\(\int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx\) [2135]

Optimal result

Integrand size = 19, antiderivative size = 155 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx=x-2 \sqrt {x+\sqrt {1+x}}-\log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right )+4 \text {RootSum}\left [1+3 \text {$\#$1}-5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log \left (\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^3}{3-10 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx \]

[In]

Int[x/(x + Sqrt[x + Sqrt[1 + x]]),x]

[Out]

x + Log[2 - Sqrt[1 + x] - 3*(1 + x) + (1 + x)^2]/2 + Defer[Subst][Defer[Int][(2 - x - 3*x^2 + x^4)^(-1), x], x
, Sqrt[1 + x]]/2 + Defer[Subst][Defer[Int][x/(2 - x - 3*x^2 + x^4), x], x, Sqrt[1 + x]] + 2*Defer[Subst][Defer
[Int][x^2/(2 - x - 3*x^2 + x^4), x], x, Sqrt[1 + x]] + 2*Defer[Subst][Defer[Int][(x*Sqrt[-1 + x + x^2])/(2 - x
- 3*x^2 + x^4), x], x, Sqrt[1 + x]] - 2*Defer[Subst][Defer[Int][(x^3*Sqrt[-1 + x + x^2])/(2 - x - 3*x^2 + x^4
), x], x, Sqrt[1 + x]]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x \left (-1+x^2\right )}{-1+x^2+\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {x}{-1+x^2+\sqrt {-1+x+x^2}}+\frac {x^3}{-1+x^2+\sqrt {-1+x+x^2}}\right ) \, dx,x,\sqrt {1+x}\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {x}{-1+x^2+\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\right )+2 \text {Subst}\left (\int \frac {x^3}{-1+x^2+\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ & = -\left (2 \text {Subst}\left (\int \left (\frac {x \left (-1+x^2\right )}{2-x-3 x^2+x^4}-\frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+x}\right )\right )+2 \text {Subst}\left (\int \left (x-\frac {x^3 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4}+\frac {x \left (-2+x+2 x^2\right )}{2-x-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+x}\right ) \\ & = x-2 \text {Subst}\left (\int \frac {x \left (-1+x^2\right )}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \text {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-2 \text {Subst}\left (\int \frac {x^3 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \text {Subst}\left (\int \frac {x \left (-2+x+2 x^2\right )}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right ) \\ & = x+\frac {1}{2} \log \left (2-\sqrt {1+x}-3 (1+x)+(1+x)^2\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1+2 x}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {2+4 x+4 x^2}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \text {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-2 \text {Subst}\left (\int \frac {x^3 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right ) \\ & = x+\frac {1}{2} \log \left (2-\sqrt {1+x}-3 (1+x)+(1+x)^2\right )-\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{2-x-3 x^2+x^4}+\frac {2 x}{2-x-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} \text {Subst}\left (\int \left (\frac {2}{2-x-3 x^2+x^4}+\frac {4 x}{2-x-3 x^2+x^4}+\frac {4 x^2}{2-x-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+x}\right )+2 \text {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-2 \text {Subst}\left (\int \frac {x^3 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right ) \\ & = x+\frac {1}{2} \log \left (2-\sqrt {1+x}-3 (1+x)+(1+x)^2\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \text {Subst}\left (\int \frac {x}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \text {Subst}\left (\int \frac {x^2}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \text {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-2 \text {Subst}\left (\int \frac {x^3 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+\text {Subst}\left (\int \frac {1}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-\text {Subst}\left (\int \frac {x}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.03 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx=x-2 \sqrt {x+\sqrt {1+x}}-\log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+4 \text {RootSum}\left [1+3 \text {$\#$1}-5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{3-10 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]

[In]

Integrate[x/(x + Sqrt[x + Sqrt[1 + x]]),x]

[Out]

x - 2*Sqrt[x + Sqrt[1 + x]] - Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] + 4*RootSum[1 + 3*#1 - 5*#1^2
+ 2*#1^3 + #1^4 & , (Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 +
x]] - #1]*#1^3)/(3 - 10*#1 + 6*#1^2 + 4*#1^3) & ]

Maple [N/A] (verified)

Time = 0.18 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.24

method	result	size
derivativedivides	\(-\sqrt {x +\sqrt {1+x}}+\sqrt {1+x}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}+\textit {\_R}^{2}-\textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}-3 \textit {\_R}^{2}-\textit {\_R} +6\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )+\frac {5}{2 \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )}+\ln \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )-4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+1+x +2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\left (3 \textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R}^{2}+2\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )\)	\(502\)
default	\(-\sqrt {x +\sqrt {1+x}}+\sqrt {1+x}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}+\textit {\_R}^{2}-\textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}-3 \textit {\_R}^{2}-\textit {\_R} +6\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )+\frac {5}{2 \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )}+\ln \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )-4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+1+x +2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\left (3 \textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R}^{2}+2\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )\)	\(502\)

[In]

int(x/(x+(x+(1+x)^(1/2))^(1/2)),x,method=_RETURNVERBOSE)

[Out]

-(x+(1+x)^(1/2))^(1/2)+(1+x)^(1/2)+2*sum((2*_R^3+_R^2-_R)/(4*_R^3+6*_R^2-10*_R+3)*ln((x+(1+x)^(1/2))^(1/2)-(1+
x)^(1/2)-_R),_R=RootOf(_Z^4+2*_Z^3-5*_Z^2+3*_Z+1))-2*sum((2*_R^3-3*_R^2-_R+6)/(4*_R^3-6*_R^2+2*_R+5)*ln((x+(1+
x)^(1/2))^(1/2)-(1+x)^(1/2)-_R),_R=RootOf(_Z^4-2*_Z^3+_Z^2+5*_Z-1))+5/2/(-1-2*(1+x)^(1/2)+2*(x+(1+x)^(1/2))^(1
/2))+ln(-1-2*(1+x)^(1/2)+2*(x+(1+x)^(1/2))^(1/2))+2*sum(_R/(4*_R^3-6*_R-1)*ln((1+x)^(1/2)-_R),_R=RootOf(_Z^4-3
*_Z^2-_Z+2))-4*sum(_R^3/(4*_R^3-6*_R-1)*ln((1+x)^(1/2)-_R),_R=RootOf(_Z^4-3*_Z^2-_Z+2))+1+x+2*sum((3*_R^3+_R^2
-2*_R)/(4*_R^3-6*_R-1)*ln((1+x)^(1/2)-_R),_R=RootOf(_Z^4-3*_Z^2-_Z+2))-2*sum((_R^3+_R^2-2*_R)/(4*_R^3+6*_R^2-1
0*_R+3)*ln((x+(1+x)^(1/2))^(1/2)-(1+x)^(1/2)-_R),_R=RootOf(_Z^4+2*_Z^3-5*_Z^2+3*_Z+1))+2*sum((_R^3-_R^2+2)/(4*
_R^3-6*_R^2+2*_R+5)*ln((x+(1+x)^(1/2))^(1/2)-(1+x)^(1/2)-_R),_R=RootOf(_Z^4-2*_Z^3+_Z^2+5*_Z-1))

Fricas [F(-1)]

Timed out. \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\text {Timed out} \]

[In]

integrate(x/(x+(x+(1+x)^(1/2))^(1/2)),x, algorithm="fricas")

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 2.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.10 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {x}{x + \sqrt {x + \sqrt {x + 1}}}\, dx \]

[In]

integrate(x/(x+(x+(1+x)**(1/2))**(1/2)),x)

[Out]

Integral(x/(x + sqrt(x + sqrt(x + 1))), x)

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.11 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {x}{x + \sqrt {x + \sqrt {x + 1}}} \,d x } \]

[In]

integrate(x/(x+(x+(1+x)^(1/2))^(1/2)),x, algorithm="maxima")

[Out]

integrate(x/(x + sqrt(x + sqrt(x + 1))), x)

Giac [N/A]

Not integrable

Time = 1.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.11 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {x}{x + \sqrt {x + \sqrt {x + 1}}} \,d x } \]

[In]

integrate(x/(x+(x+(1+x)^(1/2))^(1/2)),x, algorithm="giac")

[Out]

integrate(x/(x + sqrt(x + sqrt(x + 1))), x)

Mupad [N/A]

Not integrable

Time = 5.70 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.11 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {x}{x+\sqrt {x+\sqrt {x+1}}} \,d x \]

[In]

int(x/(x + (x + (x + 1)^(1/2))^(1/2)),x)

[Out]

int(x/(x + (x + (x + 1)^(1/2))^(1/2)), x)