Integrand size = 39, antiderivative size = 275 \[ \int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\frac {2}{3} (1+x)^{3/2}+4 \sqrt {1+\sqrt {1+x}}+\frac {2}{55} \left (25+9 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {2}{55} \left (-25+9 \sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )-\frac {4}{11} \text {RootSum}\left [-1+\text {$\#$1}-\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^5\&,\frac {-9 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3+5 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4}{1-2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+5 \text {$\#$1}^4}\&\right ] \]
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\[ \int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^2 \left (-1+x^2\right )^2}{-x \sqrt {1+x}+\left (-1+x^2\right )^2} \, dx,x,\sqrt {1+x}\right ) \\ & = 4 \text {Subst}\left (\int \frac {x^4 \left (2-3 x^2+x^4\right )^2}{1-x^2+4 x^3-4 x^5+x^7} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 4 \text {Subst}\left (\int \left (1+x-2 x^3+x^5-\frac {1+x-x^2+x^3-x^5}{1-x^2+4 x^3-4 x^5+x^7}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 2 \sqrt {1+x}+4 \sqrt {1+\sqrt {1+x}}-2 \left (1+\sqrt {1+x}\right )^2+\frac {2}{3} \left (1+\sqrt {1+x}\right )^3-4 \text {Subst}\left (\int \frac {1+x-x^2+x^3-x^5}{1-x^2+4 x^3-4 x^5+x^7} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 2 \sqrt {1+x}+4 \sqrt {1+\sqrt {1+x}}-2 \left (1+\sqrt {1+x}\right )^2+\frac {2}{3} \left (1+\sqrt {1+x}\right )^3-4 \text {Subst}\left (\int \left (\frac {-2-5 x}{11 \left (-1-x+x^2\right )}+\frac {-9+x-2 x^2+x^3+5 x^4}{11 \left (-1+x-x^2-2 x^3+x^4+x^5\right )}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 2 \sqrt {1+x}+4 \sqrt {1+\sqrt {1+x}}-2 \left (1+\sqrt {1+x}\right )^2+\frac {2}{3} \left (1+\sqrt {1+x}\right )^3-\frac {4}{11} \text {Subst}\left (\int \frac {-2-5 x}{-1-x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-\frac {4}{11} \text {Subst}\left (\int \frac {-9+x-2 x^2+x^3+5 x^4}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 2 \sqrt {1+x}+4 \sqrt {1+\sqrt {1+x}}-2 \left (1+\sqrt {1+x}\right )^2+\frac {2}{3} \left (1+\sqrt {1+x}\right )^3-\frac {4}{11} \log \left (2+\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}+2 \left (1+\sqrt {1+x}\right )^{3/2}-\left (1+\sqrt {1+x}\right )^2-\left (1+\sqrt {1+x}\right )^{5/2}\right )-\frac {4}{55} \text {Subst}\left (\int \frac {-50+15 x+20 x^2-15 x^3}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {1}{55} \left (2 \left (25-9 \sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {1}{55} \left (2 \left (25+9 \sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 2 \sqrt {1+x}+4 \sqrt {1+\sqrt {1+x}}-2 \left (1+\sqrt {1+x}\right )^2+\frac {2}{3} \left (1+\sqrt {1+x}\right )^3+\frac {2}{55} \left (25-9 \sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {2}{55} \left (25+9 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {4}{11} \log \left (2+\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}+2 \left (1+\sqrt {1+x}\right )^{3/2}-\left (1+\sqrt {1+x}\right )^2-\left (1+\sqrt {1+x}\right )^{5/2}\right )-\frac {4}{55} \text {Subst}\left (\int \left (-\frac {50}{-1+x-x^2-2 x^3+x^4+x^5}+\frac {15 x}{-1+x-x^2-2 x^3+x^4+x^5}+\frac {20 x^2}{-1+x-x^2-2 x^3+x^4+x^5}-\frac {15 x^3}{-1+x-x^2-2 x^3+x^4+x^5}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 2 \sqrt {1+x}+4 \sqrt {1+\sqrt {1+x}}-2 \left (1+\sqrt {1+x}\right )^2+\frac {2}{3} \left (1+\sqrt {1+x}\right )^3+\frac {2}{55} \left (25-9 \sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {2}{55} \left (25+9 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {4}{11} \log \left (2+\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}+2 \left (1+\sqrt {1+x}\right )^{3/2}-\left (1+\sqrt {1+x}\right )^2-\left (1+\sqrt {1+x}\right )^{5/2}\right )-\frac {12}{11} \text {Subst}\left (\int \frac {x}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {12}{11} \text {Subst}\left (\int \frac {x^3}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-\frac {16}{11} \text {Subst}\left (\int \frac {x^2}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {40}{11} \text {Subst}\left (\int \frac {1}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.99 \[ \int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\frac {2}{165} \left (330 \sqrt {1+\sqrt {1+x}}+55 \left (-2+(1+x)^{3/2}\right )+3 \left (25+9 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-3 \left (-25+9 \sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )-30 \text {RootSum}\left [-1+\text {$\#$1}-\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^5\&,\frac {-9 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3+5 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4}{1-2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+5 \text {$\#$1}^4}\&\right ]\right ) \]
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Time = 0.09 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.60
method | result | size |
derivativedivides | \(\frac {2 \left (1+\sqrt {1+x}\right )^{3}}{3}-2 \left (1+\sqrt {1+x}\right )^{2}+2 \sqrt {1+x}+2+4 \sqrt {1+\sqrt {1+x}}-\frac {4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{5}+\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-\textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\frac {\left (5 \textit {\_R}^{4}+\textit {\_R}^{3}-2 \textit {\_R}^{2}+\textit {\_R} -9\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{5 \textit {\_R}^{4}+4 \textit {\_R}^{3}-6 \textit {\_R}^{2}-2 \textit {\_R} +1}\right )}{11}+\frac {10 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )}{11}-\frac {36 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{55}\) | \(165\) |
default | \(\frac {2 \left (1+\sqrt {1+x}\right )^{3}}{3}-2 \left (1+\sqrt {1+x}\right )^{2}+2 \sqrt {1+x}+2+4 \sqrt {1+\sqrt {1+x}}-\frac {4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{5}+\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-\textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\frac {\left (5 \textit {\_R}^{4}+\textit {\_R}^{3}-2 \textit {\_R}^{2}+\textit {\_R} -9\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{5 \textit {\_R}^{4}+4 \textit {\_R}^{3}-6 \textit {\_R}^{2}-2 \textit {\_R} +1}\right )}{11}+\frac {10 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )}{11}-\frac {36 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{55}\) | \(165\) |
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Timed out. \[ \int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.12 \[ \int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\int { \frac {\sqrt {x + 1} x^{2}}{x^{2} - \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}} \,d x } \]
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Not integrable
Time = 0.41 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.12 \[ \int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\int { \frac {\sqrt {x + 1} x^{2}}{x^{2} - \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.13 \[ \int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=-\int \frac {x^2\,\sqrt {x+1}}{\sqrt {\sqrt {x+1}+1}\,\sqrt {x+1}-x^2} \,d x \]
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