Integrand size = 26, antiderivative size = 192 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=-\frac {1}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \left (\frac {2 (4+x)}{3}-\sqrt {x+\sqrt {1+x}}\right )+\frac {3}{4} \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right )+4 \text {RootSum}\left [1+3 \text {$\#$1}-5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-3 \log \left (\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^3}{3-10 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
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\[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^2 \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^2}{-1+x^2+\sqrt {-1+x+x^2}}+\frac {x^4}{-1+x^2+\sqrt {-1+x+x^2}}\right ) \, dx,x,\sqrt {1+x}\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {x^2}{-1+x^2+\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\right )+2 \text {Subst}\left (\int \frac {x^4}{-1+x^2+\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ & = -\left (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 )\right )+2 \text {Subst}\left (\int \left (2+x^2-\sqrt {-1+x+x^2}+\frac {2 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4}-\frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4}-\frac {3 x^2 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4}+\frac {-4+2 x+4 x^2+x^3}{2-x-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \sqrt {1+x}+\frac {2}{3} (1+x)^{3/2}-2 \text {Subst}\left (\int \sqrt {-1+x+x^2} \, 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^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 \text {Subst}\left (\int \frac {-4+2 x+4 x^2+x^3}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+4 \text {Subst}\left (\int \frac {\sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-6 \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 \sqrt {1+x}+\frac {2}{3} (1+x)^{3/2}-\frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )+\frac {1}{2} \log \left (2-\sqrt {1+x}-3 (1+x)+(1+x)^2\right )+\frac {1}{2} \text {Subst}\left (\int \frac {-15+14 x+16 x^2}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+\frac {5}{4} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2}} \, 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^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 )+4 \text {Subst}\left (\int \frac {\sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-6 \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 \sqrt {1+x}+\frac {2}{3} (1+x)^{3/2}-\frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )+\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 {15}{2-x-3 x^2+x^4}+\frac {14 x}{2-x-3 x^2+x^4}+\frac {16 x^2}{2-x-3 x^2+x^4}\right ) \, 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 \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^2 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+\frac {5}{2} \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {1+x}}{\sqrt {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 )+4 \text {Subst}\left (\int \frac {\sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-6 \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 \sqrt {1+x}+\frac {2}{3} (1+x)^{3/2}-\frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )+\frac {5}{4} \text {arctanh}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )+\frac {1}{2} \log \left (2-\sqrt {1+x}-3 (1+x)+(1+x)^2\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 \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^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 )+4 \text {Subst}\left (\int \frac {\sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-6 \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 )+7 \text {Subst}\left (\int \frac {x}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-\frac {15}{2} \text {Subst}\left (\int \frac {1}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+8 \text {Subst}\left (\int \frac {x^2}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right ) \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.99 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\frac {2}{3} \sqrt {1+x} (4+x)+\frac {1}{2} \left (-1-2 \sqrt {1+x}\right ) \sqrt {x+\sqrt {1+x}}+\frac {3}{4} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+4 \text {RootSum}\left [1+3 \text {$\#$1}-5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-3 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{3-10 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
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Time = 0.22 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.74
method | result | size |
derivativedivides | \(-\frac {\left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{2}+\frac {5 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}+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}-6 \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 \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}+2 \textit {\_R}^{2}-4 \textit {\_R} -9\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 )+4 \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}^{2} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+2 \sqrt {1+x}+4 \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 )+\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\left (-\textit {\_R}^{3}-7 \textit {\_R}^{2}-3 \textit {\_R} +6\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 (-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 \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 )\) | \(526\) |
default | \(-\frac {\left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{2}+\frac {5 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}+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}-6 \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 \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}+2 \textit {\_R}^{2}-4 \textit {\_R} -9\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 )+4 \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}^{2} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+2 \sqrt {1+x}+4 \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 )+\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\left (-\textit {\_R}^{3}-7 \textit {\_R}^{2}-3 \textit {\_R} +6\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 (-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 \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 )\) | \(526\) |
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Timed out. \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\text {Timed out} \]
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Not integrable
Time = 1.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {x \sqrt {x + 1}}{x + \sqrt {x + \sqrt {x + 1}}}\, dx \]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {\sqrt {x + 1} x}{x + \sqrt {x + \sqrt {x + 1}}} \,d x } \]
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Not integrable
Time = 1.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {\sqrt {x + 1} x}{x + \sqrt {x + \sqrt {x + 1}}} \,d x } \]
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Not integrable
Time = 6.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {x\,\sqrt {x+1}}{x+\sqrt {x+\sqrt {x+1}}} \,d x \]
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