Integrand size = 35, antiderivative size = 307 \[ \int \frac {x^3 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx=-\frac {1}{28} \left (35-9 i \sqrt {7}\right ) x-\frac {1}{28} \left (35+9 i \sqrt {7}\right ) x+\frac {1}{28} \left (7-5 i \sqrt {7}\right ) x^2+\frac {1}{28} \left (7+5 i \sqrt {7}\right ) x^2+\frac {1}{42} \left (7-5 i \sqrt {7}\right ) x^3+\frac {1}{42} \left (7+5 i \sqrt {7}\right ) x^3+\frac {11 \left (9 i+5 \sqrt {7}\right ) \arctan \left (\frac {1-i \sqrt {7}+8 x}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{4 \sqrt {14 \left (35+i \sqrt {7}\right )}}-\frac {11 \left (9 i-5 \sqrt {7}\right ) \arctan \left (\frac {1+i \sqrt {7}+8 x}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{4 \sqrt {14 \left (35-i \sqrt {7}\right )}}+\frac {3}{112} \left (7-11 i \sqrt {7}\right ) \log \left (4+\left (1-i \sqrt {7}\right ) x+4 x^2\right )+\frac {3}{112} \left (7+11 i \sqrt {7}\right ) \log \left (4+\left (1+i \sqrt {7}\right ) x+4 x^2\right ) \]
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Time = 0.40 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2112, 814, 648, 632, 210, 642} \[ \int \frac {x^3 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx=\frac {11 \left (5 \sqrt {7}+9 i\right ) \arctan \left (\frac {8 x-i \sqrt {7}+1}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{4 \sqrt {14 \left (35+i \sqrt {7}\right )}}-\frac {11 \left (-5 \sqrt {7}+9 i\right ) \arctan \left (\frac {8 x+i \sqrt {7}+1}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{4 \sqrt {14 \left (35-i \sqrt {7}\right )}}+\frac {1}{42} \left (7+5 i \sqrt {7}\right ) x^3+\frac {1}{42} \left (7-5 i \sqrt {7}\right ) x^3+\frac {1}{28} \left (7+5 i \sqrt {7}\right ) x^2+\frac {1}{28} \left (7-5 i \sqrt {7}\right ) x^2+\frac {3}{112} \left (7-11 i \sqrt {7}\right ) \log \left (4 x^2+\left (1-i \sqrt {7}\right ) x+4\right )+\frac {3}{112} \left (7+11 i \sqrt {7}\right ) \log \left (4 x^2+\left (1+i \sqrt {7}\right ) x+4\right )-\frac {1}{28} \left (35+9 i \sqrt {7}\right ) x-\frac {1}{28} \left (35-9 i \sqrt {7}\right ) x \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 814
Rule 2112
Rubi steps \begin{align*} \text {integral}& = \frac {i \int \frac {x^3 \left (9-5 i \sqrt {7}+\left (10-2 i \sqrt {7}\right ) x\right )}{4+\left (1-i \sqrt {7}\right ) x+4 x^2} \, dx}{\sqrt {7}}-\frac {i \int \frac {x^3 \left (9+5 i \sqrt {7}+\left (10+2 i \sqrt {7}\right ) x\right )}{4+\left (1+i \sqrt {7}\right ) x+4 x^2} \, dx}{\sqrt {7}} \\ & = \frac {i \int \left (\frac {1}{4} \left (-9+5 i \sqrt {7}\right )+\frac {1}{2} \left (5-i \sqrt {7}\right ) x+\frac {1}{2} \left (5-i \sqrt {7}\right ) x^2+\frac {2 \left (9-5 i \sqrt {7}\right )-3 \left (11+i \sqrt {7}\right ) x}{2 \left (4+\left (1-i \sqrt {7}\right ) x+4 x^2\right )}\right ) \, dx}{\sqrt {7}}-\frac {i \int \left (\frac {1}{4} \left (-9-5 i \sqrt {7}\right )+\frac {1}{2} \left (5+i \sqrt {7}\right ) x+\frac {1}{2} \left (5+i \sqrt {7}\right ) x^2+\frac {2 \left (9+5 i \sqrt {7}\right )-3 \left (11-i \sqrt {7}\right ) x}{2 \left (4+\left (1+i \sqrt {7}\right ) x+4 x^2\right )}\right ) \, dx}{\sqrt {7}} \\ & = -\frac {1}{28} \left (35-9 i \sqrt {7}\right ) x-\frac {1}{28} \left (35+9 i \sqrt {7}\right ) x+\frac {1}{28} \left (7-5 i \sqrt {7}\right ) x^2+\frac {1}{28} \left (7+5 i \sqrt {7}\right ) x^2+\frac {1}{42} \left (7-5 i \sqrt {7}\right ) x^3+\frac {1}{42} \left (7+5 i \sqrt {7}\right ) x^3+\frac {i \int \frac {2 \left (9-5 i \sqrt {7}\right )-3 \left (11+i \sqrt {7}\right ) x}{4+\left (1-i \sqrt {7}\right ) x+4 x^2} \, dx}{2 \sqrt {7}}-\frac {i \int \frac {2 \left (9+5 i \sqrt {7}\right )-3 \left (11-i \sqrt {7}\right ) x}{4+\left (1+i \sqrt {7}\right ) x+4 x^2} \, dx}{2 \sqrt {7}} \\ & = -\frac {1}{28} \left (35-9 i \sqrt {7}\right ) x-\frac {1}{28} \left (35+9 i \sqrt {7}\right ) x+\frac {1}{28} \left (7-5 i \sqrt {7}\right ) x^2+\frac {1}{28} \left (7+5 i \sqrt {7}\right ) x^2+\frac {1}{42} \left (7-5 i \sqrt {7}\right ) x^3+\frac {1}{42} \left (7+5 i \sqrt {7}\right ) x^3+\frac {1}{56} \left (11 \left (35-9 i \sqrt {7}\right )\right ) \int \frac {1}{4+\left (1+i \sqrt {7}\right ) x+4 x^2} \, dx+\frac {1}{56} \left (11 \left (35+9 i \sqrt {7}\right )\right ) \int \frac {1}{4+\left (1-i \sqrt {7}\right ) x+4 x^2} \, dx+\frac {1}{112} \left (3 \left (7-11 i \sqrt {7}\right )\right ) \int \frac {1-i \sqrt {7}+8 x}{4+\left (1-i \sqrt {7}\right ) x+4 x^2} \, dx+\frac {1}{112} \left (3 \left (7+11 i \sqrt {7}\right )\right ) \int \frac {1+i \sqrt {7}+8 x}{4+\left (1+i \sqrt {7}\right ) x+4 x^2} \, dx \\ & = -\frac {1}{28} \left (35-9 i \sqrt {7}\right ) x-\frac {1}{28} \left (35+9 i \sqrt {7}\right ) x+\frac {1}{28} \left (7-5 i \sqrt {7}\right ) x^2+\frac {1}{28} \left (7+5 i \sqrt {7}\right ) x^2+\frac {1}{42} \left (7-5 i \sqrt {7}\right ) x^3+\frac {1}{42} \left (7+5 i \sqrt {7}\right ) x^3+\frac {3}{112} \left (7-11 i \sqrt {7}\right ) \log \left (4+\left (1-i \sqrt {7}\right ) x+4 x^2\right )+\frac {3}{112} \left (7+11 i \sqrt {7}\right ) \log \left (4+\left (1+i \sqrt {7}\right ) x+4 x^2\right )-\frac {1}{28} \left (11 \left (35-9 i \sqrt {7}\right )\right ) \text {Subst}\left (\int \frac {1}{-2 \left (35-i \sqrt {7}\right )-x^2} \, dx,x,1+i \sqrt {7}+8 x\right )-\frac {1}{28} \left (11 \left (35+9 i \sqrt {7}\right )\right ) \text {Subst}\left (\int \frac {1}{-2 \left (35+i \sqrt {7}\right )-x^2} \, dx,x,1-i \sqrt {7}+8 x\right ) \\ & = -\frac {1}{28} \left (35-9 i \sqrt {7}\right ) x-\frac {1}{28} \left (35+9 i \sqrt {7}\right ) x+\frac {1}{28} \left (7-5 i \sqrt {7}\right ) x^2+\frac {1}{28} \left (7+5 i \sqrt {7}\right ) x^2+\frac {1}{42} \left (7-5 i \sqrt {7}\right ) x^3+\frac {1}{42} \left (7+5 i \sqrt {7}\right ) x^3+\frac {11 \left (9 i+5 \sqrt {7}\right ) \tan ^{-1}\left (\frac {1-i \sqrt {7}+8 x}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{4 \sqrt {14 \left (35+i \sqrt {7}\right )}}-\frac {11 \left (9 i-5 \sqrt {7}\right ) \tan ^{-1}\left (\frac {1+i \sqrt {7}+8 x}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{4 \sqrt {14 \left (35-i \sqrt {7}\right )}}+\frac {3}{112} \left (7-11 i \sqrt {7}\right ) \log \left (4+\left (1-i \sqrt {7}\right ) x+4 x^2\right )+\frac {3}{112} \left (7+11 i \sqrt {7}\right ) \log \left (4+\left (1+i \sqrt {7}\right ) x+4 x^2\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.36 \[ \int \frac {x^3 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx=\frac {1}{6} \left (x \left (-15+3 x+2 x^2\right )+3 \text {RootSum}\left [2+\text {$\#$1}+5 \text {$\#$1}^2+\text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {10 \log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}+19 \log (x-\text {$\#$1}) \text {$\#$1}^2+3 \log (x-\text {$\#$1}) \text {$\#$1}^3}{1+10 \text {$\#$1}+3 \text {$\#$1}^2+8 \text {$\#$1}^3}\&\right ]\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.24
method | result | size |
default | \(\frac {x^{3}}{3}+\frac {x^{2}}{2}-\frac {5 x}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )}{\sum }\frac {\left (3 \textit {\_R}^{3}+19 \textit {\_R}^{2}+\textit {\_R} +10\right ) \ln \left (x -\textit {\_R} \right )}{8 \textit {\_R}^{3}+3 \textit {\_R}^{2}+10 \textit {\_R} +1}\right )}{2}\) | \(74\) |
risch | \(\frac {x^{3}}{3}+\frac {x^{2}}{2}-\frac {5 x}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )}{\sum }\frac {\left (3 \textit {\_R}^{3}+19 \textit {\_R}^{2}+\textit {\_R} +10\right ) \ln \left (x -\textit {\_R} \right )}{8 \textit {\_R}^{3}+3 \textit {\_R}^{2}+10 \textit {\_R} +1}\right )}{2}\) | \(74\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1202 vs. \(2 (199) = 398\).
Time = 1.06 (sec) , antiderivative size = 1202, normalized size of antiderivative = 3.92 \[ \int \frac {x^3 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx=\text {Too large to display} \]
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Time = 0.58 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.20 \[ \int \frac {x^3 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx=\frac {x^{3}}{3} + \frac {x^{2}}{2} - \frac {5 x}{2} + \operatorname {RootSum} {\left (1372 t^{4} - 1029 t^{3} + 3136 t^{2} + 2688 t + 512, \left ( t \mapsto t \log {\left (\frac {5831 t^{3}}{1936} - \frac {23765 t^{2}}{7744} + \frac {2065 t}{242} + x + \frac {415}{121} \right )} \right )\right )} \]
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\[ \int \frac {x^3 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx=\int { \frac {{\left (2 \, x^{3} + 3 \, x^{2} + x + 5\right )} x^{3}}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2} \,d x } \]
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\[ \int \frac {x^3 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx=\int { \frac {{\left (2 \, x^{3} + 3 \, x^{2} + x + 5\right )} x^{3}}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2} \,d x } \]
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Time = 9.86 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.42 \[ \int \frac {x^3 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx=\left (\sum _{k=1}^4\ln \left (-29\,x+\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )\,\left (-\frac {289\,x}{4}+\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )\,\left (\frac {581\,x}{16}-\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )\,\left (\frac {147\,x}{4}-\frac {49}{16}\right )+\frac {1141}{64}\right )+\frac {47}{4}\right )+7\right )\,\mathrm {root}\left (z^4-\frac {3\,z^3}{4}+\frac {16\,z^2}{7}+\frac {96\,z}{49}+\frac {128}{343},z,k\right )\right )-\frac {5\,x}{2}+\frac {x^2}{2}+\frac {x^3}{3} \]
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