\(\int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx\) [2353]

Optimal result

Integrand size = 25, antiderivative size = 186 \[ \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=2 \sqrt {1+x}+2 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right )+2 \text {RootSum}\left [25-2 \text {$\#$1}^2-8 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-5 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )-2 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}+3 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\text {$\#$1}-6 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

2*Sqrt[1 + x] - 4*Defer[Subst][Defer[Int][(2 - x - 3*x^2 + x^4)^(-1), x], x, Sqrt[1 + x]] + 2*Defer[Subst][Def
er[Int][x/(2 - x - 3*x^2 + x^4), x], x, Sqrt[1 + x]] + 4*Defer[Subst][Defer[Int][x^2/(2 - x - 3*x^2 + x^4), x]
, x, Sqrt[1 + x]] - 2*Defer[Subst][Defer[Int][(x^2*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^2}{-1+x^2+\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \text {Subst}\left (\int \left (1-\frac {x^2 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4}+\frac {-2+x+2 x^2}{2-x-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \sqrt {1+x}-2 \text {Subst}\left (\int \frac {x^2 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \text {Subst}\left (\int \frac {-2+x+2 x^2}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \sqrt {1+x}-2 \text {Subst}\left (\int \frac {x^2 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \text {Subst}\left (\int \left (-\frac {2}{2-x-3 x^2+x^4}+\frac {x}{2-x-3 x^2+x^4}+\frac {2 x^2}{2-x-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \sqrt {1+x}+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 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-4 \text {Subst}\left (\int \frac {1}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+4 \text {Subst}\left (\int \frac {x^2}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=2 \sqrt {1+x}+2 \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )-4 \text {RootSum}\left [1+3 \text {$\#$1}-5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )-2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}+3 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{3-10 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 0.14 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.46

method	result	size
derivativedivides	\(-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (-3 \textit {\_R}^{2}+2 \textit {\_R} +1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )+2 \ln \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} -3\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )+2 \sqrt {1+x}-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\left (-3 \textit {\_R}^{2}-\textit {\_R} +2\right ) \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )\)	\(271\)
default	\(-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (-3 \textit {\_R}^{2}+2 \textit {\_R} +1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )+2 \ln \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} -3\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )+2 \sqrt {1+x}-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\left (-3 \textit {\_R}^{2}-\textit {\_R} +2\right ) \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )\)	\(271\)

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

Sympy [N/A]

Not integrable

Time = 1.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {\sqrt {x + 1}}{x + \sqrt {x + \sqrt {x + 1}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {\sqrt {x + 1}}{x + \sqrt {x + \sqrt {x + 1}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.80 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {\sqrt {x + 1}}{x + \sqrt {x + \sqrt {x + 1}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {\sqrt {x+1}}{x+\sqrt {x+\sqrt {x+1}}} \,d x \]

[In]

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

[Out]

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