Integrand size = 26, antiderivative size = 395 \[ \int \frac {x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=-\frac {\sqrt [3]{-2} \left (1+\sqrt [3]{-2} 3^{2/3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{3^{5/6} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\sqrt [6]{\frac {3}{2}} \left (1-(-3)^{2/3} \sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{\left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (1-\sqrt [3]{2} 3^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{\sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {1}{216} \left (36+2^{2/3} \sqrt [3]{3} \left (1+i \sqrt {3}\right )\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )+\frac {1}{108} \left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )+\frac {1}{108} \left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right ) \]
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Time = 1.12 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2122, 648, 632, 210, 642, 212} \[ \int \frac {x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=-\frac {\sqrt [3]{-2} \left (1+\sqrt [3]{-2} 3^{2/3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{3^{5/6} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\sqrt [6]{\frac {3}{2}} \left (1-(-3)^{2/3} \sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{\left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (1-\sqrt [3]{2} 3^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{\sqrt [6]{2} 3^{5/6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}+\frac {1}{216} \left (36+2^{2/3} \sqrt [3]{3} \left (1+i \sqrt {3}\right )\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )+\frac {1}{108} \left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )+\frac {1}{108} \left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right ) \]
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Rule 210
Rule 212
Rule 632
Rule 642
Rule 648
Rule 2122
Rubi steps \begin{align*} \text {integral}& = 1259712 \int \left (\frac {(-1)^{2/3} \left (3 \sqrt [3]{-3} 2^{2/3}+\left (1-3 (-3)^{2/3} \sqrt [3]{2}\right ) x\right )}{3779136 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac {(-1)^{2/3} \left (3 (-2)^{2/3} \sqrt [3]{3}-\left (1+3 \sqrt [3]{-2} 3^{2/3}\right ) x\right )}{3779136 \sqrt [3]{2} 3^{2/3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {6 \sqrt [3]{2} 3^{2/3}+\left (18-2^{2/3} \sqrt [3]{3}\right ) x}{68024448 \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx \\ & = \frac {1}{54} \int \frac {6 \sqrt [3]{2} 3^{2/3}+\left (18-2^{2/3} \sqrt [3]{3}\right ) x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx+\frac {(-1)^{2/3} \int \frac {3 (-2)^{2/3} \sqrt [3]{3}-\left (1+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{9 \sqrt [3]{2} 3^{2/3}}+\frac {(-1)^{2/3} \int \frac {3 \sqrt [3]{-3} 2^{2/3}+\left (1-3 (-3)^{2/3} \sqrt [3]{2}\right ) x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{3 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2} \\ & = \frac {\left ((-1)^{2/3} \left (1-3 (-3)^{2/3} \sqrt [3]{2}\right )\right ) \int \frac {-3 \sqrt [3]{-3} 2^{2/3}+2 x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{6 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {\left ((-1)^{2/3} \sqrt [3]{\frac {3}{2}} \left (6+\sqrt [3]{-3} 2^{2/3}\right )\right ) \int \frac {1}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{2 \left (1+\sqrt [3]{-1}\right )^2}-\frac {\left ((-1)^{2/3} \left (6-(-2)^{2/3} \sqrt [3]{3}\right )\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{2 \sqrt [3]{2} 3^{2/3}}+\frac {1}{108} \left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \int \frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx+\frac {1}{108} \left (18-2^{2/3} \sqrt [3]{3}\right ) \int \frac {3\ 2^{2/3} \sqrt [3]{3}+2 x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx+\frac {\left (1-\sqrt [3]{2} 3^{2/3}\right ) \int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{2^{2/3} \sqrt [3]{3}} \\ & = \frac {1}{216} \left (36+2^{2/3} \sqrt [3]{3}+i 2^{2/3} 3^{5/6}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )+\frac {1}{108} \left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )+\frac {1}{108} \left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )-\frac {\left ((-1)^{2/3} \sqrt [3]{6} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\right )\right ) \text {Subst}\left (\int \frac {1}{-6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )-x^2} \, dx,x,-3 \sqrt [3]{-3} 2^{2/3}+2 x\right )}{\left (1+\sqrt [3]{-1}\right )^2}-\frac {\left ((-1)^{2/3} \left ((-2)^{2/3}-2\ 3^{2/3}\right )\right ) \text {Subst}\left (\int \frac {1}{-6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )-x^2} \, dx,x,3 (-2)^{2/3} \sqrt [3]{3}+2 x\right )}{\sqrt [3]{6}}-\left (\sqrt [3]{\frac {2}{3}} \left (1-\sqrt [3]{2} 3^{2/3}\right )\right ) \text {Subst}\left (\int \frac {1}{-6 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )-x^2} \, dx,x,3\ 2^{2/3} \sqrt [3]{3}+2 x\right ) \\ & = \frac {(-1)^{2/3} \left ((-2)^{2/3}-2\ 3^{2/3}\right ) \tan ^{-1}\left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{6^{5/6} \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {(-1)^{2/3} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\right ) \tan ^{-1}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{\sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (1-\sqrt [3]{2} 3^{2/3}\right ) \tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{\sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {1}{216} \left (36+2^{2/3} \sqrt [3]{3}+i 2^{2/3} 3^{5/6}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )+\frac {1}{108} \left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )+\frac {1}{108} \left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+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 = 61, normalized size of antiderivative = 0.15 \[ \int \frac {x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {1}{6} \text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}^4}{36+162 \text {$\#$1}+12 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.14
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R}^{5} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(56\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R}^{5} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(56\) |
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Timed out. \[ \int \frac {x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\text {Timed out} \]
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Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.18 \[ \int \frac {x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\operatorname {RootSum} {\left (72662865048 t^{6} - 72662865048 t^{5} + 24163559388 t^{4} - 2646786132 t^{3} - 6626610 t^{2} - 4374 t - 1, \left ( t \mapsto t \log {\left (- \frac {89236417131047376 t^{5}}{833243797} + \frac {89301949532998128 t^{4}}{833243797} - \frac {29740560281805852 t^{3}}{833243797} + \frac {192466080408420 t^{2}}{49014341} + \frac {5867255361684 t}{833243797} + x + \frac {5365044886}{2499731391} \right )} \right )\right )} \]
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\[ \int \frac {x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x^{5}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \]
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\[ \int \frac {x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x^{5}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \]
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Time = 0.38 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.08 \[ \int \frac {x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\sum _{k=1}^6\ln \left (\frac {362797056\,\left (19236852\,x\,\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )-19131876\,x-6482268\,x\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^2+742851\,x\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^3-4130\,x\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^4+x\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^5-154944576\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^2+17047422\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^3+27054\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^4+9\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^5+465542316\,\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )-465542316\right )}{{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^5}\right )\,\mathrm {root}\left (z^6-z^5+\frac {421\,z^4}{1266}-\frac {100853\,z^3}{2768742}-\frac {505\,z^2}{5537484}-\frac {z}{16612452}-\frac {1}{72662865048},z,k\right ) \]
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