Optimal. Leaf size=75 \[ -\log \left (\sqrt [3]{x^6+x}-x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6+x}+x}\right )+\frac {1}{2} \log \left (\sqrt [3]{x^6+x} x+\left (x^6+x\right )^{2/3}+x^2\right ) \]
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Rubi [F] time = 1.24, 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 {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \int \frac {2-3 x^5}{\sqrt [3]{x} \sqrt [3]{1+x^5} \left (1-x^2+x^5\right )} \, dx}{\sqrt [3]{x+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x \left (2-3 x^{15}\right )}{\sqrt [3]{1+x^{15}} \left (1-x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \operatorname {Subst}\left (\int \left (-\frac {3 x}{\sqrt [3]{1+x^{15}}}+\frac {x \left (5-3 x^6\right )}{\sqrt [3]{1+x^{15}} \left (1-x^6+x^{15}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x \left (5-3 x^6\right )}{\sqrt [3]{1+x^{15}} \left (1-x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^6}}-\frac {\left (9 \sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1+x^{15}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^6}}\\ &=-\frac {9 x \sqrt [3]{1+x^5} \, _2F_1\left (\frac {2}{15},\frac {1}{3};\frac {17}{15};-x^5\right )}{2 \sqrt [3]{x+x^6}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {5 x}{\sqrt [3]{1+x^{15}} \left (1-x^6+x^{15}\right )}-\frac {3 x^7}{\sqrt [3]{1+x^{15}} \left (1-x^6+x^{15}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^6}}\\ &=-\frac {9 x \sqrt [3]{1+x^5} \, _2F_1\left (\frac {2}{15},\frac {1}{3};\frac {17}{15};-x^5\right )}{2 \sqrt [3]{x+x^6}}-\frac {\left (9 \sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{1+x^{15}} \left (1-x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^6}}+\frac {\left (15 \sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1+x^{15}} \left (1-x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^6}}\\ \end {align*}
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Mathematica [F] time = 0.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.69, size = 75, normalized size = 1.00 \begin {gather*} -\log \left (\sqrt [3]{x^6+x}-x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6+x}+x}\right )+\frac {1}{2} \log \left (\sqrt [3]{x^6+x} x+\left (x^6+x\right )^{2/3}+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.35, size = 100, normalized size = 1.33 \begin {gather*} \sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{6} + x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{5} + 1\right )} - 2 \, \sqrt {3} {\left (x^{6} + x\right )}^{\frac {2}{3}}}{x^{5} + 8 \, x^{2} + 1}\right ) - \frac {1}{2} \, \log \left (\frac {x^{5} - x^{2} + 3 \, {\left (x^{6} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{6} + x\right )}^{\frac {2}{3}} + 1}{x^{5} - x^{2} + 1}\right ) \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 {3 \, x^{5} - 2}{{\left (x^{6} + x\right )}^{\frac {1}{3}} {\left (x^{5} - x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.94, size = 347, normalized size = 4.63 \begin {gather*} -\ln \left (\frac {-34406523 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}-43368040 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}+50890012 x^{5}+68813046 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+70252591 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {2}{3}}+92818507 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {1}{3}} x -154109581 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-163071098 \left (x^{6}+x \right )^{\frac {2}{3}}+70252591 x \left (x^{6}+x \right )^{\frac {1}{3}}-34406523 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+76335018 x^{2}-43368040 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+50890012}{x^{5}-x^{2}+1}\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-25445006 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}+110741541 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-103219569 x^{5}+50890012 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+70252591 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {2}{3}}-163071098 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {1}{3}} x -43368040 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+92818507 \left (x^{6}+x \right )^{\frac {2}{3}}+70252591 x \left (x^{6}+x \right )^{\frac {1}{3}}-25445006 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-34406523 x^{2}+110741541 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-103219569}{x^{5}-x^{2}+1}\right ) \end {gather*}
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 {3 \, x^{5} - 2}{{\left (x^{6} + x\right )}^{\frac {1}{3}} {\left (x^{5} - x^{2} + 1\right )}}\,{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 {3\,x^5-2}{{\left (x^6+x\right )}^{1/3}\,\left (x^5-x^2+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 {3 x^{5}}{x^{5} \sqrt [3]{x^{6} + x} - x^{2} \sqrt [3]{x^{6} + x} + \sqrt [3]{x^{6} + x}}\, dx - \int \left (- \frac {2}{x^{5} \sqrt [3]{x^{6} + x} - x^{2} \sqrt [3]{x^{6} + x} + \sqrt [3]{x^{6} + x}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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