Integrand size = 35, antiderivative size = 245 \[ \int \frac {5+x+3 x^2+2 x^3}{x \left (2+x+5 x^2+x^3+2 x^4\right )} \, dx=-\frac {\left (53+i \sqrt {7}\right ) \text {arctanh}\left (\frac {i-\sqrt {7}+8 i x}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{2 \sqrt {14 \left (35-i \sqrt {7}\right )}}+\frac {\left (53-i \sqrt {7}\right ) \text {arctanh}\left (\frac {i+\sqrt {7}+8 i x}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{2 \sqrt {14 \left (35+i \sqrt {7}\right )}}+\frac {1}{28} \left (35-9 i \sqrt {7}\right ) \log (x)+\frac {1}{28} \left (35+9 i \sqrt {7}\right ) \log (x)-\frac {1}{56} \left (35-9 i \sqrt {7}\right ) \log \left (4 i+\left (i-\sqrt {7}\right ) x+4 i x^2\right )-\frac {1}{56} \left (35+9 i \sqrt {7}\right ) \log \left (4 i+\left (i+\sqrt {7}\right ) x+4 i x^2\right ) \]
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Time = 0.35 (sec) , antiderivative size = 245, 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, 212, 642} \[ \int \frac {5+x+3 x^2+2 x^3}{x \left (2+x+5 x^2+x^3+2 x^4\right )} \, dx=-\frac {\left (53+i \sqrt {7}\right ) \text {arctanh}\left (\frac {8 i x-\sqrt {7}+i}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{2 \sqrt {14 \left (35-i \sqrt {7}\right )}}+\frac {\left (53-i \sqrt {7}\right ) \text {arctanh}\left (\frac {8 i x+\sqrt {7}+i}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{2 \sqrt {14 \left (35+i \sqrt {7}\right )}}-\frac {1}{56} \left (35-9 i \sqrt {7}\right ) \log \left (4 i x^2+\left (-\sqrt {7}+i\right ) x+4 i\right )-\frac {1}{56} \left (35+9 i \sqrt {7}\right ) \log \left (4 i x^2+\left (\sqrt {7}+i\right ) x+4 i\right )+\frac {1}{28} \left (35+9 i \sqrt {7}\right ) \log (x)+\frac {1}{28} \left (35-9 i \sqrt {7}\right ) \log (x) \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 814
Rule 2112
Rubi steps \begin{align*} \text {integral}& = \frac {i \int \frac {9-5 i \sqrt {7}+\left (10-2 i \sqrt {7}\right ) x}{x \left (4+\left (1-i \sqrt {7}\right ) x+4 x^2\right )} \, dx}{\sqrt {7}}-\frac {i \int \frac {9+5 i \sqrt {7}+\left (10+2 i \sqrt {7}\right ) x}{x \left (4+\left (1+i \sqrt {7}\right ) x+4 x^2\right )} \, dx}{\sqrt {7}} \\ & = -\frac {i \int \left (\frac {9+5 i \sqrt {7}}{4 x}+\frac {3 \left (11 i+\sqrt {7}\right )-2 \left (9 i-5 \sqrt {7}\right ) x}{2 \left (4 i+\left (i-\sqrt {7}\right ) x+4 i x^2\right )}\right ) \, dx}{\sqrt {7}}+\frac {i \int \left (\frac {9-5 i \sqrt {7}}{4 x}+\frac {3 \left (11 i-\sqrt {7}\right )-2 \left (9 i+5 \sqrt {7}\right ) x}{2 \left (4 i+\left (i+\sqrt {7}\right ) x+4 i x^2\right )}\right ) \, dx}{\sqrt {7}} \\ & = \frac {1}{28} \left (35-9 i \sqrt {7}\right ) \log (x)+\frac {1}{28} \left (35+9 i \sqrt {7}\right ) \log (x)-\frac {i \int \frac {3 \left (11 i+\sqrt {7}\right )-2 \left (9 i-5 \sqrt {7}\right ) x}{4 i+\left (i-\sqrt {7}\right ) x+4 i x^2} \, dx}{2 \sqrt {7}}+\frac {i \int \frac {3 \left (11 i-\sqrt {7}\right )-2 \left (9 i+5 \sqrt {7}\right ) x}{4 i+\left (i+\sqrt {7}\right ) x+4 i x^2} \, dx}{2 \sqrt {7}} \\ & = \frac {1}{28} \left (35-9 i \sqrt {7}\right ) \log (x)+\frac {1}{28} \left (35+9 i \sqrt {7}\right ) \log (x)-\frac {1}{56} \left (35-9 i \sqrt {7}\right ) \int \frac {i-\sqrt {7}+8 i x}{4 i+\left (i-\sqrt {7}\right ) x+4 i x^2} \, dx-\frac {1}{56} \left (35+9 i \sqrt {7}\right ) \int \frac {i+\sqrt {7}+8 i x}{4 i+\left (i+\sqrt {7}\right ) x+4 i x^2} \, dx-\frac {1}{28} \left (-7 i+53 \sqrt {7}\right ) \int \frac {1}{4 i+\left (i+\sqrt {7}\right ) x+4 i x^2} \, dx+\frac {1}{28} \left (7 i+53 \sqrt {7}\right ) \int \frac {1}{4 i+\left (i-\sqrt {7}\right ) x+4 i x^2} \, dx \\ & = \frac {1}{28} \left (35-9 i \sqrt {7}\right ) \log (x)+\frac {1}{28} \left (35+9 i \sqrt {7}\right ) \log (x)-\frac {1}{56} \left (35-9 i \sqrt {7}\right ) \log \left (4 i+\left (i-\sqrt {7}\right ) x+4 i x^2\right )-\frac {1}{56} \left (35+9 i \sqrt {7}\right ) \log \left (4 i+\left (i+\sqrt {7}\right ) x+4 i x^2\right )-\frac {1}{14} \left (7 i-53 \sqrt {7}\right ) \text {Subst}\left (\int \frac {1}{2 \left (35+i \sqrt {7}\right )-x^2} \, dx,x,i+\sqrt {7}+8 i x\right )-\frac {1}{14} \left (7 i+53 \sqrt {7}\right ) \text {Subst}\left (\int \frac {1}{2 \left (35-i \sqrt {7}\right )-x^2} \, dx,x,i-\sqrt {7}+8 i x\right ) \\ & = -\frac {\left (53+i \sqrt {7}\right ) \tanh ^{-1}\left (\frac {i-\sqrt {7}+8 i x}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{2 \sqrt {14 \left (35-i \sqrt {7}\right )}}+\frac {\left (53-i \sqrt {7}\right ) \tanh ^{-1}\left (\frac {i+\sqrt {7}+8 i x}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{2 \sqrt {14 \left (35+i \sqrt {7}\right )}}+\frac {1}{28} \left (35-9 i \sqrt {7}\right ) \log (x)+\frac {1}{28} \left (35+9 i \sqrt {7}\right ) \log (x)-\frac {1}{56} \left (35-9 i \sqrt {7}\right ) \log \left (4 i+\left (i-\sqrt {7}\right ) x+4 i x^2\right )-\frac {1}{56} \left (35+9 i \sqrt {7}\right ) \log \left (4 i+\left (i+\sqrt {7}\right ) x+4 i 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 = 101, normalized size of antiderivative = 0.41 \[ \int \frac {5+x+3 x^2+2 x^3}{x \left (2+x+5 x^2+x^3+2 x^4\right )} \, dx=\frac {5 \log (x)}{2}-\frac {1}{2} \text {RootSum}\left [2+\text {$\#$1}+5 \text {$\#$1}^2+\text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {3 \log (x-\text {$\#$1})+19 \log (x-\text {$\#$1}) \text {$\#$1}+\log (x-\text {$\#$1}) \text {$\#$1}^2+10 \log (x-\text {$\#$1}) \text {$\#$1}^3}{1+10 \text {$\#$1}+3 \text {$\#$1}^2+8 \text {$\#$1}^3}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.21
method | result | size |
risch | \(\frac {5 \ln \left (x \right )}{2}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (686 \textit {\_Z}^{4}+1715 \textit {\_Z}^{3}+1372 \textit {\_Z}^{2}+448 \textit {\_Z} +256\right )}{\sum }\textit {\_R} \ln \left (2058 \textit {\_R}^{3}+20825 \textit {\_R}^{2}+25844 \textit {\_R} +8384 x +6816\right )\right )\) | \(51\) |
default | \(\frac {5 \ln \left (x \right )}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )}{\sum }\frac {\left (-10 \textit {\_R}^{3}-\textit {\_R}^{2}-19 \textit {\_R} -3\right ) \ln \left (x -\textit {\_R} \right )}{8 \textit {\_R}^{3}+3 \textit {\_R}^{2}+10 \textit {\_R} +1}\right )}{2}\) | \(67\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1143 vs. \(2 (148) = 296\).
Time = 1.02 (sec) , antiderivative size = 1143, normalized size of antiderivative = 4.67 \[ \int \frac {5+x+3 x^2+2 x^3}{x \left (2+x+5 x^2+x^3+2 x^4\right )} \, dx=\text {Too large to display} \]
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Time = 8.49 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.24 \[ \int \frac {5+x+3 x^2+2 x^3}{x \left (2+x+5 x^2+x^3+2 x^4\right )} \, dx=\frac {5 \log {\left (x \right )}}{2} + \operatorname {RootSum} {\left (686 t^{4} + 1715 t^{3} + 1372 t^{2} + 448 t + 256, \left ( t \mapsto t \log {\left (- \frac {160344611 t^{4}}{532759184} - \frac {16880402 t^{3}}{33297449} + \frac {4010520787 t^{2}}{2131036736} + \frac {1537535671 t}{532759184} + x + \frac {46660495}{66594898} \right )} \right )\right )} \]
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\[ \int \frac {5+x+3 x^2+2 x^3}{x \left (2+x+5 x^2+x^3+2 x^4\right )} \, dx=\int { \frac {2 \, x^{3} + 3 \, x^{2} + x + 5}{{\left (2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2\right )} x} \,d x } \]
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\[ \int \frac {5+x+3 x^2+2 x^3}{x \left (2+x+5 x^2+x^3+2 x^4\right )} \, dx=\int { \frac {2 \, x^{3} + 3 \, x^{2} + x + 5}{{\left (2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2\right )} x} \,d x } \]
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Time = 9.70 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.97 \[ \int \frac {5+x+3 x^2+2 x^3}{x \left (2+x+5 x^2+x^3+2 x^4\right )} \, dx=\frac {5\,\ln \left (x\right )}{2}+\left (\sum _{k=1}^4\ln \left (\frac {223\,\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}{8}-\frac {31\,x}{2}+\frac {\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )\,x\,71}{16}-\frac {{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^2\,x\,4463}{64}+\frac {{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^3\,x\,1449}{16}+\frac {{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^4\,x\,3675}{32}+\frac {257\,{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^2}{32}+\frac {1673\,{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^3}{64}-\frac {441\,{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^4}{32}+10\right )\,\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )\right ) \]
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