the whole integral. Hope this helped!