Optimal. Leaf size=873 \[ \frac {\sqrt [3]{-6} \left (2 \sqrt [3]{-3}+9 \sqrt [3]{2}\right )-3 x}{157464 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}-\frac {\left (9 i-\sqrt [3]{3} \left (2 i 2^{2/3}+9 \sqrt [6]{3}+2\ 2^{2/3} \sqrt {3}\right )\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 )}{209952 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac {\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 )}{26244 \sqrt {3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\left (9 i+\sqrt [3]{3} \left (4 i 2^{2/3}-9 \sqrt [6]{3}\right )\right ) \tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{209952 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {2 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}-\frac {\tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{26244 \sqrt {3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\left (2\ 2^{2/3}-3\ 3^{2/3}\right ) \tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{944784 \sqrt [6]{3} \sqrt {2 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{52488 \sqrt {6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {i \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{23328\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\left (i+\sqrt {3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{46656\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{629856\ 2^{2/3} \sqrt [3]{3}}-\frac {3 x+\sqrt [3]{-6} \left (9 \sqrt [3]{-2}+2 \sqrt [3]{3}\right )}{157464 \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 {-\sqrt [3]{3} x-3\ 6^{2/3}+2 \sqrt [3]{2}}{104976 \left (9 \sqrt [3]{2}-4 \sqrt [3]{3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )} \]
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Rubi [A] time = 1.92, antiderivative size = 873, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2097, 638, 618, 204, 634, 628, 206} \[ \frac {\sqrt [3]{-6} \left (2 \sqrt [3]{-3}+9 \sqrt [3]{2}\right )-3 x}{157464 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}-\frac {\left (9 i-\sqrt [3]{3} \left (2 i 2^{2/3}+9 \sqrt [6]{3}+2\ 2^{2/3} \sqrt {3}\right )\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 )}{209952 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac {\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 )}{26244 \sqrt {3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\left (9 i+\sqrt [3]{3} \left (4 i 2^{2/3}-9 \sqrt [6]{3}\right )\right ) \tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{209952 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {2 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}-\frac {\tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{26244 \sqrt {3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\left (2\ 2^{2/3}-3\ 3^{2/3}\right ) \tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{944784 \sqrt [6]{3} \sqrt {2 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{52488 \sqrt {6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {i \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{23328\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\left (i+\sqrt {3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{46656\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{629856\ 2^{2/3} \sqrt [3]{3}}-\frac {3 x+\sqrt [3]{-6} \left (9 \sqrt [3]{-2}+2 \sqrt [3]{3}\right )}{157464 \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 {-\sqrt [3]{3} x-3\ 6^{2/3}+2 \sqrt [3]{2}}{104976 \left (9 \sqrt [3]{2}-4 \sqrt [3]{3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )} \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 618
Rule 628
Rule 634
Rule 638
Rule 2097
Rubi steps
\begin {align*} \int \frac {x^3}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx &=1586874322944 \int \left (-\frac {9 (-2)^{2/3}-\sqrt [3]{-1} 3^{2/3} x}{27763953154228224\ 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 {27+3\ 2^{2/3} \sqrt [3]{3}-9 i \sqrt {3}+i 2^{2/3} 3^{5/6}+3 i \sqrt [3]{2} \sqrt [6]{3} x}{333167437850738688 \left (1+\sqrt [3]{-1}\right )^5 \left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )}-\frac {9\ 2^{2/3}-\sqrt [3]{-1} 3^{2/3} x}{27763953154228224\ 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 {-2 \left (27-i \left (9 \sqrt {3}+2\ 2^{2/3} 3^{5/6}\right )\right )+3 \sqrt [3]{2} \sqrt [6]{3} \left (i+\sqrt {3}\right ) x}{666334875701477376 \left (1+\sqrt [3]{-1}\right )^5 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac {3\ 2^{2/3} \sqrt [3]{3}+x}{9254651051409408\ 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 {18-2\ 2^{2/3} \sqrt [3]{3}+\sqrt [3]{2} 3^{2/3} x}{2998506940656648192 \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {18-2\ 2^{2/3} \sqrt [3]{3}+\sqrt [3]{2} 3^{2/3} x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1889568}-\frac {\int \frac {9\ 2^{2/3}-\sqrt [3]{-1} 3^{2/3} x}{\left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{157464\ 2^{2/3}}-\frac {\int \frac {3\ 2^{2/3} \sqrt [3]{3}+x}{\left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{52488\ 2^{2/3} \sqrt [3]{3}}+\frac {\int \frac {-2 \left (27-i \left (9 \sqrt {3}+2\ 2^{2/3} 3^{5/6}\right )\right )+3 \sqrt [3]{2} \sqrt [6]{3} \left (i+\sqrt {3}\right ) x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{419904 \left (1+\sqrt [3]{-1}\right )^5}+\frac {\int \frac {27+3\ 2^{2/3} \sqrt [3]{3}-9 i \sqrt {3}+i 2^{2/3} 3^{5/6}+3 i \sqrt [3]{2} \sqrt [6]{3} x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{209952 \left (1+\sqrt [3]{-1}\right )^5}-\frac {\int \frac {9 (-2)^{2/3}-\sqrt [3]{-1} 3^{2/3} x}{\left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )^2} \, dx}{17496\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4}\\ &=\frac {\sqrt [3]{-6} \left (2 \sqrt [3]{-3}+9 \sqrt [3]{2}\right )-3 x}{157464 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}-\frac {\sqrt [3]{-6} \left (9 \sqrt [3]{-2}+2 \sqrt [3]{3}\right )+3 x}{314928 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {2 \sqrt [3]{2}-3\ 6^{2/3}-\sqrt [3]{3} x}{104976 \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 {\int \frac {3\ 2^{2/3} \sqrt [3]{3}+2 x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{629856\ 2^{2/3} \sqrt [3]{3}}-\frac {i \int \frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{23328\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\left (9-2\ 2^{2/3} \sqrt [3]{3}\right ) \int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1889568}+\frac {\left (i+\sqrt {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}{46656\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{104976 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}-\frac {\int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{104976 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )}+\frac {\int \frac {1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{52488 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )}+\frac {\left (18 (-1)^{5/6} \sqrt {3}+2 \left (27+3\ 2^{2/3} \sqrt [3]{3}-9 i \sqrt {3}+i 2^{2/3} 3^{5/6}\right )\right ) \int \frac {1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{419904 \left (1+\sqrt [3]{-1}\right )^5}-\frac {\left (18 (-1)^{2/3} \sqrt {3} \left (i+\sqrt {3}\right )+4 \left (27-i \left (9 \sqrt {3}+2\ 2^{2/3} 3^{5/6}\right )\right )\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{839808 \left (1+\sqrt [3]{-1}\right )^5}\\ &=\frac {\sqrt [3]{-6} \left (2 \sqrt [3]{-3}+9 \sqrt [3]{2}\right )-3 x}{157464 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}-\frac {\sqrt [3]{-6} \left (9 \sqrt [3]{-2}+2 \sqrt [3]{3}\right )+3 x}{314928 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {2 \sqrt [3]{2}-3\ 6^{2/3}-\sqrt [3]{3} x}{104976 \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 {i \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{23328\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\left (i+\sqrt {3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{46656\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{629856\ 2^{2/3} \sqrt [3]{3}}+\frac {\left (-9+2\ 2^{2/3} \sqrt [3]{3}\right ) \operatorname {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 )}{944784}+\frac {\operatorname {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 )}{52488 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}+\frac {\operatorname {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 )}{52488 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )}-\frac {\operatorname {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 )}{26244 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )}-\frac {\left (18 (-1)^{5/6} \sqrt {3}+2 \left (27+3\ 2^{2/3} \sqrt [3]{3}-9 i \sqrt {3}+i 2^{2/3} 3^{5/6}\right )\right ) \operatorname {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 )}{209952 \left (1+\sqrt [3]{-1}\right )^5}+\frac {\left (18 (-1)^{2/3} \sqrt {3} \left (i+\sqrt {3}\right )+4 \left (27-i \left (9 \sqrt {3}+2\ 2^{2/3} 3^{5/6}\right )\right )\right ) \operatorname {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 )}{419904 \left (1+\sqrt [3]{-1}\right )^5}\\ &=\frac {\sqrt [3]{-6} \left (2 \sqrt [3]{-3}+9 \sqrt [3]{2}\right )-3 x}{157464 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}-\frac {\sqrt [3]{-6} \left (9 \sqrt [3]{-2}+2 \sqrt [3]{3}\right )+3 x}{314928 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {2 \sqrt [3]{2}-3\ 6^{2/3}-\sqrt [3]{3} x}{104976 \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 {\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 )}{26244 \sqrt {3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\left (27+6\ 2^{2/3} \sqrt [3]{3}-9 i \sqrt {3}+2 i 2^{2/3} 3^{5/6}\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 )}{209952 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}-\frac {\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 )}{52488 \sqrt {6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac {\left (27-9 i \sqrt {3}-4 i 2^{2/3} 3^{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 )}{209952 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}-\frac {\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 )}{52488 \sqrt {6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\left (2\ 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 )}{944784 \sqrt [6]{3} \sqrt {2 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {i \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{23328\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\left (i+\sqrt {3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{46656\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{629856\ 2^{2/3} \sqrt [3]{3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 167, normalized size = 0.19 \[ \frac {\text {RootSum}\left [\text {$\#$1}^6+18 \text {$\#$1}^4+324 \text {$\#$1}^3+108 \text {$\#$1}^2+216\& ,\frac {2 \text {$\#$1}^4 \log (x-\text {$\#$1})-27 \text {$\#$1}^3 \log (x-\text {$\#$1})+72 \text {$\#$1}^2 \log (x-\text {$\#$1})-162 \text {$\#$1} \log (x-\text {$\#$1})+1971 \log (x-\text {$\#$1})}{\text {$\#$1}^5+12 \text {$\#$1}^3+162 \text {$\#$1}^2+36 \text {$\#$1}}\& \right ]}{11074968}+\frac {4 x^5-27 x^4+96 x^3+648 x^2-3942 x+972}{3691656 \left (x^6+18 x^4+324 x^3+108 x^2+216\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 122, normalized size = 0.14 \[ \frac {\left (2 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{4}-27 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}+72 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}-162 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )+1971\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )+x \right )}{11074968 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{5}+132899616 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}+1794144816 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}+398698848 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}+\frac {\frac {1}{922914} x^{5}-\frac {1}{136728} x^{4}+\frac {4}{153819} x^{3}+\frac {1}{5697} x^{2}-\frac {73}{68364} x +\frac {1}{3798}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {4 \, x^{5} - 27 \, x^{4} + 96 \, x^{3} + 648 \, x^{2} - 3942 \, x + 972}{3691656 \, {\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}} + \frac {1}{1845828} \, \int \frac {2 \, x^{4} - 27 \, x^{3} + 72 \, x^{2} - 162 \, x + 1971}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.42, size = 387, normalized size = 0.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 112, normalized size = 0.13 \[ \operatorname {RootSum} {\left (1282755170017893101915524820582750453426552832 t^{6} - 906388465775544244426251149770752 t^{4} - 4300873166389987741684137984 t^{3} - 717000908921644962816 t^{2} + 135354162312576 t - 7197829, \left (t \mapsto t \log {\left (\frac {17257935592810449901409556597891882995604001083339368041361480613888 t^{5}}{154206009791052044490694380303237521} + \frac {2389607400620985524376358853572652207181956324560587684052992 t^{4}}{154206009791052044490694380303237521} - \frac {12286072160883283930711715948878260078996992193488388096 t^{3}}{154206009791052044490694380303237521} - \frac {59490553573959173161125496013527909754156558410752 t^{2}}{154206009791052044490694380303237521} - \frac {17520149679836691112367064197713753004827200 t}{154206009791052044490694380303237521} + x + \frac {766422988707229615055855287040887332}{154206009791052044490694380303237521} \right )} \right )\right )} + \frac {4 x^{5} - 27 x^{4} + 96 x^{3} + 648 x^{2} - 3942 x + 972}{3691656 x^{6} + 66449808 x^{4} + 1196096544 x^{3} + 398698848 x^{2} + 797397696} \]
Verification of antiderivative is not currently implemented for this CAS.
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