Optimal. Leaf size=157 \[ -\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{\sqrt [4]{3}}+\frac {\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{\sqrt [4]{3}}+\frac {\log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2} \sqrt [4]{3}} \]
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Rubi [A]
time = 0.08, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {708, 335, 303,
1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\sqrt {2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt [4]{3}}+\frac {\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt [4]{3}}+\frac {\log \left (2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\log \left (2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{\sqrt {2} \sqrt [4]{3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 708
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {\sqrt {1+2 x}}{1+x+x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {x}}{\frac {3}{4}+\frac {x^2}{4}} \, dx,x,1+2 x\right )\\ &=\text {Subst}\left (\int \frac {x^2}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {3}-x^2}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )\right )+\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {3}+x^2}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )\\ &=\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{3}+2 x}{-\sqrt {3}-\sqrt {2} \sqrt [4]{3} x-x^2} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {2} \sqrt [4]{3}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{3}-2 x}{-\sqrt {3}+\sqrt {2} \sqrt [4]{3} x-x^2} \, dx,x,\sqrt {1+2 x}\right )}{\sqrt {2} \sqrt [4]{3}}+\text {Subst}\left (\int \frac {1}{\sqrt {3}-\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt {1+2 x}\right )+\text {Subst}\left (\int \frac {1}{\sqrt {3}+\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt {1+2 x}\right )\\ &=\frac {\log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2} \sqrt [4]{3}}+\frac {\sqrt {2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2+4 x}}{\sqrt [4]{3}}\right )}{\sqrt [4]{3}}-\frac {\sqrt {2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2+4 x}}{\sqrt [4]{3}}\right )}{\sqrt [4]{3}}\\ &=-\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{\sqrt [4]{3}}+\frac {\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{\sqrt [4]{3}}+\frac {\log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2} \sqrt [4]{3}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 78, normalized size = 0.50 \begin {gather*} \frac {\sqrt {2} \left (\tan ^{-1}\left (\frac {-3+\sqrt {3}+2 \sqrt {3} x}{3^{3/4} \sqrt {2+4 x}}\right )-\tanh ^{-1}\left (\frac {3^{3/4} \sqrt {2+4 x}}{3+\sqrt {3}+2 \sqrt {3} x}\right )\right )}{\sqrt [4]{3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.23, size = 99, normalized size = 0.63
method | result | size |
derivativedivides | \(\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}{1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )\right )}{6}\) | \(99\) |
default | \(\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}{1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )\right )}{6}\) | \(99\) |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+27\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+27\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+27\right )^{4} x -6 \RootOf \left (\textit {\_Z}^{4}+27\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+27\right )^{2}\right )-9 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+27\right )^{2}\right ) x -108 \sqrt {2 x +1}-18 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+27\right )^{2}\right )}{6+\RootOf \left (\textit {\_Z}^{4}+27\right )^{2} x +3 x}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{4}+27\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+27\right )^{5} x +6 \RootOf \left (\textit {\_Z}^{4}+27\right )^{3}-9 \RootOf \left (\textit {\_Z}^{4}+27\right ) x -18 \RootOf \left (\textit {\_Z}^{4}+27\right )+108 \sqrt {2 x +1}}{-6+\RootOf \left (\textit {\_Z}^{4}+27\right )^{2} x -3 x}\right )}{3}\) | \(198\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 132, normalized size = 0.84 \begin {gather*} \frac {1}{3} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right )}\right ) + \frac {1}{3} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (-\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} - 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) + \frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.11, size = 188, normalized size = 1.20 \begin {gather*} -\frac {2}{3} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{3} \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1} - \frac {1}{3} \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x + 1} - 1\right ) - \frac {2}{3} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {-4 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 8 \, x + 4 \, \sqrt {3} + 4} - \frac {1}{3} \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 1\right ) - \frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (4 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 8 \, x + 4 \, \sqrt {3} + 4\right ) + \frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (-4 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 8 \, x + 4 \, \sqrt {3} + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.13, size = 155, normalized size = 0.99 \begin {gather*} \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (2 x - \sqrt {2} \cdot \sqrt [4]{3} \sqrt {2 x + 1} + 1 + \sqrt {3} \right )}}{6} - \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (2 x + \sqrt {2} \cdot \sqrt [4]{3} \sqrt {2 x + 1} + 1 + \sqrt {3} \right )}}{6} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \sqrt {2 x + 1}}{3} - 1 \right )}}{3} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \sqrt {2 x + 1}}{3} + 1 \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.50, size = 120, normalized size = 0.76 \begin {gather*} \frac {1}{3} \cdot 108^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right )}\right ) + \frac {1}{3} \cdot 108^{\frac {1}{4}} \arctan \left (-\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} - 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{6} \cdot 108^{\frac {1}{4}} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) + \frac {1}{6} \cdot 108^{\frac {1}{4}} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 57, normalized size = 0.36 \begin {gather*} \sqrt {2}\,3^{3/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (\frac {1}{3}-\frac {1}{3}{}\mathrm {i}\right )+\sqrt {2}\,3^{3/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (\frac {1}{3}+\frac {1}{3}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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