Optimal. Leaf size=112 \[ \frac {1}{12} \text {RootSum}\left [4 \text {$\#$1}^6-6 \text {$\#$1}^3+3\& ,\frac {-8 \text {$\#$1}^3 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )+8 \text {$\#$1}^3 \log (x)+9 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )-9 \log (x)}{4 \text {$\#$1}^4-3 \text {$\#$1}}\& \right ]+\frac {\left (x^3+1\right )^{2/3} \left (-23 x^3-8\right )}{20 x^5} \]
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Rubi [C] time = 0.85, antiderivative size = 203, normalized size of antiderivative = 1.81, number of steps used = 9, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {6728, 264, 277, 239, 429} \begin {gather*} -\frac {\left (\sqrt {3}+9 i\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {x^3}{1-i \sqrt {3}}\right )}{12 \left (\sqrt {3}+i\right )}-\frac {\left (-\sqrt {3}+9 i\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {x^3}{1+i \sqrt {3}}\right )}{12 \left (-\sqrt {3}+i\right )}-\frac {3}{4} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )-\frac {2 \left (x^3+1\right )^{5/3}}{5 x^5}-\frac {3 \left (x^3+1\right )^{2/3}}{4 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 239
Rule 264
Rule 277
Rule 429
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right ) \left (4+3 x^3\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx &=\int \left (\frac {2 \left (1+x^3\right )^{2/3}}{x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{2 x^3}+\frac {\left (-4-3 x^3\right ) \left (1+x^3\right )^{2/3}}{2 \left (4+2 x^3+x^6\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\left (-4-3 x^3\right ) \left (1+x^3\right )^{2/3}}{4+2 x^3+x^6} \, dx+\frac {3}{2} \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx+2 \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {1}{2} \int \left (\frac {\left (-3+\frac {i}{\sqrt {3}}\right ) \left (1+x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3}+\frac {\left (-3-\frac {i}{\sqrt {3}}\right ) \left (1+x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3}\right ) \, dx+\frac {3}{2} \int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {3}{4} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{6} \left (-9+i \sqrt {3}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3} \, dx-\frac {1}{6} \left (9+i \sqrt {3}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3} \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}-\frac {\left (9 i+\sqrt {3}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {x^3}{1-i \sqrt {3}}\right )}{12 \left (i+\sqrt {3}\right )}-\frac {\left (9 i-\sqrt {3}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {x^3}{1+i \sqrt {3}}\right )}{12 \left (i-\sqrt {3}\right )}+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {3}{4} \log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 2.55, size = 464, normalized size = 4.14 \begin {gather*} \left (x^3+1\right )^{2/3} \left (-\frac {2}{5 x^5}-\frac {23}{20 x^2}\right )+\frac {1}{72} \left (-\frac {2\ 3^{2/3} \sqrt [3]{3-i \sqrt {3}} \left (3 \sqrt {3}+i\right ) \log \left (-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+\sqrt [3]{3-i \sqrt {3}}\right )}{\sqrt {3}-i}+\frac {2\ 3^{2/3} \left (-3 \sqrt {3}+i\right ) \sqrt [3]{3+i \sqrt {3}} \log \left (-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+\sqrt [3]{3+i \sqrt {3}}\right )}{\sqrt {3}+i}+\frac {\left (3 \sqrt {3}+i\right ) \left (6 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 x}{\sqrt [6]{3} \sqrt [3]{3-i \sqrt {3}} \sqrt [3]{x^3+1}}\right )+\sqrt {3} \log \left (\frac {\sqrt [3]{9-3 i \sqrt {3}} x}{\sqrt [3]{x^3+1}}+\frac {3^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (3-i \sqrt {3}\right )^{2/3}\right )\right )}{\left (1-\frac {i}{\sqrt {3}}\right )^{2/3}}+\frac {\sqrt [6]{3} \left (9-i \sqrt {3}\right ) \left (6 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 x}{\sqrt [6]{3} \sqrt [3]{3+i \sqrt {3}} \sqrt [3]{x^3+1}}\right )+\sqrt {3} \log \left (\frac {\sqrt [3]{9+3 i \sqrt {3}} x}{\sqrt [3]{x^3+1}}+\frac {3^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (3+i \sqrt {3}\right )^{2/3}\right )\right )}{\left (3+i \sqrt {3}\right )^{2/3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 112, normalized size = 1.00 \begin {gather*} \frac {\left (-8-23 x^3\right ) \left (1+x^3\right )^{2/3}}{20 x^5}+\frac {1}{12} \text {RootSum}\left [3-6 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-9 \log (x)+9 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+8 \log (x) \text {$\#$1}^3-8 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {{\left (3 \, x^{3} + 4\right )} {\left (x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 2 \, x^{3} + 4\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (x^{3}+2\right ) \left (3 x^{3}+4\right )}{x^{6} \left (x^{6}+2 x^{3}+4\right )}\, dx\]
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 \begin {gather*} \int \frac {{\left (3 \, x^{3} + 4\right )} {\left (x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 2 \, x^{3} + 4\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^3+2\right )\,\left (3\,x^3+4\right )}{x^6\,\left (x^6+2\,x^3+4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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