\(\int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx\) [2734]

Optimal result

Integrand size = 32, antiderivative size = 252 \[ \int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=x+\frac {4}{55} \left (5+4 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {4}{55} \left (-5+4 \sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )-\frac {4}{11} \text {RootSum}\left [-1+\text {$\#$1}-\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^5\&,\frac {3 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )+7 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-3 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3+2 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4}{1-2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+5 \text {$\#$1}^4}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx \]

[In]

Int[x^2/(x^2 - Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]]),x]

[Out]

-2*Sqrt[1 + x] + (1 + Sqrt[1 + x])^2 + (4*(5 - 4*Sqrt[5])*Log[1 - Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/55 + (4*
(5 + 4*Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (8*Log[2 + Sqrt[1 + x] - Sqrt[1 + Sqrt[1 + x]
] + 2*(1 + Sqrt[1 + x])^(3/2) - (1 + Sqrt[1 + x])^2 - (1 + Sqrt[1 + x])^(5/2)])/55 - (52*Defer[Subst][Defer[In
t][(-1 + x - x^2 - 2*x^3 + x^4 + x^5)^(-1), x], x, Sqrt[1 + Sqrt[1 + x]]])/55 - (156*Defer[Subst][Defer[Int][x
/(-1 + x - x^2 - 2*x^3 + x^4 + x^5), x], x, Sqrt[1 + Sqrt[1 + x]]])/55 + (12*Defer[Subst][Defer[Int][x^2/(-1 +
x - x^2 - 2*x^3 + x^4 + x^5), x], x, Sqrt[1 + Sqrt[1 + x]]])/55 + (112*Defer[Subst][Defer[Int][x^3/(-1 + x -
x^2 - 2*x^3 + x^4 + x^5), x], x, Sqrt[1 + Sqrt[1 + x]]])/55

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x \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+x^2\right )^2 \left (-1+x^2\right )}{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 (-x+x^3+\frac {x \left (1-2 x^2+x^4\right )}{1-x^2+4 x^3-4 x^5+x^7}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = -2 \sqrt {1+x}+\left (1+\sqrt {1+x}\right )^2+4 \text {Subst}\left (\int \frac {x \left (1-2 x^2+x^4\right )}{1-x^2+4 x^3-4 x^5+x^7} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = -2 \sqrt {1+x}+\left (1+\sqrt {1+x}\right )^2+4 \text {Subst}\left (\int \frac {x \left (-1+x^2\right )^2}{1-x^2+4 x^3-4 x^5+x^7} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = -2 \sqrt {1+x}+\left (1+\sqrt {1+x}\right )^2+4 \text {Subst}\left (\int \left (\frac {3+2 x}{11 \left (-1-x+x^2\right )}+\frac {-3-7 x+3 x^2+4 x^3-2 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}+\left (1+\sqrt {1+x}\right )^2+\frac {4}{11} \text {Subst}\left (\int \frac {3+2 x}{-1-x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {4}{11} \text {Subst}\left (\int \frac {-3-7 x+3 x^2+4 x^3-2 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}+\left (1+\sqrt {1+x}\right )^2-\frac {8}{55} \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 {-13-39 x+3 x^2+28 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 (4 \left (5-4 \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 (4 \left (5+4 \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}+\left (1+\sqrt {1+x}\right )^2+\frac {4}{55} \left (5-4 \sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {4}{55} \left (5+4 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {8}{55} \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 {13}{-1+x-x^2-2 x^3+x^4+x^5}-\frac {39 x}{-1+x-x^2-2 x^3+x^4+x^5}+\frac {3 x^2}{-1+x-x^2-2 x^3+x^4+x^5}+\frac {28 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}+\left (1+\sqrt {1+x}\right )^2+\frac {4}{55} \left (5-4 \sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {4}{55} \left (5+4 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {8}{55} \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}{55} \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 {52}{55} \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 )+\frac {112}{55} \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 {156}{55} \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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=x+\frac {4}{55} \left (5+4 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {4}{55} \left (-5+4 \sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )-\frac {4}{11} \text {RootSum}\left [-1+\text {$\#$1}-\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^5\&,\frac {3 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )+7 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-3 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3+2 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4}{1-2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+5 \text {$\#$1}^4}\&\right ] \]

[In]

Integrate[x^2/(x^2 - Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]]),x]

[Out]

x + (4*(5 + 4*Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (4*(-5 + 4*Sqrt[5])*Log[-1 + Sqrt[5] +
2*Sqrt[1 + Sqrt[1 + x]]])/55 - (4*RootSum[-1 + #1 - #1^2 - 2*#1^3 + #1^4 + #1^5 & , (3*Log[Sqrt[1 + Sqrt[1 +
x]] - #1] + 7*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 3*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 - 4*Log[Sqrt[1 + Sqr
t[1 + x]] - #1]*#1^3 + 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 - 2*#1 - 6*#1^2 + 4*#1^3 + 5*#1^4) & ])/11

Maple [N/A] (verified)

Time = 0.05 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.58

method	result	size
derivativedivides	\(\left (1+\sqrt {1+x}\right )^{2}-2-2 \sqrt {1+x}+\frac {4 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )}{11}-\frac {32 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{55}-\frac {4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{5}+\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-\textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\frac {\left (2 \textit {\_R}^{4}-4 \textit {\_R}^{3}-3 \textit {\_R}^{2}+7 \textit {\_R} +3\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{5 \textit {\_R}^{4}+4 \textit {\_R}^{3}-6 \textit {\_R}^{2}-2 \textit {\_R} +1}\right )}{11}\)	\(145\)
default	\(\left (1+\sqrt {1+x}\right )^{2}-2-2 \sqrt {1+x}+\frac {4 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )}{11}-\frac {32 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{55}-\frac {4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{5}+\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-\textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\frac {\left (2 \textit {\_R}^{4}-4 \textit {\_R}^{3}-3 \textit {\_R}^{2}+7 \textit {\_R} +3\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{5 \textit {\_R}^{4}+4 \textit {\_R}^{3}-6 \textit {\_R}^{2}-2 \textit {\_R} +1}\right )}{11}\)	\(145\)

[In]

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

[Out]

(1+(1+x)^(1/2))^2-2-2*(1+x)^(1/2)+4/11*ln((1+x)^(1/2)-(1+(1+x)^(1/2))^(1/2))-32/55*5^(1/2)*arctanh(1/5*(2*(1+(
1+x)^(1/2))^(1/2)-1)*5^(1/2))-4/11*sum((2*_R^4-4*_R^3-3*_R^2+7*_R+3)/(5*_R^4+4*_R^3-6*_R^2-2*_R+1)*ln((1+(1+x)
^(1/2))^(1/2)-_R),_R=RootOf(_Z^5+_Z^4-2*_Z^3-_Z^2+_Z-1))

Fricas [F(-1)]

Timed out. \[ \int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 162.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.10 \[ \int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\int \frac {x^{2}}{x^{2} - \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}}\, dx \]

[In]

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

[Out]

Integral(x**2/(x**2 - sqrt(x + 1)*sqrt(sqrt(x + 1) + 1)), x)

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.11 \[ \int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\int { \frac {x^{2}}{x^{2} - \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}} \,d x } \]

[In]

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

[Out]

integrate(x^2/(x^2 - sqrt(x + 1)*sqrt(sqrt(x + 1) + 1)), x)

Giac [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.11 \[ \int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\int { \frac {x^{2}}{x^{2} - \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}} \,d x } \]

[In]

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

[Out]

integrate(x^2/(x^2 - sqrt(x + 1)*sqrt(sqrt(x + 1) + 1)), x)

Mupad [N/A]

Not integrable

Time = 7.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.12 \[ \int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=-\int \frac {x^2}{\sqrt {\sqrt {x+1}+1}\,\sqrt {x+1}-x^2} \,d x \]

[In]

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

[Out]

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