Optimal. Leaf size=275 \[ -\frac {4}{11} \text {RootSum}\left [\text {$\#$1}^5+\text {$\#$1}^4-2 \text {$\#$1}^3-\text {$\#$1}^2+\text {$\#$1}-1\& ,\frac {5 \text {$\#$1}^4 \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )+\text {$\#$1}^3 \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )-2 \text {$\#$1}^2 \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )+\text {$\#$1} \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )-9 \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )}{5 \text {$\#$1}^4+4 \text {$\#$1}^3-6 \text {$\#$1}^2-2 \text {$\#$1}+1}\& \right ]+\frac {2}{3} (x+1)^{3/2}+4 \sqrt {\sqrt {x+1}+1}+\frac {2}{55} \left (25+9 \sqrt {5}\right ) \log \left (-2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}+1\right )-\frac {2}{55} \left (9 \sqrt {5}-25\right ) \log \left (2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}-1\right ) \]
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Rubi [F] time = 1.27, 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 {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {1+x}}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx &=2 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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} \operatorname {Subst}\left (\int \frac {-2-5 x}{-1-x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-\frac {4}{11} \operatorname {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} \operatorname {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 ) \operatorname {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 ) \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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*}
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Mathematica [B] time = 3.29, size = 1912, normalized size = 6.95
result too large to display
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.00, size = 277, normalized size = 1.01 \begin {gather*} 4 \sqrt {1+\sqrt {1+x}}+\frac {2}{3} \left (-2+(1+x)^{3/2}\right )+\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 ] \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x + 1} x^{2}}{x^{2} - \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 169, normalized size = 0.61
method | result | size |
derivativedivides | \(\frac {2 \left (1+\sqrt {1+x}\right )^{3}}{3}-2 \left (1+\sqrt {1+x}\right )^{2}+2 \sqrt {1+x}+2+4 \sqrt {1+\sqrt {1+x}}+\frac {10 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )}{11}-\frac {36 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{55}+\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{5}+\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-\textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\frac {\left (-5 \textit {\_R}^{4}-\textit {\_R}^{3}+2 \textit {\_R}^{2}-\textit {\_R} +9\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{5 \textit {\_R}^{4}+4 \textit {\_R}^{3}-6 \textit {\_R}^{2}-2 \textit {\_R} +1}\right )}{11}\) | \(169\) |
default | \(\frac {2 \left (1+\sqrt {1+x}\right )^{3}}{3}-2 \left (1+\sqrt {1+x}\right )^{2}+2 \sqrt {1+x}+2+4 \sqrt {1+\sqrt {1+x}}+\frac {10 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )}{11}-\frac {36 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{55}+\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{5}+\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-\textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\frac {\left (-5 \textit {\_R}^{4}-\textit {\_R}^{3}+2 \textit {\_R}^{2}-\textit {\_R} +9\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{5 \textit {\_R}^{4}+4 \textit {\_R}^{3}-6 \textit {\_R}^{2}-2 \textit {\_R} +1}\right )}{11}\) | \(169\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x + 1} x^{2}}{x^{2} - \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {x^2\,\sqrt {x+1}}{\sqrt {\sqrt {x+1}+1}\,\sqrt {x+1}-x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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