integrate(sqrt(x+sqrt(x)),x)
- Subject: integrate(sqrt(x+sqrt(x)),x)
- From: Aleksas Domarkas
- Date: Sun, 30 Dec 2012 23:02:21 +0200
(%i1) S:'integrate(sqrt(x+sqrt(x)),x)$
(%i2) assume(y>0)$
(%i3) changevar(S, y^4=x, y, x);
(%o3) 4*integrate(y^4*sqrt(y^2+1),y)
(%i4) ev(%, nouns);
(%o4)
4*(asinh(y)/16+(y^3*(y^2+1)^(3/2))/6-(y*(y^2+1)^(3/2))/8+(y*sqrt(y^2+1))/16)
(%i5) subst(y=x^(1/4),%),expand,ratsimp;
(%o5)
(2*sqrt(sqrt(x)+1)*x^(5/4))/3+(sqrt(sqrt(x)+1)*x^(3/4))/6-(sqrt(sqrt(x)+1)*x^(1/4))/4+asinh(x^(1/4))/4
(%i6) sol:factor(%-asinh(x^(1/4))/4)+asinh(x^(1/4))/4;
(%o6) (sqrt(sqrt(x)+1)*x^(1/4)*(8*x+2*sqrt(x)-3))/12+asinh(x^(1/4))/4
Test:
(%i7) diff(sol,x)-sqrt(x+sqrt(x)),radcan;
(%o7) sqrt(sqrt(x)+1)*x^(1/4)-sqrt(x+sqrt(x))
(%i8) rootscontract(%);
(%o8) 0
best
Aleksas D