Optimal. Leaf size=361 \[ -\frac {(-1)^{2/3} \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{36 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {(-1)^{2/3} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{108 \sqrt [3]{2} 3^{2/3}}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{108 \sqrt [3]{2} 3^{2/3}}-\frac {\tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{6 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\sqrt [3]{-1} \tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{9\ 2^{2/3} 3^{5/6} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{18 \sqrt [6]{2} 3^{5/6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}} \]
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Rubi [A] time = 0.51, antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2097, 634, 618, 204, 628, 206} \[ -\frac {(-1)^{2/3} \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{36 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {(-1)^{2/3} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{108 \sqrt [3]{2} 3^{2/3}}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{108 \sqrt [3]{2} 3^{2/3}}-\frac {\tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{6 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\sqrt [3]{-1} \tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{9\ 2^{2/3} 3^{5/6} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{18 \sqrt [6]{2} 3^{5/6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 618
Rule 628
Rule 634
Rule 2097
Rubi steps
\begin {align*} \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx &=1259712 \int \left (\frac {(-1)^{2/3} x}{22674816 \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} x}{22674816 \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 {x}{68024448 \sqrt [3]{2} 3^{2/3} \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{54 \sqrt [3]{2} 3^{2/3}}+\frac {(-1)^{2/3} \int \frac {x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{54 \sqrt [3]{2} 3^{2/3}}+\frac {(-1)^{2/3} \int \frac {x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{18 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}\\ &=\frac {\sqrt [3]{-\frac {1}{3}} \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{18\ 2^{2/3}}+\frac {\int \frac {3\ 2^{2/3} \sqrt [3]{3}+2 x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{108 \sqrt [3]{2} 3^{2/3}}+\frac {(-1)^{2/3} \int \frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{108 \sqrt [3]{2} 3^{2/3}}-\frac {\int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{18\ 2^{2/3} \sqrt [3]{3}}-\frac {(-1)^{2/3} \int \frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{36 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\int \frac {1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{6\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}\\ &=-\frac {(-1)^{2/3} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{36 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {(-1)^{2/3} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{108 \sqrt [3]{2} 3^{2/3}}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{108 \sqrt [3]{2} 3^{2/3}}-\frac {\sqrt [3]{-\frac {1}{3}} \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 )}{9\ 2^{2/3}}+\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 )}{9\ 2^{2/3} \sqrt [3]{3}}+\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 )}{3\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}\\ &=-\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 )}{6 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\sqrt [3]{-1} \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 )}{18 \sqrt [6]{2} 3^{5/6} \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\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 )}{18 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}-\frac {(-1)^{2/3} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{36 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {(-1)^{2/3} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{108 \sqrt [3]{2} 3^{2/3}}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{108 \sqrt [3]{2} 3^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 61, normalized size = 0.17 \[ \frac {1}{6} \text {RootSum}\left [\text {$\#$1}^6+18 \text {$\#$1}^4+324 \text {$\#$1}^3+108 \text {$\#$1}^2+216\& ,\frac {\text {$\#$1}^2 \log (x-\text {$\#$1})}{\text {$\#$1}^4+12 \text {$\#$1}^2+162 \text {$\#$1}+36}\& \right ] \]
Antiderivative was successfully verified.
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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 \[ \int \frac {x^{3}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 56, normalized size = 0.16 \[ \frac {\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3} \ln \left (-\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )+x \right )}{6 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{5}+72 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}+972 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}+216 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )} \]
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 \[ \int \frac {x^{3}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 276, normalized size = 0.76 \[ \sum _{k=1}^6\ln \left (-\frac {23328\,\left (297538935552\,x-7992872640\,x\,\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )+52488\,x\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^3+2904\,x\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^4+x\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^5-153055008\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^2-2764368\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^3-1620\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^4-3673320192\,\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )+7240114098432\right )}{{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^5}\right )\,\mathrm {root}\left (z^6-\frac {z^4}{45576}-\frac {235\,z^3}{598048272}-\frac {z^2}{2392193088}-\frac {1}{3390158631679488},z,k\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 61, normalized size = 0.17 \[ \operatorname {RootSum} {\left (3390158631679488 t^{6} - 74384733888 t^{4} - 1332145440 t^{3} - 1417176 t^{2} - 1, \left (t \mapsto t \log {\left (- \frac {8482372214243328 t^{5}}{415817} + \frac {2216055910930560 t^{4}}{415817} - \frac {2062546612992 t^{3}}{415817} - \frac {57027208896 t^{2}}{415817} - \frac {416583756 t}{415817} + x - \frac {89938}{415817} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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