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

Optimal result

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

Unintegrable
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=x+\frac {4}{55} \left (5+4 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {4}{55} \left (-5+4 \sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )-\frac {4}{11} \text {RootSum}\left [-1+\text {$\#$1}-\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^5\&,\frac {3 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )+7 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-3 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3+2 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4}{1-2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+5 \text {$\#$1}^4}\&\right ] \] Input:

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

Output:

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*Roo 
tSum[-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 + Sqrt[1 + x]] - #1]*#1^3 + 2*Log[Sqrt[1 + Sq 
rt[1 + x]] - #1]*#1^4)/(1 - 2*#1 - 6*#1^2 + 4*#1^3 + 5*#1^4) & ])/11
 

Rubi [F]

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.

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

\(\Big \downarrow \) 7267

\(\displaystyle 2 \int -\frac {x^2 \sqrt {x+1}}{\sqrt {x+1} \sqrt {\sqrt {x+1}+1}-x^2}d\sqrt {x+1}\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \int \frac {x^2 \sqrt {x+1}}{\sqrt {x+1} \sqrt {\sqrt {x+1}+1}-x^2}d\sqrt {x+1}\)

\(\Big \downarrow \) 7267

\(\displaystyle 4 \int \frac {(1-x)^2 x (x+1)^2}{(x+1)^{7/2}-4 (x+1)^{5/2}+4 (x+1)^{3/2}-x}d\sqrt {\sqrt {x+1}+1}\)

\(\Big \downarrow \) 25

\(\displaystyle -4 \int -\frac {(1-x)^2 x (x+1)^2}{(x+1)^{7/2}-4 (x+1)^{5/2}+4 (x+1)^{3/2}-x}d\sqrt {\sqrt {x+1}+1}\)

\(\Big \downarrow \) 2462

\(\displaystyle -4 \int \left (-(x+1)^{3/2}+\frac {2 (x+1)^2-4 (x+1)^{3/2}-3 (x+1)+7 \sqrt {\sqrt {x+1}+1}+3}{11 \left ((x+1)^{5/2}+(x+1)^2-2 (x+1)^{3/2}-x+\sqrt {\sqrt {x+1}+1}-2\right )}+\sqrt {\sqrt {x+1}+1}+\frac {-2 \sqrt {\sqrt {x+1}+1}-3}{11 \left (x-\sqrt {\sqrt {x+1}+1}\right )}\right )d\sqrt {\sqrt {x+1}+1}\)

\(\Big \downarrow \) 2009

\(\displaystyle 4 \left (-\frac {13}{55} \int \frac {1}{(x+1)^{5/2}+(x+1)^2-2 (x+1)^{3/2}-x+\sqrt {\sqrt {x+1}+1}-2}d\sqrt {\sqrt {x+1}+1}+\frac {3}{55} \int \frac {x+1}{(x+1)^{5/2}+(x+1)^2-2 (x+1)^{3/2}-x+\sqrt {\sqrt {x+1}+1}-2}d\sqrt {\sqrt {x+1}+1}+\frac {28}{55} \int \frac {(x+1)^{3/2}}{(x+1)^{5/2}+(x+1)^2-2 (x+1)^{3/2}-x+\sqrt {\sqrt {x+1}+1}-2}d\sqrt {\sqrt {x+1}+1}-\frac {39}{55} \int \frac {\sqrt {\sqrt {x+1}+1}}{(x+1)^{5/2}+(x+1)^2-2 (x+1)^{3/2}-x+\sqrt {\sqrt {x+1}+1}-2}d\sqrt {\sqrt {x+1}+1}+\frac {1}{4} (x+1)^2+\frac {1}{2} (-x-1)+\frac {1}{55} \left (5-4 \sqrt {5}\right ) \log \left (-2 \sqrt {\sqrt {x+1}+1}-\sqrt {5}+1\right )-\frac {2}{55} \log \left ((x+1)^{5/2}+(x+1)^2-2 (x+1)^{3/2}-x+\sqrt {\sqrt {x+1}+1}-2\right )+\frac {1}{55} \left (5+4 \sqrt {5}\right ) \log \left (-2 \sqrt {5} \sqrt {\sqrt {x+1}+1}+\sqrt {5}+5\right )\right )\)

Input:

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

Output:

$Aborted
 

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

Input:

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

Output:

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.11 \[ \int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\int { \frac {x^{2}}{x^{2} - \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}} \,d x } \] Input:

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

Output:

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

Giac [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.11 \[ \int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\int { \frac {x^{2}}{x^{2} - \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}} \,d x } \] Input:

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

Output:

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

Mupad [N/A]

Not integrable

Time = 8.95 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.12 \[ \int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=-\int \frac {x^2}{\sqrt {\sqrt {x+1}+1}\,\sqrt {x+1}-x^2} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 200.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.11 \[ \int \frac {x^2}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\int \frac {x^{2}}{x^{2}-\sqrt {x +1}\, \sqrt {1+\sqrt {x +1}}}d x \] Input:

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

Output:

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