Integrand size = 25, antiderivative size = 186 \[ \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=2 \sqrt {1+x}+2 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right )+2 \text {RootSum}\left [25-2 \text {$\#$1}^2-8 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-5 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )-2 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}+3 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\text {$\#$1}-6 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \]
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\[ \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx \]
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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*}
Time = 0.00 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=2 \sqrt {1+x}+2 \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )-4 \text {RootSum}\left [1+3 \text {$\#$1}-5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )-2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}+3 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{3-10 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
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Time = 0.09 (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 | \(-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (-3 \textit {\_R}^{2}+2 \textit {\_R} +1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )+2 \ln \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} -3\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )+2 \sqrt {1+x}-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\left (-3 \textit {\_R}^{2}-\textit {\_R} +2\right ) \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )\) | \(271\) |
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Exception generated. \[ \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 1.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {\sqrt {x + 1}}{x + \sqrt {x + \sqrt {x + 1}}}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {\sqrt {x + 1}}{x + \sqrt {x + \sqrt {x + 1}}} \,d x } \]
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Not integrable
Time = 0.79 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {\sqrt {x + 1}}{x + \sqrt {x + \sqrt {x + 1}}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {\sqrt {x+1}}{x+\sqrt {x+\sqrt {x+1}}} \,d x \]
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