Integrand size = 26, antiderivative size = 192 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=-\frac {1}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \left (\frac {2 (4+x)}{3}-\sqrt {x+\sqrt {1+x}}\right )+\frac {3}{4} \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right )+4 \text {RootSum}\left [1+3 \text {$\#$1}-5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-3 \log \left (\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^3}{3-10 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
Time = 0.39 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.99 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\frac {2}{3} \sqrt {1+x} (4+x)+\frac {1}{2} \left (-1-2 \sqrt {1+x}\right ) \sqrt {x+\sqrt {1+x}}+\frac {3}{4} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+4 \text {RootSum}\left [1+3 \text {$\#$1}-5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-3 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{3-10 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
(2*Sqrt[1 + x]*(4 + x))/3 + ((-1 - 2*Sqrt[1 + x])*Sqrt[x + Sqrt[1 + x]])/2 + (3*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/4 + 4*RootSum[1 + 3*#1 - 5*#1^2 + 2*#1^3 + #1^4 & , (-3*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^2 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^3)/(3 - 10*#1 + 6*#1^2 + 4*#1^3) & ]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x \sqrt {x+1}}{x+\sqrt {x+\sqrt {x+1}}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int -\frac {x (x+1)}{-x-\sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {(x+1)^2}{x+\sqrt {x+\sqrt {x+1}}}-\frac {x+1}{x+\sqrt {x+\sqrt {x+1}}}\right )d\sqrt {x+1}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-\frac {7}{4} \int \frac {1}{(x+1)^2-3 (x+1)-\sqrt {x+1}+2}d\sqrt {x+1}+\frac {5}{2} \int \frac {\sqrt {x+1}}{(x+1)^2-3 (x+1)-\sqrt {x+1}+2}d\sqrt {x+1}+2 \int \frac {x+1}{(x+1)^2-3 (x+1)-\sqrt {x+1}+2}d\sqrt {x+1}+2 \int \frac {\sqrt {x+\sqrt {x+1}}}{(x+1)^2-3 (x+1)-\sqrt {x+1}+2}d\sqrt {x+1}-\int \frac {\sqrt {x+1} \sqrt {x+\sqrt {x+1}}}{(x+1)^2-3 (x+1)-\sqrt {x+1}+2}d\sqrt {x+1}-2 \int \frac {(x+1) \sqrt {x+\sqrt {x+1}}}{(x+1)^2-3 (x+1)-\sqrt {x+1}+2}d\sqrt {x+1}+\frac {5}{8} \text {arctanh}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )+\frac {1}{3} (x+1)^{3/2}+\sqrt {x+1}-\frac {1}{4} \sqrt {x+\sqrt {x+1}} \left (2 \sqrt {x+1}+1\right )+\frac {1}{4} \log \left ((x+1)^2-3 (x+1)-\sqrt {x+1}+2\right )\right )\) |
3.24.94.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
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 )\) | \(526\) |
-1/2*(2*(1+x)^(1/2)+1)*(x+(1+x)^(1/2))^(1/2)+5/4*ln(1/2+(1+x)^(1/2)+(x+(1+ x)^(1/2))^(1/2))+2*sum((_R^3-6*_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 *sum((-_R^3+2*_R^2-4*_R-9)/(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))+4*ln(-1-2*(1+x)^(1/2) +2*(x+(1+x)^(1/2))^(1/2))+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*(1+x)^(1/2)+4*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))+2/3*(1+x)^(3/2)-2*sum( (-_R^3-7*_R^2-3*_R+6)/(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*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))
Timed out. \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\text {Timed out} \]
Not integrable
Time = 1.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {x \sqrt {x + 1}}{x + \sqrt {x + \sqrt {x + 1}}}\, dx \]
Not integrable
Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {\sqrt {x + 1} x}{x + \sqrt {x + \sqrt {x + 1}}} \,d x } \]
Not integrable
Time = 1.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {\sqrt {x + 1} x}{x + \sqrt {x + \sqrt {x + 1}}} \,d x } \]
Not integrable
Time = 6.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {x\,\sqrt {x+1}}{x+\sqrt {x+\sqrt {x+1}}} \,d x \]