Optimal. Leaf size=112 \[ \frac {1}{16} \text {RootSum}\left [8 \text {$\#$1}^6-10 \text {$\#$1}^3+5\& ,\frac {-6 \text {$\#$1}^3 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )+6 \text {$\#$1}^3 \log (x)+5 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )-5 \log (x)}{8 \text {$\#$1}^4-5 \text {$\#$1}}\& \right ]+\frac {\left (x^3+1\right )^{2/3} \left (-23 x^3-8\right )}{80 x^5} \]
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Rubi [C] time = 0.56, antiderivative size = 203, normalized size of antiderivative = 1.81, number of steps used = 9, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {6728, 264, 277, 239, 429} \begin {gather*} -\frac {3 \left (-\sqrt {15}+15 i\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {3 x^3}{3-i \sqrt {15}}\right )}{80 \left (\sqrt {15}+3 i\right )}-\frac {3 \left (\sqrt {15}+15 i\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {3 x^3}{3+i \sqrt {15}}\right )}{80 \left (-\sqrt {15}+3 i\right )}-\frac {3}{16} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {1}{8} \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )-\frac {\left (x^3+1\right )^{5/3}}{10 x^5}-\frac {3 \left (x^3+1\right )^{2/3}}{16 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 (4+6 x^3+3 x^6\right )}{x^6 \left (8+6 x^3+3 x^6\right )} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{2 x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{8 x^3}-\frac {3 \left (1+x^3\right )^{2/3} \left (2+3 x^3\right )}{8 \left (8+6 x^3+3 x^6\right )}\right ) \, dx\\ &=\frac {3}{8} \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx-\frac {3}{8} \int \frac {\left (1+x^3\right )^{2/3} \left (2+3 x^3\right )}{8+6 x^3+3 x^6} \, dx+\frac {1}{2} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{16 x^2}-\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {3}{8} \int \frac {1}{\sqrt [3]{1+x^3}} \, dx-\frac {3}{8} \int \left (\frac {\left (3+i \sqrt {\frac {3}{5}}\right ) \left (1+x^3\right )^{2/3}}{6-2 i \sqrt {15}+6 x^3}+\frac {\left (3-i \sqrt {\frac {3}{5}}\right ) \left (1+x^3\right )^{2/3}}{6+2 i \sqrt {15}+6 x^3}\right ) \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{16 x^2}-\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{8} \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {3}{16} \log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {1}{40} \left (3 \left (15-i \sqrt {15}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{6+2 i \sqrt {15}+6 x^3} \, dx-\frac {1}{40} \left (3 \left (15+i \sqrt {15}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{6-2 i \sqrt {15}+6 x^3} \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{16 x^2}-\frac {\left (1+x^3\right )^{5/3}}{10 x^5}-\frac {3 \left (15 i-\sqrt {15}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {3 x^3}{3-i \sqrt {15}}\right )}{80 \left (3 i+\sqrt {15}\right )}-\frac {3 \left (15 i+\sqrt {15}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {3 x^3}{3+i \sqrt {15}}\right )}{80 \left (3 i-\sqrt {15}\right )}+\frac {1}{8} \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {3}{16} \log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 1.17, size = 446, normalized size = 3.98 \begin {gather*} \left (x^3+1\right )^{2/3} \left (-\frac {1}{10 x^5}-\frac {23}{80 x^2}\right )+\frac {-\frac {6 \sqrt [3]{5-i \sqrt {15}} \left (\sqrt {15}-i\right ) \log \left (-\frac {\sqrt [3]{5} x}{\sqrt [3]{x^3+1}}+\sqrt [3]{5-i \sqrt {15}}\right )}{\sqrt {15}-3 i}-\frac {6 \sqrt [3]{5+i \sqrt {15}} \left (\sqrt {15}+i\right ) \log \left (-\frac {\sqrt [3]{5} x}{\sqrt [3]{x^3+1}}+\sqrt [3]{5+i \sqrt {15}}\right )}{\sqrt {15}+3 i}+\frac {\left (15-i \sqrt {15}\right ) \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1-i \sqrt {\frac {3}{5}}} \sqrt [3]{x^3+1}}}{\sqrt {3}}\right )+\log \left (\frac {\sqrt [3]{25-5 i \sqrt {15}} x}{\sqrt [3]{x^3+1}}+\frac {5^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (5-i \sqrt {15}\right )^{2/3}\right )\right )}{\left (5-i \sqrt {15}\right )^{2/3}}+\frac {\left (15+i \sqrt {15}\right ) \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+i \sqrt {\frac {3}{5}}} \sqrt [3]{x^3+1}}}{\sqrt {3}}\right )+\log \left (\frac {\sqrt [3]{25+5 i \sqrt {15}} x}{\sqrt [3]{x^3+1}}+\frac {5^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (5+i \sqrt {15}\right )^{2/3}\right )\right )}{\left (5+i \sqrt {15}\right )^{2/3}}}{96 \sqrt [3]{5}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 112, normalized size = 1.00 \begin {gather*} \frac {\left (-8-23 x^3\right ) \left (1+x^3\right )^{2/3}}{80 x^5}+\frac {1}{16} \text {RootSum}\left [5-10 \text {$\#$1}^3+8 \text {$\#$1}^6\&,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+8 \text {$\#$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^{6} + 6 \, x^{3} + 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (3 \, x^{6} + 6 \, x^{3} + 8\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 200.88, size = 8187, normalized size = 73.10
method | result | size |
risch | \(\text {Expression too large to display}\) | \(8187\) |
trager | \(\text {Expression too large to display}\) | \(10431\) |
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^{6} + 6 \, x^{3} + 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (3 \, x^{6} + 6 \, x^{3} + 8\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 (3\,x^6+6\,x^3+4\right )}{x^6\,\left (3\,x^6+6\,x^3+8\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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