Optimal. Leaf size=248 \[ \frac {(-1)^{2/3} \tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{27\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{81 \sqrt [3]{2} \sqrt [6]{3} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{81\ 2^{5/6} \sqrt [6]{3} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}} \]
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Rubi [A] time = 0.32, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2097, 618, 204, 206} \[ \frac {(-1)^{2/3} \tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{27\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{81 \sqrt [3]{2} \sqrt [6]{3} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{81\ 2^{5/6} \sqrt [6]{3} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 618
Rule 2097
Rubi steps
\begin {align*} \int \frac {x^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx &=1259712 \int \left (\frac {(-1)^{2/3}}{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}}{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 {1}{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 {1}{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 {1}{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 {1}{-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 {\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 )}{27 \sqrt [3]{2} 3^{2/3}}-\frac {(-1)^{2/3} \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 )}{27 \sqrt [3]{2} 3^{2/3}}-\frac {(-1)^{2/3} \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 )}{9 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}\\ &=\frac {(-1)^{2/3} \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 )}{27\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {(-1)^{2/3} \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 )}{81\ 2^{5/6} \sqrt [6]{3} \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 )}{81\ 2^{5/6} \sqrt [6]{3} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 59, normalized size = 0.24 \[ \frac {1}{6} \text {RootSum}\left [\text {$\#$1}^6+18 \text {$\#$1}^4+324 \text {$\#$1}^3+108 \text {$\#$1}^2+216\& ,\frac {\text {$\#$1} \log (x-\text {$\#$1})}{\text {$\#$1}^4+12 \text {$\#$1}^2+162 \text {$\#$1}+36}\& \right ] \]
Antiderivative was successfully verified.
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fricas [B] time = 2.00, size = 1277, normalized size = 5.15 \[ \text {result too large to display} \]
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 \[ \int \frac {x^{2}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 56, normalized size = 0.23 \[ \frac {\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2} \ln \left (-\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )+x \right )}{6 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{5}+72 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}+972 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}+216 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )} \]
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 \[ \int \frac {x^{2}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.68, size = 247, normalized size = 1.00 \[ \sum _{k=1}^6\ln \left (-\frac {216\,\left (32134205039616\,x-1836660096\,{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^2-1889568\,{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^3+972\,{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^4+{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^5+132239526912\,x\,\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )+204073344\,x\,{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^2+139968\,x\,{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^3+36\,x\,{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^4+863230245120\,\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )+781932322630656\right )}{{\mathrm {root}\left (z^6-2834352\,z^4+2677850419968\,z^2-732274264442769408,z,k\right )}^5}\right )\,\mathrm {root}\left (z^6-\frac {z^4}{273456}+\frac {z^2}{258356853504}-\frac {1}{732274264442769408},z,k\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 48, normalized size = 0.19 \[ \operatorname {RootSum} {\left (732274264442769408 t^{6} - 2677850419968 t^{4} + 2834352 t^{2} - 1, \left (t \mapsto t \log {\left (10170475895038464 t^{5} - 5231726283456 t^{4} - 31809932496 t^{3} + 19131876 t^{2} + 19683 t + x - \frac {27}{2} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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