Optimal. Leaf size=100 \[ \frac {3 \sqrt [3]{2 x^7+1}}{x}-\log \left (\sqrt [3]{2 x^7+1}+x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2 x^7+1}-x}\right )+\frac {1}{2} \log \left (-\sqrt [3]{2 x^7+1} x+\left (2 x^7+1\right )^{2/3}+x^2\right ) \]
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Rubi [F] time = 0.87, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx &=\int \left (-\frac {3 \sqrt [3]{1+2 x^7}}{x^2}+\frac {x \left (3+14 x^4\right ) \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7}\right ) \, dx\\ &=-\left (3 \int \frac {\sqrt [3]{1+2 x^7}}{x^2} \, dx\right )+\int \frac {x \left (3+14 x^4\right ) \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {1}{3},-\frac {1}{7};\frac {6}{7};-2 x^7\right )}{x}+\int \left (\frac {3 x \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7}+\frac {14 x^5 \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7}\right ) \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {1}{3},-\frac {1}{7};\frac {6}{7};-2 x^7\right )}{x}+3 \int \frac {x \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7} \, dx+14 \int \frac {x^5 \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7} \, dx\\ \end {align*}
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Mathematica [F] time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 17.79, size = 100, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{1+2 x^7}}{x}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+2 x^7}}\right )-\log \left (x+\sqrt [3]{1+2 x^7}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+2 x^7}+\left (1+2 x^7\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 12.40, size = 141, normalized size = 1.41 \begin {gather*} \frac {2 \, \sqrt {3} x \arctan \left (\frac {8377128467638 \, \sqrt {3} {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}} x^{2} + 15171948325814 \, \sqrt {3} {\left (2 \, x^{7} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (2102123379894 \, x^{7} + 4448471619035 \, x^{3} + 1051061689947\right )}}{60468559237154 \, x^{7} - 5089335571601 \, x^{3} + 30234279618577}\right ) - x \log \left (\frac {2 \, x^{7} + x^{3} + 3 \, {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (2 \, x^{7} + 1\right )}^{\frac {2}{3}} x + 1}{2 \, x^{7} + x^{3} + 1}\right ) + 6 \, {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}}}{2 \, x} \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 (8 \, x^{7} - 3\right )} {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (2 \, x^{7} + x^{3} + 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 35.09, size = 632, normalized size = 6.32
method | result | size |
trager | \(\frac {3 \left (2 x^{7}+1\right )^{\frac {1}{3}}}{x}+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {-88928146808112798 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{7}-55447617558383298 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{7}+2558542431350156 x^{7}+133392220212169197 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+66431161712710197 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (2 x^{7}+1\right )^{\frac {2}{3}} x -66431161712710197 \left (2 x^{7}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-20578078271889984 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-20401446009640383 \left (2 x^{7}+1\right )^{\frac {2}{3}} x +20401446009640383 \left (2 x^{7}+1\right )^{\frac {1}{3}} x^{2}+639635607837539 x^{3}-44464073404056399 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-27723808779191649 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+1279271215675078}{2 x^{7}+x^{3}+1}\right )-3 \ln \left (-\frac {88928146808112798 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{7}-114733048763791830 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{7}+25804901955679032 x^{7}-133392220212169197 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+66431161712710197 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (2 x^{7}+1\right )^{\frac {2}{3}} x -66431161712710197 \left (2 x^{7}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+68350068536222814 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-1742274561263016 \left (2 x^{7}+1\right )^{\frac {2}{3}} x +1742274561263016 \left (2 x^{7}+1\right )^{\frac {1}{3}} x^{2}-8601633985226344 x^{3}+44464073404056399 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-57366524381895915 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+12902450977839516}{2 x^{7}+x^{3}+1}\right ) \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+\ln \left (-\frac {88928146808112798 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{7}-114733048763791830 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{7}+25804901955679032 x^{7}-133392220212169197 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+66431161712710197 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (2 x^{7}+1\right )^{\frac {2}{3}} x -66431161712710197 \left (2 x^{7}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+68350068536222814 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-1742274561263016 \left (2 x^{7}+1\right )^{\frac {2}{3}} x +1742274561263016 \left (2 x^{7}+1\right )^{\frac {1}{3}} x^{2}-8601633985226344 x^{3}+44464073404056399 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-57366524381895915 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+12902450977839516}{2 x^{7}+x^{3}+1}\right )\) | \(632\) |
risch | \(\frac {3 \left (2 x^{7}+1\right )^{\frac {1}{3}}}{x}+\frac {\left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {8 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{14}-4 x^{14}-4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{10}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{10}-6 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x^{8}+6 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x^{8}+8 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-4 x^{7}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {2}{3}} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {2}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x +3 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x +2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{\left (2 x^{7}+1\right ) \left (2 x^{7}+x^{3}+1\right )}\right )-\ln \left (\frac {8 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{14}-4 x^{14}+4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{10}-6 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{10}-6 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x^{8}+2 x^{10}+8 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-4 x^{7}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {2}{3}} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x +x^{3}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{\left (2 x^{7}+1\right ) \left (2 x^{7}+x^{3}+1\right )}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (\frac {8 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{14}-4 x^{14}+4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{10}-6 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{10}-6 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x^{8}+2 x^{10}+8 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-4 x^{7}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {2}{3}} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x +x^{3}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{\left (2 x^{7}+1\right ) \left (2 x^{7}+x^{3}+1\right )}\right )\right ) \left (\left (2 x^{7}+1\right )^{2}\right )^{\frac {1}{3}}}{\left (2 x^{7}+1\right )^{\frac {2}{3}}}\) | \(778\) |
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 (8 \, x^{7} - 3\right )} {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (2 \, x^{7} + x^{3} + 1\right )} x^{2}}\,{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 (2\,x^7+1\right )}^{1/3}\,\left (8\,x^7-3\right )}{x^2\,\left (2\,x^7+x^3+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{2 x^{7} + 1} \left (8 x^{7} - 3\right )}{x^{2} \left (2 x^{7} + x^{3} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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