Integrand size = 28, antiderivative size = 76 \[ \int \frac {3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt {4+2 x+x^2}} \, dx=-\frac {(3-x) \sqrt {4+2 x+x^2}}{4 \left (3+2 x+x^2\right )}-\frac {\arctan \left (\frac {1+x}{\sqrt {2} \sqrt {4+2 x+x^2}}\right )}{4 \sqrt {2}}+\text {arctanh}\left (\sqrt {4+2 x+x^2}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1030, 1039, 996, 210, 1038, 212} \[ \int \frac {3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt {4+2 x+x^2}} \, dx=-\frac {\arctan \left (\frac {x+1}{\sqrt {2} \sqrt {x^2+2 x+4}}\right )}{4 \sqrt {2}}+\text {arctanh}\left (\sqrt {x^2+2 x+4}\right )-\frac {\sqrt {x^2+2 x+4} (3-x)}{4 \left (x^2+2 x+3\right )} \]
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Rule 210
Rule 212
Rule 996
Rule 1030
Rule 1038
Rule 1039
Rubi steps \begin{align*} \text {integral}& = -\frac {(3-x) \sqrt {4+2 x+x^2}}{4 \left (3+2 x+x^2\right )}+\frac {1}{8} \int \frac {-10-8 x}{\left (3+2 x+x^2\right ) \sqrt {4+2 x+x^2}} \, dx \\ & = -\frac {(3-x) \sqrt {4+2 x+x^2}}{4 \left (3+2 x+x^2\right )}-\frac {1}{4} \int \frac {1}{\left (3+2 x+x^2\right ) \sqrt {4+2 x+x^2}} \, dx-\frac {1}{2} \int \frac {2+2 x}{\left (3+2 x+x^2\right ) \sqrt {4+2 x+x^2}} \, dx \\ & = -\frac {(3-x) \sqrt {4+2 x+x^2}}{4 \left (3+2 x+x^2\right )}+2 \text {Subst}\left (\int \frac {1}{2-2 x^2} \, dx,x,\sqrt {4+2 x+x^2}\right )+\text {Subst}\left (\int \frac {1}{-16-2 x^2} \, dx,x,\frac {2+2 x}{\sqrt {4+2 x+x^2}}\right ) \\ & = -\frac {(3-x) \sqrt {4+2 x+x^2}}{4 \left (3+2 x+x^2\right )}-\frac {\arctan \left (\frac {2+2 x}{2 \sqrt {2} \sqrt {4+2 x+x^2}}\right )}{4 \sqrt {2}}+\text {arctanh}\left (\sqrt {4+2 x+x^2}\right ) \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.11 \[ \int \frac {3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt {4+2 x+x^2}} \, dx=\frac {1}{8} \left (\frac {2 (-3+x) \sqrt {4+2 x+x^2}}{3+2 x+x^2}+\sqrt {2} \arctan \left (\frac {3+2 x+x^2-(1+x) \sqrt {4+2 x+x^2}}{\sqrt {2}}\right )\right )+\text {arctanh}\left (\sqrt {4+2 x+x^2}\right ) \]
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Time = 0.84 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {\left (-3+x \right ) \sqrt {x^{2}+2 x +4}}{4 x^{2}+8 x +12}+\operatorname {arctanh}\left (\sqrt {x^{2}+2 x +4}\right )-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (2 x +2\right )}{4 \sqrt {x^{2}+2 x +4}}\right )}{8}\) | \(64\) |
default | \(-\frac {1}{2 \left (\sqrt {x^{2}+2 x +4}+1\right )}+\frac {\ln \left (\sqrt {x^{2}+2 x +4}+1\right )}{2}-\frac {1}{2 \left (\sqrt {x^{2}+2 x +4}-1\right )}-\frac {\ln \left (\sqrt {x^{2}+2 x +4}-1\right )}{2}+\frac {\frac {3}{4}+\frac {3 x}{4}}{\sqrt {x^{2}+2 x +4}\, \left (\frac {\left (1+x \right )^{2}}{x^{2}+2 x +4}+2\right )}-\frac {\arctan \left (\frac {\left (1+x \right ) \sqrt {2}}{2 \sqrt {x^{2}+2 x +4}}\right ) \sqrt {2}}{8}\) | \(123\) |
trager | \(\frac {\left (-3+x \right ) \sqrt {x^{2}+2 x +4}}{4 x^{2}+8 x +12}-3 \ln \left (-\frac {48384 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )^{2} x +960 \sqrt {x^{2}+2 x +4}\, \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )+15312 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right ) x +143 \sqrt {x^{2}+2 x +4}-3696 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )+1210 x -605}{48 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right ) x +7 x -3}\right ) \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )+3 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right ) \ln \left (\frac {-16128 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )^{2} x +320 \sqrt {x^{2}+2 x +4}\, \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )-5648 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right ) x +59 \sqrt {x^{2}+2 x +4}-1232 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )-494 x -209}{16 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right ) x +3 x +1}\right )-\ln \left (-\frac {48384 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )^{2} x +960 \sqrt {x^{2}+2 x +4}\, \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )+15312 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right ) x +143 \sqrt {x^{2}+2 x +4}-3696 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right )+1210 x -605}{48 \operatorname {RootOf}\left (384 \textit {\_Z}^{2}+128 \textit {\_Z} +11\right ) x +7 x -3}\right )\) | \(375\) |
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Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (61) = 122\).
Time = 0.24 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.29 \[ \int \frac {3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt {4+2 x+x^2}} \, dx=\frac {\sqrt {2} {\left (x^{2} + 2 \, x + 3\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (x + 2\right )} + \frac {1}{2} \, \sqrt {2} \sqrt {x^{2} + 2 \, x + 4}\right ) - \sqrt {2} {\left (x^{2} + 2 \, x + 3\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} x + \frac {1}{2} \, \sqrt {2} \sqrt {x^{2} + 2 \, x + 4}\right ) + 2 \, x^{2} - 4 \, {\left (x^{2} + 2 \, x + 3\right )} \log \left (x^{2} - \sqrt {x^{2} + 2 \, x + 4} {\left (x + 2\right )} + 3 \, x + 5\right ) + 4 \, {\left (x^{2} + 2 \, x + 3\right )} \log \left (x^{2} - \sqrt {x^{2} + 2 \, x + 4} x + x + 3\right ) + 2 \, \sqrt {x^{2} + 2 \, x + 4} {\left (x - 3\right )} + 4 \, x + 6}{8 \, {\left (x^{2} + 2 \, x + 3\right )}} \]
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\[ \int \frac {3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt {4+2 x+x^2}} \, dx=\int \frac {2 x + 3}{\left (x^{2} + 2 x + 3\right )^{2} \sqrt {x^{2} + 2 x + 4}}\, dx \]
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\[ \int \frac {3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt {4+2 x+x^2}} \, dx=\int { \frac {2 \, x + 3}{\sqrt {x^{2} + 2 \, x + 4} {\left (x^{2} + 2 \, x + 3\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (61) = 122\).
Time = 0.29 (sec) , antiderivative size = 235, normalized size of antiderivative = 3.09 \[ \int \frac {3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt {4+2 x+x^2}} \, dx=\frac {1}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (x - \sqrt {x^{2} + 2 \, x + 4} + 2\right )}\right ) - \frac {1}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}\right ) + \frac {4 \, {\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}^{3} + 13 \, {\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}^{2} + 26 \, x - 26 \, \sqrt {x^{2} + 2 \, x + 4} + 26}{2 \, {\left ({\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}^{4} + 4 \, {\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}^{3} + 8 \, {\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}^{2} + 8 \, x - 8 \, \sqrt {x^{2} + 2 \, x + 4} + 12\right )}} - \frac {1}{2} \, \log \left ({\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}^{2} + 4 \, x - 4 \, \sqrt {x^{2} + 2 \, x + 4} + 6\right ) + \frac {1}{2} \, \log \left ({\left (x - \sqrt {x^{2} + 2 \, x + 4}\right )}^{2} + 2\right ) \]
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Timed out. \[ \int \frac {3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt {4+2 x+x^2}} \, dx=\int \frac {2\,x+3}{{\left (x^2+2\,x+3\right )}^2\,\sqrt {x^2+2\,x+4}} \,d x \]
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