Integrand size = 26, antiderivative size = 677 \[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {9 (-2)^{2/3}+\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{2916\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac {9\ 2^{2/3}+\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{13122\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {3\ 2^{2/3} \sqrt [3]{3}-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{8748\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {\sqrt [3]{-1} \left (3 (-3)^{2/3}-2^{2/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{486\ 6^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )^{3/2}}+\frac {\left (3 (-3)^{2/3}+\sqrt [3]{-1} 2^{2/3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{486\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{486\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\sqrt [6]{-\frac {1}{3}} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{5832 \sqrt [3]{2} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{5832 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{52488 \sqrt [3]{2} 3^{2/3}} \]
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Time = 1.56 (sec) , antiderivative size = 677, 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, 652, 632, 210, 642, 212} \[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {\sqrt [3]{-1} \left (3 (-3)^{2/3}-2^{2/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{486\ 6^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )^{3/2}}+\frac {\left (3 (-3)^{2/3}+\sqrt [3]{-1} 2^{2/3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{486\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{486\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )^{3/2}}+\frac {\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x+9 (-2)^{2/3}}{2916\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac {\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x+9\ 2^{2/3}}{13122\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {3\ 2^{2/3} \sqrt [3]{3}-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{8748\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}+\frac {\sqrt [6]{-\frac {1}{3}} \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{5832 \sqrt [3]{2} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{5832 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{52488 \sqrt [3]{2} 3^{2/3}} \]
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Rule 210
Rule 212
Rule 632
Rule 642
Rule 652
Rule 2122
Rubi steps \begin{align*} \text {integral}& = 1586874322944 \int \left (-\frac {-2 \sqrt [3]{-1} 3^{2/3}+3 (-2)^{2/3} x}{1542441841901568\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )^2}+\frac {-3 i 3^{5/6}+\left (\sqrt [3]{-2}+\sqrt [3]{2}\right ) x}{4627325525704704\ 6^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}-\frac {-2 \sqrt [3]{-1} 3^{2/3}+3\ 2^{2/3} x}{1542441841901568\ 2^{2/3} \left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2}+\frac {3+3 \sqrt [3]{-1}-i \sqrt [3]{2} \sqrt [6]{3} x}{4627325525704704\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac {2+2^{2/3} \sqrt [3]{3} x}{514147280633856\ 2^{2/3} \sqrt [3]{3} \left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2}+\frac {3 \sqrt [3]{3}+\sqrt [3]{2} x}{41645929731342336\ 6^{2/3} \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx \\ & = -\frac {\int \frac {-2 \sqrt [3]{-1} 3^{2/3}+3\ 2^{2/3} x}{\left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{8748\ 2^{2/3}}-\frac {\int \frac {2+2^{2/3} \sqrt [3]{3} x}{\left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{2916\ 2^{2/3} \sqrt [3]{3}}+\frac {\int \frac {3 \sqrt [3]{3}+\sqrt [3]{2} x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{26244\ 6^{2/3}}+\frac {\int \frac {3+3 \sqrt [3]{-1}-i \sqrt [3]{2} \sqrt [6]{3} x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{2916\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\int \frac {-3 i 3^{5/6}+\left (\sqrt [3]{-2}+\sqrt [3]{2}\right ) x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{2916\ 6^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\int \frac {-2 \sqrt [3]{-1} 3^{2/3}+3 (-2)^{2/3} x}{\left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )^2} \, dx}{972\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4} \\ & = \frac {9 (-2)^{2/3}+\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{2916\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac {9\ 2^{2/3}+\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{26244\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {3\ 2^{2/3} \sqrt [3]{3}-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{8748\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {\sqrt [6]{-\frac {1}{3}} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{5832 \sqrt [3]{2} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{5832 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{52488 \sqrt [3]{2} 3^{2/3}}-\frac {\left (-18 \sqrt [3]{-6} (-1)^{2/3}+4 \sqrt [3]{-1} 3^{2/3}\right ) \int \frac {1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{972\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (24-18 (-3)^{2/3} \sqrt [3]{2}\right )}-\frac {\left (2\ 3^{2/3}-9 \sqrt [3]{6}\right ) \int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{52488 \left (2\ 2^{2/3}-3\ 3^{2/3}\right )}+-\frac {\left (-4 \sqrt [3]{-1} 3^{2/3}-18 (-1)^{2/3} \sqrt [3]{6}\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{8748\ 2^{2/3} \left (24+18 \sqrt [3]{-2} 3^{2/3}\right )} \\ & = \frac {9 (-2)^{2/3}+\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{2916\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac {9\ 2^{2/3}+\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{26244\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {3\ 2^{2/3} \sqrt [3]{3}-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{8748\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {\sqrt [6]{-\frac {1}{3}} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{5832 \sqrt [3]{2} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{5832 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{52488 \sqrt [3]{2} 3^{2/3}}+\frac {\left (-18 \sqrt [3]{-6} (-1)^{2/3}+4 \sqrt [3]{-1} 3^{2/3}\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 )}{486\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (24-18 (-3)^{2/3} \sqrt [3]{2}\right )}+\frac {\left (2\ 3^{2/3}-9 \sqrt [3]{6}\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 )}{26244 \left (2\ 2^{2/3}-3\ 3^{2/3}\right )}--\frac {\left (-4 \sqrt [3]{-1} 3^{2/3}-18 (-1)^{2/3} \sqrt [3]{6}\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 )}{4374\ 2^{2/3} \left (24+18 \sqrt [3]{-2} 3^{2/3}\right )} \\ & = \frac {9 (-2)^{2/3}+\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{2916\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac {9\ 2^{2/3}+\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{26244\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {3\ 2^{2/3} \sqrt [3]{3}-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{8748\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {\sqrt [3]{-1} \left (3 (-3)^{2/3}-2^{2/3}\right ) \tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{486\ 6^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )^{3/2}}+\frac {\left (3 (-3)^{2/3} \sqrt [6]{2}+\sqrt [3]{-1} 2^{5/6}\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 )}{8748\ 3^{5/6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac {\left (2^{2/3}-3\ 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 )}{4374\ 6^{5/6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\sqrt [6]{-\frac {1}{3}} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{5832 \sqrt [3]{2} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{5832 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{52488 \sqrt [3]{2} 3^{2/3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.25 \[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {-96+108 x-64 x^2-72 x^3+73 x^4-3 x^5}{68364 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )}-\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {108 \log (x-\text {$\#$1})-32 \log (x-\text {$\#$1}) \text {$\#$1}+108 \log (x-\text {$\#$1}) \text {$\#$1}^2-146 \log (x-\text {$\#$1}) \text {$\#$1}^3+3 \log (x-\text {$\#$1}) \text {$\#$1}^4}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{410184} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.18
method | result | size |
default | \(\frac {-\frac {1}{22788} x^{5}+\frac {73}{68364} x^{4}-\frac {2}{1899} x^{3}-\frac {16}{17091} x^{2}+\frac {1}{633} x -\frac {8}{5697}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-3 \textit {\_R}^{4}+146 \textit {\_R}^{3}-108 \textit {\_R}^{2}+32 \textit {\_R} -108\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{410184}\) | \(122\) |
risch | \(\frac {-\frac {1}{22788} x^{5}+\frac {73}{68364} x^{4}-\frac {2}{1899} x^{3}-\frac {16}{17091} x^{2}+\frac {1}{633} x -\frac {8}{5697}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-3 \textit {\_R}^{4}+146 \textit {\_R}^{3}-108 \textit {\_R}^{2}+32 \textit {\_R} -108\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{410184}\) | \(122\) |
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Timed out. \[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Timed out} \]
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Time = 0.25 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.17 \[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\operatorname {RootSum} {\left (3977704731623097128039995515166457856 t^{6} - 1010314319415295961050951680 t^{4} - 20168224477093957151232 t^{3} - 112582856818899648 t^{2} - 50648453064 t - 880007, \left ( t \mapsto t \log {\left (- \frac {273655567090018991570649941414395560986199688040644608 t^{5}}{49797855396139900267573395695} + \frac {11837008470196046085308646230764354292805044570112 t^{4}}{49797855396139900267573395695} - \frac {10570581900446717266374077482873315047787008 t^{3}}{49797855396139900267573395695} - \frac {1552547411569469872387563218792789323392 t^{2}}{49797855396139900267573395695} - \frac {12542923791159140826909003250295928 t}{49797855396139900267573395695} + x - \frac {23066533870320322410834348296}{49797855396139900267573395695} \right )} \right )\right )} + \frac {- 3 x^{5} + 73 x^{4} - 72 x^{3} - 64 x^{2} + 108 x - 96}{68364 x^{6} + 1230552 x^{4} + 22149936 x^{3} + 7383312 x^{2} + 14766624} \]
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\[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{6}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \]
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\[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{6}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \]
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Time = 9.05 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.57 \[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Too large to display} \]
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