Let, I=int(x^2+x+5)/sqrt(x^2+1)dx=int{(x^2+1)+(x+4)}/sqrt(x^2+1)dx.
:. I=int{(x^2+1)/sqrt(x^2+1)+(x+4)/sqrt(x^2+1)}dx,
=intsqrt(x^2+1)dx+int(x+4)/sqrt(x^2+1)dx,
=I_1+1/2int(2x)/sqrt(x^2+1)dx+4int1/sqrt(x^2+1)dx,
:. I=I_1+1/2I_2+4I_3..........................(star), where,
I_1=intsqrt(x^2+1)dx,
:. I_1=x/2sqrt(x^2+1)+1/2ln|(x+sqrt(x^2+1))|...........(star_1);
I_2=int(2x)/sqrt(x^2+1)dx,
=int(x^2+1)^(-1/2)d/dx(x^2+1)dx,
=(x^2+1)^(-1/2+1)/(-1/2+1),
:. I_2=2sqrt(x^2+1)....................................(star_2);
I_3=int1/sqrt(x^2+1)dx,
:. I_3=ln|x+sqrt(x^2+1)|..............................(star_3).
"Using "(star_1), (star_2)" and "(star_3)" in "(star)," we have,"
I=x/2sqrt(x^2+1)+1/2ln|(x+sqrt(x^2+1))|+1/2*2sqrt(x^2+1)
+4*ln|x+sqrt(x^2+1)|, i.e.,
I=1/2[(x+2)sqrt(x^2+1)+9ln|(x+sqrt(x^2+1))|]+C.
Enjoy Maths.!