Integrand size = 18, antiderivative size = 180 \[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=-\frac {4}{9 (1+2 x)^{3/2}}+\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}-\frac {\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}+\frac {\log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} 3^{3/4}}-\frac {\log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} 3^{3/4}} \]
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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, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )}{3\ 3^{3/4}}-\frac {4}{9 (2 x+1)^{3/2}}+\frac {\log \left (2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{3 \sqrt {2} 3^{3/4}}-\frac {\log \left (2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{3 \sqrt {2} 3^{3/4}} \]
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Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 707
Rule 708
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = -\frac {4}{9 (1+2 x)^{3/2}}-\frac {1}{3} \int \frac {1}{\sqrt {1+2 x} \left (1+x+x^2\right )} \, dx \\ & = -\frac {4}{9 (1+2 x)^{3/2}}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (\frac {3}{4}+\frac {x^2}{4}\right )} \, dx,x,1+2 x\right ) \\ & = -\frac {4}{9 (1+2 x)^{3/2}}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right ) \\ & = -\frac {4}{9 (1+2 x)^{3/2}}-\frac {\text {Subst}\left (\int \frac {\sqrt {3}-x^2}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )}{6 \sqrt {3}}-\frac {\text {Subst}\left (\int \frac {\sqrt {3}+x^2}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right )}{6 \sqrt {3}} \\ & = -\frac {4}{9 (1+2 x)^{3/2}}+\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} 3^{3/4}}+\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} 3^{3/4}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {3}-\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{3 \sqrt {3}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {3}+\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{3 \sqrt {3}} \\ & = -\frac {4}{9 (1+2 x)^{3/2}}+\frac {\log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} 3^{3/4}}-\frac {\log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} 3^{3/4}}-\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\ 3^{3/4}}+\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\ 3^{3/4}} \\ & = -\frac {4}{9 (1+2 x)^{3/2}}+\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}-\frac {\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )}{3\ 3^{3/4}}+\frac {\log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} 3^{3/4}}-\frac {\log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{3 \sqrt {2} 3^{3/4}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.58 \[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=\frac {1}{9} \left (-\frac {4}{(1+2 x)^{3/2}}-\sqrt {2} \sqrt [4]{3} \arctan \left (\frac {-3+\sqrt {3}+2 \sqrt {3} x}{3^{3/4} \sqrt {2+4 x}}\right )-\sqrt {2} \sqrt [4]{3} \text {arctanh}\left (\frac {3^{3/4} \sqrt {2+4 x}}{3+\sqrt {3}+2 \sqrt {3} x}\right )\right ) \]
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Time = 2.55 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.61
method | result | size |
derivativedivides | \(-\frac {3^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}{1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )\right )}{18}-\frac {4}{9 \left (1+2 x \right )^{\frac {3}{2}}}\) | \(109\) |
default | \(-\frac {3^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}{1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )\right )}{18}-\frac {4}{9 \left (1+2 x \right )^{\frac {3}{2}}}\) | \(109\) |
pseudoelliptic | \(-\frac {8+3^{\frac {1}{4}} \left (1+2 x \right )^{\frac {3}{2}} \left (\ln \left (\frac {1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}{1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )\right ) \sqrt {2}}{18 \left (1+2 x \right )^{\frac {3}{2}}}\) | \(116\) |
trager | \(-\frac {4}{9 \left (1+2 x \right )^{\frac {3}{2}}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{5} x -4 x \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{3}+3 x \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )+12 \sqrt {1+2 x}+6 \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )}{x \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}+x +2}\right )}{9}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{4} x +4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right ) x +2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right )+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right ) x -12 \sqrt {1+2 x}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right )}{x \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}-x -2}\right )}{9}\) | \(240\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=\frac {27^{\frac {3}{4}} \sqrt {2} {\left (-\left (4 i + 4\right ) \, x^{2} - \left (4 i + 4\right ) \, x - i - 1\right )} \log \left (\left (i + 1\right ) \cdot 27^{\frac {3}{4}} \sqrt {2} + 18 \, \sqrt {2 \, x + 1}\right ) + 27^{\frac {3}{4}} \sqrt {2} {\left (\left (4 i - 4\right ) \, x^{2} + \left (4 i - 4\right ) \, x + i - 1\right )} \log \left (-\left (i - 1\right ) \cdot 27^{\frac {3}{4}} \sqrt {2} + 18 \, \sqrt {2 \, x + 1}\right ) + 27^{\frac {3}{4}} \sqrt {2} {\left (-\left (4 i - 4\right ) \, x^{2} - \left (4 i - 4\right ) \, x - i + 1\right )} \log \left (\left (i - 1\right ) \cdot 27^{\frac {3}{4}} \sqrt {2} + 18 \, \sqrt {2 \, x + 1}\right ) + 27^{\frac {3}{4}} \sqrt {2} {\left (\left (4 i + 4\right ) \, x^{2} + \left (4 i + 4\right ) \, x + i + 1\right )} \log \left (-\left (i + 1\right ) \cdot 27^{\frac {3}{4}} \sqrt {2} + 18 \, \sqrt {2 \, x + 1}\right ) - 72 \, \sqrt {2 \, x + 1}}{162 \, {\left (4 \, x^{2} + 4 \, x + 1\right )}} \]
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\[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=\int \frac {1}{\left (2 x + 1\right )^{\frac {5}{2}} \left (x^{2} + x + 1\right )}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=-\frac {1}{9} \cdot 3^{\frac {1}{4}} \sqrt {2} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{9} \cdot 3^{\frac {1}{4}} \sqrt {2} \arctan \left (-\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} - 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{18} \cdot 3^{\frac {1}{4}} \sqrt {2} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) + \frac {1}{18} \cdot 3^{\frac {1}{4}} \sqrt {2} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {4}{9 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}}} \]
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Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=-\frac {1}{9} \cdot 12^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{9} \cdot 12^{\frac {1}{4}} \arctan \left (-\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} - 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{18} \cdot 12^{\frac {1}{4}} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) + \frac {1}{18} \cdot 12^{\frac {1}{4}} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) - \frac {4}{9 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}}} \]
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Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.37 \[ \int \frac {1}{(1+2 x)^{5/2} \left (1+x+x^2\right )} \, dx=-\frac {4}{9\,{\left (2\,x+1\right )}^{3/2}}+\sqrt {2}\,3^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{9}-\frac {1}{9}{}\mathrm {i}\right )+\sqrt {2}\,3^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{9}+\frac {1}{9}{}\mathrm {i}\right ) \]
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