Integrand size = 18, antiderivative size = 180 \[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=-\frac {4}{3 \sqrt {1+2 x}}+\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3 \sqrt [4]{3}}-\frac {\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3 \sqrt [4]{3}}-\frac {\log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} \sqrt [4]{3}}+\frac {\log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} \sqrt [4]{3}} \]
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Time = 0.09 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {707, 708, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{3 \sqrt [4]{3}}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{3 \sqrt [4]{3}}-\frac {4}{3 \sqrt {2 x+1}}-\frac {\log \left (2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{3 \sqrt {2} \sqrt [4]{3}}+\frac {\log \left (2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{3 \sqrt {2} \sqrt [4]{3}} \]
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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*} \text {integral}& = -\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*}
Time = 0.19 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=\frac {1}{9} \left (-\frac {12}{\sqrt {1+2 x}}-\sqrt {2} 3^{3/4} \arctan \left (\frac {-3+\sqrt {3}+2 \sqrt {3} x}{3^{3/4} \sqrt {2+4 x}}\right )+\sqrt {2} 3^{3/4} \text {arctanh}\left (\frac {3^{3/4} \sqrt {2+4 x}}{3+\sqrt {3}+2 \sqrt {3} x}\right )\right ) \]
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Time = 2.59 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.61
method | result | size |
derivativedivides | \(-\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}{1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )\right )}{18}-\frac {4}{3 \sqrt {1+2 x}}\) | \(109\) |
default | \(-\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}{1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )\right )}{18}-\frac {4}{3 \sqrt {1+2 x}}\) | \(109\) |
risch | \(-\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}{1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )\right )}{18}-\frac {4}{3 \sqrt {1+2 x}}\) | \(109\) |
pseudoelliptic | \(-\frac {\ln \left (\frac {1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}{1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}\right ) 3^{\frac {3}{4}} \sqrt {2}\, \sqrt {1+2 x}+2 \,3^{\frac {3}{4}} \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right ) \sqrt {2}\, \sqrt {1+2 x}+2 \,3^{\frac {3}{4}} \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right ) \sqrt {2}\, \sqrt {1+2 x}+24}{18 \sqrt {1+2 x}}\) | \(141\) |
trager | \(-\frac {4}{3 \sqrt {1+2 x}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2}\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{4} x +6 \operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2}\right )+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2}\right ) x +108 \sqrt {1+2 x}+18 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2}\right )}{6+\operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2} x +3 x}\right )}{9}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{5} x +6 \operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{3}-9 \operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right ) x -18 \operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )-108 \sqrt {1+2 x}}{-6+\operatorname {RootOf}\left (\textit {\_Z}^{4}+27\right )^{2} x -3 x}\right )}{9}\) | \(208\) |
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Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=\frac {3^{\frac {3}{4}} \sqrt {2} {\left (-\left (2 i - 2\right ) \, x - i + 1\right )} \log \left (\left (i + 1\right ) \cdot 3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right ) + 3^{\frac {3}{4}} \sqrt {2} {\left (\left (2 i + 2\right ) \, x + i + 1\right )} \log \left (-\left (i - 1\right ) \cdot 3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right ) + 3^{\frac {3}{4}} \sqrt {2} {\left (-\left (2 i + 2\right ) \, x - i - 1\right )} \log \left (\left (i - 1\right ) \cdot 3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right ) + 3^{\frac {3}{4}} \sqrt {2} {\left (\left (2 i - 2\right ) \, x + i - 1\right )} \log \left (-\left (i + 1\right ) \cdot 3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right ) - 24 \, \sqrt {2 \, x + 1}}{18 \, {\left (2 \, x + 1\right )}} \]
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\[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=\int \frac {1}{\left (2 x + 1\right )^{\frac {3}{2}} \left (x^{2} + x + 1\right )}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=-\frac {1}{9} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{9} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (-\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} - 2 \, \sqrt {2 \, x + 1}\right )}\right ) + \frac {1}{18} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {1}{18} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {4}{3 \, \sqrt {2 \, x + 1}} \]
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Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=-\frac {1}{9} \cdot 108^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{9} \cdot 108^{\frac {1}{4}} \arctan \left (-\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} - 2 \, \sqrt {2 \, x + 1}\right )}\right ) + \frac {1}{18} \cdot 108^{\frac {1}{4}} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {1}{18} \cdot 108^{\frac {1}{4}} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {4}{3 \, \sqrt {2 \, x + 1}} \]
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Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.37 \[ \int \frac {1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx=-\frac {4}{3\,\sqrt {2\,x+1}}+\sqrt {2}\,3^{3/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{9}+\frac {1}{9}{}\mathrm {i}\right )+\sqrt {2}\,3^{3/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{9}-\frac {1}{9}{}\mathrm {i}\right ) \]
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