Squaring both sides:
(sqrt(x^2+66^2)+x)/x-2*66+xsqrt(x^2+66)-x^2=25;
sqrt(1+(66/x)^2)+1-132+x^2sqrt(1+(66/x)^2)-x^2=25;
sqrt(1+(66/x)^2)+x^2(sqrt(1+(66/x)^2)-1)=156 (if x^2=36/119, this is 120+36/119*119=156)
sqrt(1+(66/x)^2)+x^2sqrt(1+(66/x)^2)-x^2=156;
sqrt(1+(66/x)^2)(1+x^2)=156+x^2.
Squaring again: (1+(66/x)^2)(1+2x^2+x^4)=(156+x^2)^2=156^2+312x^2+x^4.
Multiply through by x^2 and putting y=x^2:
(y+4356)(1+2y+y^2)=24336y+312y^2+y^3;
y+2y^2+y^3+4356+8712y+4356y^2=24336y+312y^2+y^3;
y^2(312-2-4356)+y(24336-8712-1)-4356=0;
4046y^2-15623y+4356=0=(34y-121)(119y-36).
So y=121/34 or 36/119 making x=11/sqrt(34) or x=6/sqrt(119).
Both solutions need to be checked. sqrt(1+(66/x)^2)=35 or 120.
35+(121/34)34=156. OK!
120+(36/119)119=156. OK! So both results are correct for line 5.
But x=11/sqrt(34) gives -5 instead of 5 in the original equation, so the only solution is x=6/sqrt(119).
ALTERNATIVE METHOD
Let y^2=1+(66/x)^2, so x^2=66^2/(y^2-1).
sqrt((sqrt(x^2+66^2)+x)/x)=sqrt(y+1);
sqrt(x.sqrt(x^2+66^2)-x^2)=sqrt(x^2y-x^2)=x.sqrt(y-1).
So, sqrt(y+1)-x.sqrt(y-1)=5=sqrt(y+1)-(66/sqrt(y^2-1))sqrt(y-1);
sqrt(y+1)-66sqrt((y-1)/(y^2-1))=sqrt(y+1)-66/sqrt(y+1)=5.
Let z=sqrt(y+1): z-66/z=5; z^2-66=5z; z^2-5z-66=0; (z-11)(z+6)=0
So z=11 or -6 and z^2=121 or 36, making y+1=121 or 36, y=120 or 35.
x=66/sqrt(y^2-1)=sqrt(66^2/((119*121)=sqrt(36*121/((119*121)))=6/sqrt(119); or
sqrt(36*121/((34*36)))=11/sqrt(34).
We know from above that only one solution fits the original equation: x=6/sqrt(119).