\(\int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx\) [2395]

Optimal result

Integrand size = 26, antiderivative size = 192 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=-\frac {1}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \left (\frac {2 (4+x)}{3}-\sqrt {x+\sqrt {1+x}}\right )+\frac {3}{4} \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right )+4 \text {RootSum}\left [1+3 \text {$\#$1}-5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-3 \log \left (\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^3}{3-10 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

2*Sqrt[1 + x] + (2*(1 + x)^(3/2))/3 - (Sqrt[x + Sqrt[1 + x]]*(1 + 2*Sqrt[1 + x]))/2 + (5*ArcTanh[(1 + 2*Sqrt[1
+ x])/(2*Sqrt[x + Sqrt[1 + x]])])/4 + Log[2 - Sqrt[1 + x] - 3*(1 + x) + (1 + x)^2]/2 - (7*Defer[Subst][Defer[
Int][(2 - x - 3*x^2 + x^4)^(-1), x], x, Sqrt[1 + x]])/2 + 5*Defer[Subst][Defer[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]] + 4*Defer[Subst]
[Defer[Int][Sqrt[-1 + x + x^2]/(2 - x - 3*x^2 + x^4), x], x, Sqrt[1 + x]] - 2*Defer[Subst][Defer[Int][(x*Sqrt[
-1 + x + x^2])/(2 - x - 3*x^2 + x^4), x], x, Sqrt[1 + x]] - 4*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 \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*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.99 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\frac {2}{3} \sqrt {1+x} (4+x)+\frac {1}{2} \left (-1-2 \sqrt {1+x}\right ) \sqrt {x+\sqrt {1+x}}+\frac {3}{4} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+4 \text {RootSum}\left [1+3 \text {$\#$1}-5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-3 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{3-10 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]

[In]

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

[Out]

(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*Sqr
t[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) & ]

Maple [N/A] (verified)

Time = 0.09 (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 )\)	\(526\)

[In]

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

[Out]

-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=R
ootOf(_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))

Fricas [F(-1)]

Timed out. \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 1.50 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {x \sqrt {x + 1}}{x + \sqrt {x + \sqrt {x + 1}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {\sqrt {x + 1} x}{x + \sqrt {x + \sqrt {x + 1}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {\sqrt {x + 1} x}{x + \sqrt {x + \sqrt {x + 1}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {x\,\sqrt {x+1}}{x+\sqrt {x+\sqrt {x+1}}} \,d x \]

[In]

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

[Out]

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