Optimal. Leaf size=180 \[ -\frac {4}{3 \sqrt {1+2 x}}+\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3 \sqrt [4]{3}}-\frac {\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3 \sqrt [4]{3}}-\frac {\log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} \sqrt [4]{3}}+\frac {\log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} \sqrt [4]{3}} \]
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Rubi [A]
time = 0.09, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {707, 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 )}{3 \sqrt [4]{3}}-\frac {\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{3 \sqrt [4]{3}}-\frac {4}{3 \sqrt {2 x+1}}-\frac {\log \left (2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{3 \sqrt {2} \sqrt [4]{3}}+\frac {\log \left (2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{3 \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 707
Rule 708
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx &=-\frac {4}{3 \sqrt {1+2 x}}-\frac {1}{3} \int \frac {\sqrt {1+2 x}}{1+x+x^2} \, dx\\ &=-\frac {4}{3 \sqrt {1+2 x}}-\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {x}}{\frac {3}{4}+\frac {x^2}{4}} \, dx,x,1+2 x\right )\\ &=-\frac {4}{3 \sqrt {1+2 x}}-\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\frac {4}{3 \sqrt {1+2 x}}+\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {3}-x^2}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )-\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {3}+x^2}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\frac {4}{3 \sqrt {1+2 x}}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {3}-\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt {1+2 x}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {3}+\sqrt {2} \sqrt [4]{3} x+x^2} \, 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 )}{3 \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 )}{3 \sqrt {2} \sqrt [4]{3}}\\ &=-\frac {4}{3 \sqrt {1+2 x}}-\frac {\log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} \sqrt [4]{3}}+\frac {\log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \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 )}{3 \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 )}{3 \sqrt [4]{3}}\\ &=-\frac {4}{3 \sqrt {1+2 x}}+\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3 \sqrt [4]{3}}-\frac {\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3 \sqrt [4]{3}}-\frac {\log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} \sqrt [4]{3}}+\frac {\log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} \sqrt [4]{3}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 103, normalized size = 0.57 \begin {gather*} \frac {1}{9} \left (-\frac {12}{\sqrt {1+2 x}}-\sqrt {2} 3^{3/4} \tan ^{-1}\left (\frac {-3+\sqrt {3}+2 \sqrt {3} x}{3^{3/4} \sqrt {2+4 x}}\right )+\sqrt {2} 3^{3/4} \tanh ^{-1}\left (\frac {3^{3/4} \sqrt {2+4 x}}{3+\sqrt {3}+2 \sqrt {3} x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.16, size = 109, normalized size = 0.61
method | result | size |
derivativedivides | \(-\frac {4}{3 \sqrt {2 x +1}}-\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}{1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )\right )}{18}\) | \(109\) |
default | \(-\frac {4}{3 \sqrt {2 x +1}}-\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}{1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )\right )}{18}\) | \(109\) |
risch | \(-\frac {4}{3 \sqrt {2 x +1}}-\frac {3^{\frac {3}{4}} \arctan \left (1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right ) \sqrt {2}}{9}-\frac {3^{\frac {3}{4}} \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right ) \sqrt {2}}{9}-\frac {\sqrt {2}\, 3^{\frac {3}{4}} \ln \left (\frac {1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}{1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}\right )}{18}\) | \(120\) |
trager | \(-\frac {4}{3 \sqrt {2 x +1}}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+27\right )^{2}\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+27\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+27\right )^{4} x -6 \RootOf \left (\textit {\_Z}^{4}+27\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+27\right )^{2}\right )-9 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+27\right )^{2}\right ) x -18 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+27\right )^{2}\right )+108 \sqrt {2 x +1}}{6+\RootOf \left (\textit {\_Z}^{4}+27\right )^{2} x +3 x}\right )}{9}+\frac {\RootOf \left (\textit {\_Z}^{4}+27\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+27\right )^{5} x -6 \RootOf \left (\textit {\_Z}^{4}+27\right )^{3}+9 \RootOf \left (\textit {\_Z}^{4}+27\right ) x +108 \sqrt {2 x +1}+18 \RootOf \left (\textit {\_Z}^{4}+27\right )}{-6+\RootOf \left (\textit {\_Z}^{4}+27\right )^{2} x -3 x}\right )}{9}\) | \(209\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 141, normalized size = 0.78 \begin {gather*} -\frac {1}{9} \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}{9} \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}{18} \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}{18} \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 {4}{3 \, \sqrt {2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.52, size = 225, normalized size = 1.25 \begin {gather*} \frac {4 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, x + 1\right )} \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 ) + 4 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, x + 1\right )} \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 ) + 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, x + 1\right )} \log \left (4 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 8 \, x + 4 \, \sqrt {3} + 4\right ) - 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, x + 1\right )} \log \left (-4 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 8 \, x + 4 \, \sqrt {3} + 4\right ) - 24 \, \sqrt {2 \, x + 1}}{18 \, {\left (2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (2 x + 1\right )^{\frac {3}{2}} \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.23, size = 129, normalized size = 0.72 \begin {gather*} -\frac {1}{9} \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}{9} \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}{18} \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}{18} \cdot 108^{\frac {1}{4}} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {4}{3 \, \sqrt {2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.53, size = 66, normalized size = 0.37 \begin {gather*} -\frac {4}{3\,\sqrt {2\,x+1}}+\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}{9}+\frac {1}{9}{}\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}{9}-\frac {1}{9}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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