Optimal. Leaf size=168 \[ 4 \sqrt {1+2 x}+\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )-\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {1+2 x}}{\sqrt [4]{3}}\right )+\frac {\sqrt [4]{3} \log \left (1+\sqrt {3}+2 x-\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}}-\frac {\sqrt [4]{3} \log \left (1+\sqrt {3}+2 x+\sqrt {2} \sqrt [4]{3} \sqrt {1+2 x}\right )}{\sqrt {2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 168, 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 = {706, 708, 335,
217, 1179, 642, 1176, 631, 210} \begin {gather*} \sqrt {2} \sqrt [4]{3} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {2 x+1}}{\sqrt [4]{3}}\right )-\sqrt {2} \sqrt [4]{3} \text {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}} \end {gather*}
Antiderivative was successfully verified.
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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*} \int \frac {(1+2 x)^{3/2}}{1+x+x^2} \, dx &=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*}
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Mathematica [A]
time = 0.15, size = 100, normalized size = 0.60 \begin {gather*} 4 \sqrt {1+2 x}-\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (\frac {-3+\sqrt {3}+2 \sqrt {3} x}{3^{3/4} \sqrt {2+4 x}}\right )-\sqrt {2} \sqrt [4]{3} \tanh ^{-1}\left (\frac {3^{3/4} \sqrt {2+4 x}}{3+\sqrt {3}+2 \sqrt {3} x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.28, size = 109, normalized size = 0.65
method | result | size |
derivativedivides | \(4 \sqrt {2 x +1}-\frac {3^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}{1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )\right )}{2}\) | \(109\) |
default | \(4 \sqrt {2 x +1}-\frac {3^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}{1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right )\right )}{2}\) | \(109\) |
risch | \(4 \sqrt {2 x +1}-3^{\frac {1}{4}} \arctan \left (1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right ) \sqrt {2}-3^{\frac {1}{4}} \arctan \left (-1+\frac {\sqrt {2}\, \sqrt {2 x +1}\, 3^{\frac {3}{4}}}{3}\right ) \sqrt {2}-\frac {3^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {1+2 x +\sqrt {3}+3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}{1+2 x +\sqrt {3}-3^{\frac {1}{4}} \sqrt {2}\, \sqrt {2 x +1}}\right )}{2}\) | \(120\) |
trager | \(4 \sqrt {2 x +1}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+3\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+3\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+3\right )^{4} x +4 \RootOf \left (\textit {\_Z}^{4}+3\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+3\right )^{2}\right ) x +2 \RootOf \left (\textit {\_Z}^{4}+3\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+3\right )^{2}\right )+3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+3\right )^{2}\right ) x +6 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+3\right )^{2}\right )-12 \sqrt {2 x +1}}{\RootOf \left (\textit {\_Z}^{4}+3\right )^{2} x -x -2}\right )-\RootOf \left (\textit {\_Z}^{4}+3\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+3\right )^{5} x -4 \RootOf \left (\textit {\_Z}^{4}+3\right )^{3} x -2 \RootOf \left (\textit {\_Z}^{4}+3\right )^{3}+3 \RootOf \left (\textit {\_Z}^{4}+3\right ) x +6 \RootOf \left (\textit {\_Z}^{4}+3\right )+12 \sqrt {2 x +1}}{\RootOf \left (\textit {\_Z}^{4}+3\right )^{2} x +x +2}\right )\) | \(239\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 141, normalized size = 0.84 \begin {gather*} -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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.51, size = 190, normalized size = 1.13 \begin {gather*} 2 \cdot 3^{\frac {1}{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 ) + 2 \cdot 3^{\frac {1}{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 {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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 6.98, size = 162, normalized size = 0.96 \begin {gather*} 4 \sqrt {2 x + 1} + \frac {\sqrt {2} \cdot \sqrt [4]{3} \log {\left (2 x - \sqrt {2} \cdot \sqrt [4]{3} \sqrt {2 x + 1} + 1 + \sqrt {3} \right )}}{2} - \frac {\sqrt {2} \cdot \sqrt [4]{3} \log {\left (2 x + \sqrt {2} \cdot \sqrt [4]{3} \sqrt {2 x + 1} + 1 + \sqrt {3} \right )}}{2} - \sqrt {2} \cdot \sqrt [4]{3} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \sqrt {2 x + 1}}{3} - 1 \right )} - \sqrt {2} \cdot \sqrt [4]{3} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \sqrt {2 x + 1}}{3} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.13, size = 129, normalized size = 0.77 \begin {gather*} -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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 66, normalized size = 0.39 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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