3.24.54 \(\int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx\)

Optimal. Leaf size=186 \[ 2 \text {RootSum}\left [\text {$\#$1}^4-8 \text {$\#$1}^3-2 \text {$\#$1}^2+25\& ,\frac {3 \text {$\#$1}^2 \log \left (-\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )-2 \text {$\#$1} \log \left (-\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )-5 \log \left (-\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )}{\text {$\#$1}^3-6 \text {$\#$1}^2-\text {$\#$1}}\& \right ]+2 \sqrt {x+1}+2 \log \left (2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right ) \]

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Rubi [F]  time = 0.47, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

2*Sqrt[1 + x] - 4*Defer[Subst][Defer[Int][(2 - x - 3*x^2 + x^4)^(-1), x], x, Sqrt[1 + x]] + 2*Defer[Subst][Def
er[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]] - 2*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*} \int \frac {\sqrt {1+x}}{x+\sqrt {x+\sqrt {1+x}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2+\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {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 )\\ &=2 \sqrt {1+x}-2 \operatorname {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 \operatorname {Subst}\left (\int \frac {-2+x+2 x^2}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ &=2 \sqrt {1+x}-2 \operatorname {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 \operatorname {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 )\\ &=2 \sqrt {1+x}+2 \operatorname {Subst}\left (\int \frac {x}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-2 \operatorname {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 \operatorname {Subst}\left (\int \frac {1}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+4 \operatorname {Subst}\left (\int \frac {x^2}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ \end {align*}

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Mathematica [B]  time = 6.31, size = 6615, normalized size = 35.56 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.00, size = 184, normalized size = 0.99 \begin {gather*} 2 \sqrt {1+x}+2 \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )-4 \text {RootSum}\left [1+3 \text {$\#$1}-5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )-2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}+3 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{3-10 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 + x]/(x + Sqrt[x + Sqrt[1 + x]]),x]

[Out]

2*Sqrt[1 + x] + 2*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] - 4*RootSum[1 + 3*#1 - 5*#1^2 + 2*#1^3 + #
1^4 & , (-Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] - 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1
+ 3*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^2)/(3 - 10*#1 + 6*#1^2 + 4*#1^3) & ]

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x + 1}}{x + \sqrt {x + \sqrt {x + 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.12, size = 271, normalized size = 1.46

method	result	size
derivativedivides	\(-2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (-3 \textit {\_R}^{2}+2 \textit {\_R} +1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} -3\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )+2 \ln \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )+2 \sqrt {1+x}-2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\left (-3 \textit {\_R}^{2}-\textit {\_R} +2\right ) \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )\)	\(271\)
default	\(-2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (-3 \textit {\_R}^{2}+2 \textit {\_R} +1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} -3\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )+2 \ln \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )+2 \sqrt {1+x}-2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\left (-3 \textit {\_R}^{2}-\textit {\_R} +2\right ) \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )\)	\(271\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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*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))+2*ln(-1-
2*(1+x)^(1/2)+2*(x+(1+x)^(1/2))^(1/2))+2*(1+x)^(1/2)-2*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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x + 1}}{x + \sqrt {x + \sqrt {x + 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {x+1}}{x+\sqrt {x+\sqrt {x+1}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x + 1}}{x + \sqrt {x + \sqrt {x + 1}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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