Optimal. Leaf size=103 \[ \frac {1}{4} \log \left (\sqrt [3]{x^5-x}+x^3\right )+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^5-x}}{\sqrt [3]{x^5-x}-2 x^3}\right )-\frac {1}{8} \log \left (x^6+\left (x^5-x\right )^{2/3}-\sqrt [3]{x^5-x} x^3\right ) \]
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Rubi [C] time = 0.77, antiderivative size = 117, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2056, 6728, 466, 465, 511, 510} \begin {gather*} \frac {3 \sqrt [3]{1-x^4} x^3 F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,-\frac {2 x^4}{1-\sqrt {5}}\right )}{8 \sqrt [3]{x^5-x}}+\frac {3 \sqrt [3]{1-x^4} x^3 F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,-\frac {2 x^4}{1+\sqrt {5}}\right )}{8 \sqrt [3]{x^5-x}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 465
Rule 466
Rule 510
Rule 511
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \frac {x^{5/3} \left (-2+x^4\right )}{\sqrt [3]{-1+x^4} \left (-1+x^4+x^8\right )} \, dx}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \left (\frac {\left (1-\sqrt {5}\right ) x^{5/3}}{\sqrt [3]{-1+x^4} \left (1-\sqrt {5}+2 x^4\right )}+\frac {\left (1+\sqrt {5}\right ) x^{5/3}}{\sqrt [3]{-1+x^4} \left (1+\sqrt {5}+2 x^4\right )}\right ) \, dx}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (\left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \frac {x^{5/3}}{\sqrt [3]{-1+x^4} \left (1-\sqrt {5}+2 x^4\right )} \, dx}{\sqrt [3]{-x+x^5}}+\frac {\left (\left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \frac {x^{5/3}}{\sqrt [3]{-1+x^4} \left (1+\sqrt {5}+2 x^4\right )} \, dx}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{-1+x^{12}} \left (1-\sqrt {5}+2 x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{-1+x^{12}} \left (1+\sqrt {5}+2 x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^3} \left (1-\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^3} \left (1+\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^3} \left (1-\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^3} \left (1+\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}\\ &=\frac {3 x^3 \sqrt [3]{1-x^4} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,-\frac {2 x^4}{1-\sqrt {5}}\right )}{8 \sqrt [3]{-x+x^5}}+\frac {3 x^3 \sqrt [3]{1-x^4} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,-\frac {2 x^4}{1+\sqrt {5}}\right )}{8 \sqrt [3]{-x+x^5}}\\ \end {align*}
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Mathematica [F] time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.52, size = 103, normalized size = 1.00 \begin {gather*} \frac {1}{4} \log \left (\sqrt [3]{x^5-x}+x^3\right )+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^5-x}}{\sqrt [3]{x^5-x}-2 x^3}\right )-\frac {1}{8} \log \left (x^6+\left (x^5-x\right )^{2/3}-\sqrt [3]{x^5-x} x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 3.73, size = 112, normalized size = 1.09 \begin {gather*} -\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{8} + 2 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {1}{3}} x^{5} + 4 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {2}{3}} x^{2}}{x^{8} - 8 \, x^{4} + 8}\right ) + \frac {1}{8} \, \log \left (\frac {x^{8} + 3 \, {\left (x^{5} - x\right )}^{\frac {1}{3}} x^{5} + x^{4} + 3 \, {\left (x^{5} - x\right )}^{\frac {2}{3}} x^{2} - 1}{x^{8} + x^{4} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [C] time = 44.10, size = 613, normalized size = 5.95
result too large to display
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 (x^{4} - 2\right )} x^{2}}{{\left (x^{8} + x^{4} - 1\right )} {\left (x^{5} - x\right )}^{\frac {1}{3}}}\,{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 {x^2\,\left (x^4-2\right )}{{\left (x^5-x\right )}^{1/3}\,\left (x^8+x^4-1\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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