Integrand size = 18, antiderivative size = 168 \[ \int \frac {(1+2 x)^{3/2}}{1+x+x^2} \, dx=4 \sqrt {1+2 x}+\sqrt {2} \sqrt [4]{3} \arctan \left (1-\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )-\sqrt {2} \sqrt [4]{3} \arctan \left (1+\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )+\frac {\sqrt [4]{3} \log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}}-\frac {\sqrt [4]{3} \log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}} \]
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Time = 0.10 (sec) , antiderivative size = 168, 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 = {706, 708, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(1+2 x)^{3/2}}{1+x+x^2} \, dx=\sqrt {2} \sqrt [4]{3} \arctan \left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )-\sqrt {2} \sqrt [4]{3} \arctan \left (\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}+1\right )+4 \sqrt {2 x+1}+\frac {\sqrt [4]{3} \log \left (2 x-\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{\sqrt {2}}-\frac {\sqrt [4]{3} \log \left (2 x+\sqrt {2} \sqrt [4]{3} \sqrt {2 x+1}+\sqrt {3}+1\right )}{\sqrt {2}} \]
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Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 706
Rule 708
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = 4 \sqrt {1+2 x}-3 \int \frac {1}{\sqrt {1+2 x} \left (1+x+x^2\right )} \, dx \\ & = 4 \sqrt {1+2 x}-\frac {3}{2} \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (\frac {3}{4}+\frac {x^2}{4}\right )} \, dx,x,1+2 x\right ) \\ & = 4 \sqrt {1+2 x}-3 \text {Subst}\left (\int \frac {1}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right ) \\ & = 4 \sqrt {1+2 x}-\frac {1}{2} \sqrt {3} \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}{2} \sqrt {3} \text {Subst}\left (\int \frac {\sqrt {3}+x^2}{\frac {3}{4}+\frac {x^4}{4}} \, dx,x,\sqrt {1+2 x}\right ) \\ & = 4 \sqrt {1+2 x}+\frac {\sqrt [4]{3} \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}}+\frac {\sqrt [4]{3} \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 {3} \text {Subst}\left (\int \frac {1}{\sqrt {3}-\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt {1+2 x}\right )-\sqrt {3} \text {Subst}\left (\int \frac {1}{\sqrt {3}+\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\sqrt {1+2 x}\right ) \\ & = 4 \sqrt {1+2 x}+\frac {\sqrt [4]{3} \log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}}-\frac {\sqrt [4]{3} \log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}}-\left (\sqrt {2} \sqrt [4]{3}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2+4 x}}{\sqrt [4]{3}}\right )+\left (\sqrt {2} \sqrt [4]{3}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2+4 x}}{\sqrt [4]{3}}\right ) \\ & = 4 \sqrt {1+2 x}+\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )-\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )+\frac {\sqrt [4]{3} \log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}}-\frac {\sqrt [4]{3} \log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.60 \[ \int \frac {(1+2 x)^{3/2}}{1+x+x^2} \, dx=4 \sqrt {1+2 x}-\sqrt {2} \sqrt [4]{3} \arctan \left (\frac {-3+\sqrt {3}+2 \sqrt {3} x}{3^{3/4} \sqrt {2+4 x}}\right )-\sqrt {2} \sqrt [4]{3} \text {arctanh}\left (\frac {3^{3/4} \sqrt {2+4 x}}{3+\sqrt {3}+2 \sqrt {3} x}\right ) \]
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Time = 2.57 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(4 \sqrt {1+2 x}-\frac {3^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}{1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )\right )}{2}\) | \(109\) |
default | \(4 \sqrt {1+2 x}-\frac {3^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}{1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )\right )}{2}\) | \(109\) |
risch | \(4 \sqrt {1+2 x}-\frac {3^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}{1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right )\right )}{2}\) | \(109\) |
pseudoelliptic | \(4 \sqrt {1+2 x}-\frac {\ln \left (\frac {1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}{1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {1+2 x}}\right ) 3^{\frac {1}{4}} \sqrt {2}}{2}-3^{\frac {1}{4}} \arctan \left (1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right ) \sqrt {2}-3^{\frac {1}{4}} \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {1+2 x}\, 3^{\frac {3}{4}}}{3}\right ) \sqrt {2}\) | \(120\) |
trager | \(4 \sqrt {1+2 x}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{5} x -4 x \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{3}+3 x \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )+6 \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )-12 \sqrt {1+2 x}}{x \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}+x +2}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{4} x +4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right ) x +2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right )+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right ) x -12 \sqrt {1+2 x}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}\right )}{x \operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )^{2}-x -2}\right )\) | \(237\) |
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Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.49 \[ \int \frac {(1+2 x)^{3/2}}{1+x+x^2} \, dx=-\left (-3\right )^{\frac {1}{4}} \log \left (\sqrt {2 \, x + 1} + \left (-3\right )^{\frac {1}{4}}\right ) - i \, \left (-3\right )^{\frac {1}{4}} \log \left (\sqrt {2 \, x + 1} + i \, \left (-3\right )^{\frac {1}{4}}\right ) + i \, \left (-3\right )^{\frac {1}{4}} \log \left (\sqrt {2 \, x + 1} - i \, \left (-3\right )^{\frac {1}{4}}\right ) + \left (-3\right )^{\frac {1}{4}} \log \left (\sqrt {2 \, x + 1} - \left (-3\right )^{\frac {1}{4}}\right ) + 4 \, \sqrt {2 \, x + 1} \]
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\[ \int \frac {(1+2 x)^{3/2}}{1+x+x^2} \, dx=\int \frac {\left (2 x + 1\right )^{\frac {3}{2}}}{x^{2} + x + 1}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.84 \[ \int \frac {(1+2 x)^{3/2}}{1+x+x^2} \, dx=-3^{\frac {1}{4}} \sqrt {2} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right )}\right ) - 3^{\frac {1}{4}} \sqrt {2} \arctan \left (-\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} - 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{2} \cdot 3^{\frac {1}{4}} \sqrt {2} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) + \frac {1}{2} \cdot 3^{\frac {1}{4}} \sqrt {2} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) + 4 \, \sqrt {2 \, x + 1} \]
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Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.77 \[ \int \frac {(1+2 x)^{3/2}}{1+x+x^2} \, dx=-12^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} + 2 \, \sqrt {2 \, x + 1}\right )}\right ) - 12^{\frac {1}{4}} \arctan \left (-\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} - 2 \, \sqrt {2 \, x + 1}\right )}\right ) - \frac {1}{2} \cdot 12^{\frac {1}{4}} \log \left (3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) + \frac {1}{2} \cdot 12^{\frac {1}{4}} \log \left (-3^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x + 1} + 2 \, x + \sqrt {3} + 1\right ) + 4 \, \sqrt {2 \, x + 1} \]
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Time = 9.36 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.39 \[ \int \frac {(1+2 x)^{3/2}}{1+x+x^2} \, dx=4\,\sqrt {2\,x+1}+\sqrt {2}\,3^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-1-\mathrm {i}\right )+\sqrt {2}\,3^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,\sqrt {2\,x+1}\,\left (\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (-1+1{}\mathrm {i}\right ) \]
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