Optimal. Leaf size=151 \[ -\frac {1}{2} \sqrt {1+\frac {7 \sqrt {\frac {2}{5}}}{5}} \tanh ^{-1}\left (\frac {3 \left (4-\sqrt {10}\right )+\left (17-4 \sqrt {10}\right ) x}{2 \sqrt {55-17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right )+\frac {1}{2} \sqrt {1-\frac {7 \sqrt {\frac {2}{5}}}{5}} \tanh ^{-1}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (17+4 \sqrt {10}\right ) x}{2 \sqrt {55+17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right ) \]
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Rubi [A]
time = 0.14, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1046, 738, 212}
\begin {gather*} \frac {1}{2} \sqrt {1-\frac {7 \sqrt {\frac {2}{5}}}{5}} \tanh ^{-1}\left (\frac {\left (17+4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {55+17 \sqrt {10}} \sqrt {2 x^2+3 x+1}}\right )-\frac {1}{2} \sqrt {1+\frac {7 \sqrt {\frac {2}{5}}}{5}} \tanh ^{-1}\left (\frac {\left (17-4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {55-17 \sqrt {10}} \sqrt {2 x^2+3 x+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 1046
Rubi steps
\begin {align*} \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x+2 x^2}} \, dx &=\frac {1}{5} \left (5-4 \sqrt {10}\right ) \int \frac {1}{\left (4-2 \sqrt {10}-6 x\right ) \sqrt {1+3 x+2 x^2}} \, dx+\frac {1}{5} \left (5+4 \sqrt {10}\right ) \int \frac {1}{\left (4+2 \sqrt {10}-6 x\right ) \sqrt {1+3 x+2 x^2}} \, dx\\ &=-\left (\frac {1}{5} \left (2 \left (5-4 \sqrt {10}\right )\right ) \text {Subst}\left (\int \frac {1}{144+72 \left (4-2 \sqrt {10}\right )+8 \left (4-2 \sqrt {10}\right )^2-x^2} \, dx,x,\frac {-12-3 \left (4-2 \sqrt {10}\right )-\left (18+4 \left (4-2 \sqrt {10}\right )\right ) x}{\sqrt {1+3 x+2 x^2}}\right )\right )-\frac {1}{5} \left (2 \left (5+4 \sqrt {10}\right )\right ) \text {Subst}\left (\int \frac {1}{144+72 \left (4+2 \sqrt {10}\right )+8 \left (4+2 \sqrt {10}\right )^2-x^2} \, dx,x,\frac {-12-3 \left (4+2 \sqrt {10}\right )-\left (18+4 \left (4+2 \sqrt {10}\right )\right ) x}{\sqrt {1+3 x+2 x^2}}\right )\\ &=-\frac {1}{10} \sqrt {25+7 \sqrt {10}} \tanh ^{-1}\left (\frac {3 \left (4-\sqrt {10}\right )+\left (17-4 \sqrt {10}\right ) x}{2 \sqrt {55-17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right )+\frac {1}{10} \sqrt {25-7 \sqrt {10}} \tanh ^{-1}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (17+4 \sqrt {10}\right ) x}{2 \sqrt {55+17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.52, size = 109, normalized size = 0.72 \begin {gather*} -\frac {1}{5} \sqrt {25+7 \sqrt {10}} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {\frac {2}{5}}} \sqrt {1+3 x+2 x^2}}{1+2 x}\right )+\frac {1}{5} \sqrt {25-7 \sqrt {10}} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {\frac {2}{5}}} \sqrt {1+3 x+2 x^2}}{1+2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.62, size = 186, normalized size = 1.23
method | result | size |
default | \(\frac {\left (8+\sqrt {10}\right ) \sqrt {10}\, \arctanh \left (\frac {55+17 \sqrt {10}+\frac {9 \left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {55+17 \sqrt {10}}\, \sqrt {18 \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {17}{3}+\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}-\frac {\sqrt {10}}{3}\right )+55+17 \sqrt {10}}}\right )}{20 \sqrt {55+17 \sqrt {10}}}+\frac {\left (-8+\sqrt {10}\right ) \sqrt {10}\, \arctanh \left (\frac {55-17 \sqrt {10}+\frac {9 \left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )}{2}}{\sqrt {55-17 \sqrt {10}}\, \sqrt {18 \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )^{2}+9 \left (\frac {17}{3}-\frac {4 \sqrt {10}}{3}\right ) \left (x -\frac {2}{3}+\frac {\sqrt {10}}{3}\right )+55-17 \sqrt {10}}}\right )}{20 \sqrt {55-17 \sqrt {10}}}\) | \(186\) |
trager | \(-\RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right ) \ln \left (\frac {10000 x \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{5}+12600 \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{3} x +12600 \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{3}-1600 \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2} \sqrt {2 x^{2}+3 x +1}-7695 \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right ) x -5670 \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )+918 \sqrt {2 x^{2}+3 x +1}}{100 x \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-11 x +14}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-2\right ) \ln \left (-\frac {10000 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-2\right ) \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{4} x -22600 \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2} \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-2\right ) x -12600 \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2} \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-2\right )-3200 \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2} \sqrt {2 x^{2}+3 x +1}+1105 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-2\right ) x +630 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-2\right )-236 \sqrt {2 x^{2}+3 x +1}}{100 x \RootOf \left (2000 \textit {\_Z}^{4}-1000 \textit {\_Z}^{2}+27\right )^{2}-39 x -14}\right )}{2}\) | \(442\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 363 vs.
\(2 (103) = 206\).
time = 0.51, size = 363, normalized size = 2.40 \begin {gather*} \frac {1}{60} \, \sqrt {10} {\left (\frac {3 \, \sqrt {10} \log \left (\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {17 \, \sqrt {10} + 55}}{3 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{\sqrt {17 \, \sqrt {10} + 55}} + \frac {\sqrt {10} \log \left (-\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}}}{{\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} - \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{\sqrt {-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}}} + \frac {24 \, \log \left (\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {17 \, \sqrt {10} + 55}}{3 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{\sqrt {17 \, \sqrt {10} + 55}} - \frac {8 \, \log \left (-\frac {2}{9} \, \sqrt {10} + \frac {2 \, \sqrt {2 \, x^{2} + 3 \, x + 1} \sqrt {-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}}}{{\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} - \frac {34 \, \sqrt {10}}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {110}{9 \, {\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}} + \frac {17}{18}\right )}{\sqrt {-\frac {17}{9} \, \sqrt {10} + \frac {55}{9}}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 245 vs.
\(2 (103) = 206\).
time = 0.36, size = 245, normalized size = 1.62 \begin {gather*} \frac {1}{10} \, \sqrt {7 \, \sqrt {10} + 25} \log \left (-\frac {3 \, \sqrt {10} x + {\left (\sqrt {10} x - 4 \, x\right )} \sqrt {7 \, \sqrt {10} + 25} + 6 \, x - 6 \, \sqrt {2 \, x^{2} + 3 \, x + 1} + 6}{x}\right ) - \frac {1}{10} \, \sqrt {7 \, \sqrt {10} + 25} \log \left (-\frac {3 \, \sqrt {10} x - {\left (\sqrt {10} x - 4 \, x\right )} \sqrt {7 \, \sqrt {10} + 25} + 6 \, x - 6 \, \sqrt {2 \, x^{2} + 3 \, x + 1} + 6}{x}\right ) + \frac {1}{10} \, \sqrt {-7 \, \sqrt {10} + 25} \log \left (\frac {3 \, \sqrt {10} x + {\left (\sqrt {10} x + 4 \, x\right )} \sqrt {-7 \, \sqrt {10} + 25} - 6 \, x + 6 \, \sqrt {2 \, x^{2} + 3 \, x + 1} - 6}{x}\right ) - \frac {1}{10} \, \sqrt {-7 \, \sqrt {10} + 25} \log \left (\frac {3 \, \sqrt {10} x - {\left (\sqrt {10} x + 4 \, x\right )} \sqrt {-7 \, \sqrt {10} + 25} - 6 \, x + 6 \, \sqrt {2 \, x^{2} + 3 \, x + 1} - 6}{x}\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 {x}{3 x^{2} \sqrt {2 x^{2} + 3 x + 1} - 4 x \sqrt {2 x^{2} + 3 x + 1} - 2 \sqrt {2 x^{2} + 3 x + 1}}\, dx - \int \frac {2}{3 x^{2} \sqrt {2 x^{2} + 3 x + 1} - 4 x \sqrt {2 x^{2} + 3 x + 1} - 2 \sqrt {2 x^{2} + 3 x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 7.02, size = 93, normalized size = 0.62 \begin {gather*} 0.169235232112667 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} + 5.90976932712000\right ) - 0.686556214893333 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.176527156327000\right ) + 0.686556214893333 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.919278730509000\right ) - 0.169235232112667 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 1.04272727395000\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x+2}{\sqrt {2\,x^2+3\,x+1}\,\left (-3\,x^2+4\,x+2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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