Integrand size = 30, antiderivative size = 70 \[ \int \frac {1+2 x}{\sqrt {-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx=-\frac {5 \arctan \left (\frac {\sqrt {\frac {7}{2}} (2-x)}{2 \sqrt {-1+6 x+x^2}}\right )}{6 \sqrt {14}}-\frac {\text {arctanh}\left (\frac {\sqrt {7} (1+x)}{\sqrt {-1+6 x+x^2}}\right )}{3 \sqrt {7}} \]
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Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1049, 1043, 213, 209} \[ \int \frac {1+2 x}{\sqrt {-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx=-\frac {5 \arctan \left (\frac {\sqrt {\frac {7}{2}} (2-x)}{2 \sqrt {x^2+6 x-1}}\right )}{6 \sqrt {14}}-\frac {\text {arctanh}\left (\frac {\sqrt {7} (x+1)}{\sqrt {x^2+6 x-1}}\right )}{3 \sqrt {7}} \]
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Rule 209
Rule 213
Rule 1043
Rule 1049
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{42} \int \frac {-70-70 x}{\sqrt {-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx\right )+\frac {1}{42} \int \frac {-28+14 x}{\sqrt {-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx \\ & = -\left (\frac {896}{3} \text {Subst}\left (\int \frac {1}{-200704+28 x^2} \, dx,x,\frac {-224-224 x}{\sqrt {-1+6 x+x^2}}\right )\right )-\frac {2800}{3} \text {Subst}\left (\int \frac {1}{627200+28 x^2} \, dx,x,\frac {280-140 x}{\sqrt {-1+6 x+x^2}}\right ) \\ & = -\frac {5 \arctan \left (\frac {\sqrt {\frac {7}{2}} (2-x)}{2 \sqrt {-1+6 x+x^2}}\right )}{6 \sqrt {14}}-\frac {\text {arctanh}\left (\frac {\sqrt {7} (1+x)}{\sqrt {-1+6 x+x^2}}\right )}{3 \sqrt {7}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.76 \[ \int \frac {1+2 x}{\sqrt {-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx=\text {RootSum}\left [171-104 \text {$\#$1}+46 \text {$\#$1}^2-8 \text {$\#$1}^3+3 \text {$\#$1}^4\&,\frac {4 \log \left (-x+\sqrt {-1+6 x+x^2}-\text {$\#$1}\right )-\log \left (-x+\sqrt {-1+6 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-x+\sqrt {-1+6 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-26+23 \text {$\#$1}-6 \text {$\#$1}^2+3 \text {$\#$1}^3}\&\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs. \(2(53)=106\).
Time = 1.65 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.26
method | result | size |
default | \(\frac {\sqrt {-\frac {6 \left (-2+x \right )^{2}}{\left (-1-x \right )^{2}}+15}\, \left (5 \sqrt {14}\, \arctan \left (\frac {\sqrt {14}\, \sqrt {-\frac {6 \left (-2+x \right )^{2}}{\left (-1-x \right )^{2}}+15}\, \left (-2+x \right )}{4 \left (\frac {2 \left (-2+x \right )^{2}}{\left (-1-x \right )^{2}}-5\right ) \left (-1-x \right )}\right )-4 \sqrt {7}\, \operatorname {arctanh}\left (\frac {\sqrt {-\frac {6 \left (-2+x \right )^{2}}{\left (-1-x \right )^{2}}+15}\, \sqrt {7}}{21}\right )\right )}{84 \sqrt {-\frac {3 \left (\frac {2 \left (-2+x \right )^{2}}{\left (-1-x \right )^{2}}-5\right )}{\left (\frac {-2+x}{-1-x}+1\right )^{2}}}\, \left (\frac {-2+x}{-1-x}+1\right )}\) | \(158\) |
trager | \(-\frac {672 \ln \left (\frac {159860736 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{5} x +2221632 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{3} x -4044096 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{3}+352352 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{2} \sqrt {x^{2}+6 x -1}-2340 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right ) x +3978 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )+319 \sqrt {x^{2}+6 x -1}}{2016 x \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{2}+37 x +40}\right ) \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{3}}{11}-\frac {34 \ln \left (\frac {159860736 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{5} x +2221632 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{3} x -4044096 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{3}+352352 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{2} \sqrt {x^{2}+6 x -1}-2340 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right ) x +3978 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )+319 \sqrt {x^{2}+6 x -1}}{2016 x \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{2}+37 x +40}\right ) \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )}{33}+\operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right ) \ln \left (-\frac {1568802816 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{5} x +6019776 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{3} x +39686976 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{3}+3171168 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{2} \sqrt {x^{2}+6 x -1}+5768 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right ) x +73542 \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )+50611 \sqrt {x^{2}+6 x -1}}{2016 x \operatorname {RootOf}\left (451584 \textit {\_Z}^{4}+7616 \textit {\_Z}^{2}+121\right )^{2}-3 x -40}\right )\) | \(502\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 229, normalized size of antiderivative = 3.27 \[ \int \frac {1+2 x}{\sqrt {-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx=\frac {1}{168} \, \sqrt {14} \sqrt {20 i \, \sqrt {2} - 17} \log \left (\sqrt {14} \sqrt {20 i \, \sqrt {2} - 17} {\left (13 i \, \sqrt {2} + 16\right )} - 198 \, x - 132 i \, \sqrt {2} + 198 \, \sqrt {x^{2} + 6 \, x - 1} - 132\right ) - \frac {1}{168} \, \sqrt {14} \sqrt {20 i \, \sqrt {2} - 17} \log \left (\sqrt {14} \sqrt {20 i \, \sqrt {2} - 17} {\left (-13 i \, \sqrt {2} - 16\right )} - 198 \, x - 132 i \, \sqrt {2} + 198 \, \sqrt {x^{2} + 6 \, x - 1} - 132\right ) - \frac {1}{168} \, \sqrt {14} \sqrt {-20 i \, \sqrt {2} - 17} \log \left (\sqrt {14} {\left (13 i \, \sqrt {2} - 16\right )} \sqrt {-20 i \, \sqrt {2} - 17} - 198 \, x + 132 i \, \sqrt {2} + 198 \, \sqrt {x^{2} + 6 \, x - 1} - 132\right ) + \frac {1}{168} \, \sqrt {14} \sqrt {-20 i \, \sqrt {2} - 17} \log \left (\sqrt {14} {\left (-13 i \, \sqrt {2} + 16\right )} \sqrt {-20 i \, \sqrt {2} - 17} - 198 \, x + 132 i \, \sqrt {2} + 198 \, \sqrt {x^{2} + 6 \, x - 1} - 132\right ) \]
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\[ \int \frac {1+2 x}{\sqrt {-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx=\int \frac {2 x + 1}{\sqrt {x^{2} + 6 x - 1} \cdot \left (3 x^{2} + 4 x + 4\right )}\, dx \]
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\[ \int \frac {1+2 x}{\sqrt {-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx=\int { \frac {2 \, x + 1}{{\left (3 \, x^{2} + 4 \, x + 4\right )} \sqrt {x^{2} + 6 \, x - 1}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (51) = 102\).
Time = 0.31 (sec) , antiderivative size = 257, normalized size of antiderivative = 3.67 \[ \int \frac {1+2 x}{\sqrt {-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx=-\frac {5}{84} \, \sqrt {7} \sqrt {2} {\left (\arctan \left (2\right ) + \arctan \left (\frac {1}{8} \, {\left (x - \sqrt {x^{2} + 6 \, x - 1}\right )} {\left (\sqrt {14} + \sqrt {2}\right )} + \frac {1}{8} \, \sqrt {14} + \frac {3}{8} \, \sqrt {2}\right )\right )} + \frac {5}{84} \, \sqrt {7} \sqrt {2} {\left (\arctan \left (\frac {1}{2}\right ) + \arctan \left (-\frac {1}{8} \, {\left (x - \sqrt {x^{2} + 6 \, x - 1}\right )} {\left (\sqrt {14} - \sqrt {2}\right )} - \frac {1}{8} \, \sqrt {14} + \frac {3}{8} \, \sqrt {2}\right )\right )} + \frac {1}{42} \, \sqrt {7} \log \left (4 \, {\left (4 \, \sqrt {7} \sqrt {2} + 3 \, x + \sqrt {7} - 4 \, \sqrt {2} - 3 \, \sqrt {x^{2} + 6 \, x - 1} + 2\right )}^{2} + 16 \, {\left (\sqrt {7} \sqrt {2} - 3 \, x - \sqrt {7} - \sqrt {2} + 3 \, \sqrt {x^{2} + 6 \, x - 1} - 2\right )}^{2}\right ) - \frac {1}{42} \, \sqrt {7} \log \left (4 \, {\left (4 \, \sqrt {7} \sqrt {2} + 3 \, x - \sqrt {7} + 4 \, \sqrt {2} - 3 \, \sqrt {x^{2} + 6 \, x - 1} + 2\right )}^{2} + 16 \, {\left (\sqrt {7} \sqrt {2} - 3 \, x + \sqrt {7} + \sqrt {2} + 3 \, \sqrt {x^{2} + 6 \, x - 1} - 2\right )}^{2}\right ) \]
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Timed out. \[ \int \frac {1+2 x}{\sqrt {-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx=\int \frac {2\,x+1}{\sqrt {x^2+6\,x-1}\,\left (3\,x^2+4\,x+4\right )} \,d x \]
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