\(\int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx\) [2809]

Optimal result

Integrand size = 39, antiderivative size = 275 \[ \int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\frac {2}{3} (1+x)^{3/2}+4 \sqrt {1+\sqrt {1+x}}+\frac {2}{55} \left (25+9 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {2}{55} \left (-25+9 \sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )-\frac {4}{11} \text {RootSum}\left [-1+\text {$\#$1}-\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^5\&,\frac {-9 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3+5 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4}{1-2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+5 \text {$\#$1}^4}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

2*Sqrt[1 + x] + 4*Sqrt[1 + Sqrt[1 + x]] - 2*(1 + Sqrt[1 + x])^2 + (2*(1 + Sqrt[1 + x])^3)/3 + (2*(25 - 9*Sqrt[
5])*Log[1 - Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/55 + (2*(25 + 9*Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 +
x]]])/55 - (4*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)])/11 + (40*Defer[Subst][Defer[Int][(-1 + x - x^2 - 2*x^3 + x^4 + x^5)^(-1), x], x, Sq
rt[1 + Sqrt[1 + x]]])/11 - (12*Defer[Subst][Defer[Int][x/(-1 + x - x^2 - 2*x^3 + x^4 + x^5), x], x, Sqrt[1 + S
qrt[1 + x]]])/11 - (16*Defer[Subst][Defer[Int][x^2/(-1 + x - x^2 - 2*x^3 + x^4 + x^5), x], x, Sqrt[1 + Sqrt[1
+ x]]])/11 + (12*Defer[Subst][Defer[Int][x^3/(-1 + x - x^2 - 2*x^3 + x^4 + x^5), x], x, Sqrt[1 + Sqrt[1 + x]]]
)/11

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*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.99 \[ \int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\frac {2}{165} \left (330 \sqrt {1+\sqrt {1+x}}+55 \left (-2+(1+x)^{3/2}\right )+3 \left (25+9 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-3 \left (-25+9 \sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )-30 \text {RootSum}\left [-1+\text {$\#$1}-\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^5\&,\frac {-9 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3+5 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4}{1-2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+5 \text {$\#$1}^4}\&\right ]\right ) \]

[In]

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

[Out]

(2*(330*Sqrt[1 + Sqrt[1 + x]] + 55*(-2 + (1 + x)^(3/2)) + 3*(25 + 9*Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + Sqrt
[1 + x]]] - 3*(-25 + 9*Sqrt[5])*Log[-1 + Sqrt[5] + 2*Sqrt[1 + Sqrt[1 + x]]] - 30*RootSum[-1 + #1 - #1^2 - 2*#1
^3 + #1^4 + #1^5 & , (-9*Log[Sqrt[1 + Sqrt[1 + x]] - #1] + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 2*Log[Sqrt[1 +
Sqrt[1 + x]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^3 + 5*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 -
2*#1 - 6*#1^2 + 4*#1^3 + 5*#1^4) & ]))/165

Maple [N/A] (verified)

Time = 0.26 (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	\(\frac {2 \left (1+\sqrt {1+x}\right )^{3}}{3}-2 \left (1+\sqrt {1+x}\right )^{2}+2 \sqrt {1+x}+2+4 \sqrt {1+\sqrt {1+x}}-\frac {4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{5}+\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-\textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\frac {\left (5 \textit {\_R}^{4}+\textit {\_R}^{3}-2 \textit {\_R}^{2}+\textit {\_R} -9\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{5 \textit {\_R}^{4}+4 \textit {\_R}^{3}-6 \textit {\_R}^{2}-2 \textit {\_R} +1}\right )}{11}+\frac {10 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )}{11}-\frac {36 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{55}\)	\(165\)

[In]

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

[Out]

2/3*(1+(1+x)^(1/2))^3-2*(1+(1+x)^(1/2))^2+2*(1+x)^(1/2)+2+4*(1+(1+x)^(1/2))^(1/2)-4/11*sum((5*_R^4+_R^3-2*_R^2
+_R-9)/(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))+10/11
*ln((1+x)^(1/2)-(1+(1+x)^(1/2))^(1/2))-36/55*5^(1/2)*arctanh(1/5*(2*(1+(1+x)^(1/2))^(1/2)-1)*5^(1/2))

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.12 \[ \int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\int { \frac {\sqrt {x + 1} x^{2}}{x^{2} - \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.12 \[ \int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=\int { \frac {\sqrt {x + 1} x^{2}}{x^{2} - \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 7.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.13 \[ \int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx=-\int \frac {x^2\,\sqrt {x+1}}{\sqrt {\sqrt {x+1}+1}\,\sqrt {x+1}-x^2} \,d x \]

[In]

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

[Out]

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