Optimal. Leaf size=164 \[ \frac {3 \left (2-x+2 x^3\right )^{2/3} \left (2-x+7 x^3\right )}{10 x^5}-2 \sqrt [3]{2} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+\sqrt [3]{2} \sqrt [3]{2-x+2 x^3}}\right )+2 \sqrt [3]{2} \log \left (-2 x+\sqrt [3]{2} \sqrt [3]{2-x+2 x^3}\right )-\sqrt [3]{2} \log \left (4 x^2+2 \sqrt [3]{2} x \sqrt [3]{2-x+2 x^3}+2^{2/3} \left (2-x+2 x^3\right )^{2/3}\right ) \]
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Rubi [F]
time = 2.62, 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 {(-3+x) (-2+x) \left (2-x+2 x^3\right )^{2/3}}{x^6 \left (-2+x+2 x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(-3+x) (-2+x) \left (2-x+2 x^3\right )^{2/3}}{x^6 \left (-2+x+2 x^3\right )} \, dx &=\int \left (-\frac {3 \left (2-x+2 x^3\right )^{2/3}}{x^6}+\frac {\left (2-x+2 x^3\right )^{2/3}}{x^5}-\frac {3 \left (2-x+2 x^3\right )^{2/3}}{x^3}-\frac {\left (2-x+2 x^3\right )^{2/3}}{2 x^2}-\frac {\left (2-x+2 x^3\right )^{2/3}}{4 x}+\frac {\left (25+4 x+2 x^2\right ) \left (2-x+2 x^3\right )^{2/3}}{4 \left (-2+x+2 x^3\right )}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\left (2-x+2 x^3\right )^{2/3}}{x} \, dx\right )+\frac {1}{4} \int \frac {\left (25+4 x+2 x^2\right ) \left (2-x+2 x^3\right )^{2/3}}{-2+x+2 x^3} \, dx-\frac {1}{2} \int \frac {\left (2-x+2 x^3\right )^{2/3}}{x^2} \, dx-3 \int \frac {\left (2-x+2 x^3\right )^{2/3}}{x^6} \, dx-3 \int \frac {\left (2-x+2 x^3\right )^{2/3}}{x^3} \, dx+\int \frac {\left (2-x+2 x^3\right )^{2/3}}{x^5} \, dx\\ &=\frac {1}{4} \int \left (\frac {25 \left (2-x+2 x^3\right )^{2/3}}{-2+x+2 x^3}+\frac {4 x \left (2-x+2 x^3\right )^{2/3}}{-2+x+2 x^3}+\frac {2 x^2 \left (2-x+2 x^3\right )^{2/3}}{-2+x+2 x^3}\right ) \, dx-\frac {\left (2-x+2 x^3\right )^{2/3} \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x} \, dx}{4 \left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}-\frac {\left (2-x+2 x^3\right )^{2/3} \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^2} \, dx}{2 \left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}+\frac {\left (2-x+2 x^3\right )^{2/3} \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^5} \, dx}{\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}-\frac {\left (3 \left (2-x+2 x^3\right )^{2/3}\right ) \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^6} \, dx}{\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}-\frac {\left (3 \left (2-x+2 x^3\right )^{2/3}\right ) \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^3} \, dx}{\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}\\ &=\frac {1}{2} \int \frac {x^2 \left (2-x+2 x^3\right )^{2/3}}{-2+x+2 x^3} \, dx+\frac {25}{4} \int \frac {\left (2-x+2 x^3\right )^{2/3}}{-2+x+2 x^3} \, dx-\frac {\left (2-x+2 x^3\right )^{2/3} \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x} \, dx}{4 \left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}-\frac {\left (2-x+2 x^3\right )^{2/3} \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^2} \, dx}{2 \left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}+\frac {\left (2-x+2 x^3\right )^{2/3} \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^5} \, dx}{\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}-\frac {\left (3 \left (2-x+2 x^3\right )^{2/3}\right ) \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^6} \, dx}{\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}-\frac {\left (3 \left (2-x+2 x^3\right )^{2/3}\right ) \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^3} \, dx}{\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}+\int \frac {x \left (2-x+2 x^3\right )^{2/3}}{-2+x+2 x^3} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 141, normalized size = 0.86 \begin {gather*} \frac {3 \left (2-x+2 x^3\right )^{2/3} \left (2-x+7 x^3\right )}{10 x^5}-2 \sqrt [3]{2} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+\sqrt [3]{4-2 x+4 x^3}}\right )+2 \sqrt [3]{2} \log \left (-2 x+\sqrt [3]{4-2 x+4 x^3}\right )-\sqrt [3]{2} \log \left (4 x^2+2 x \sqrt [3]{4-2 x+4 x^3}+\left (4-2 x+4 x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 10.81, size = 1100, normalized size = 6.71
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1100\) |
trager | \(\text {Expression too large to display}\) | \(1515\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 424 vs.
\(2 (134) = 268\).
time = 5.90, size = 424, normalized size = 2.59 \begin {gather*} -\frac {20 \, \sqrt {3} 2^{\frac {1}{3}} x^{5} \arctan \left (\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (20 \, x^{7} + 8 \, x^{5} - 16 \, x^{4} - x^{3} + 4 \, x^{2} - 4 \, x\right )} {\left (2 \, x^{3} - x + 2\right )}^{\frac {2}{3}} - 12 \, \sqrt {3} 2^{\frac {1}{3}} {\left (76 \, x^{8} - 32 \, x^{6} + 64 \, x^{5} + x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} {\left (2 \, x^{3} - x + 2\right )}^{\frac {1}{3}} - \sqrt {3} {\left (568 \, x^{9} - 444 \, x^{7} + 888 \, x^{6} + 66 \, x^{5} - 264 \, x^{4} + 263 \, x^{3} + 6 \, x^{2} - 12 \, x + 8\right )}}{3 \, {\left (872 \, x^{9} - 420 \, x^{7} + 840 \, x^{6} + 6 \, x^{5} - 24 \, x^{4} + 25 \, x^{3} - 6 \, x^{2} + 12 \, x - 8\right )}}\right ) - 20 \cdot 2^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 2^{\frac {2}{3}} {\left (2 \, x^{3} - x + 2\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (2 \, x^{3} - x + 2\right )}^{\frac {2}{3}} x - 2^{\frac {1}{3}} {\left (2 \, x^{3} + x - 2\right )}}{2 \, x^{3} + x - 2}\right ) + 10 \cdot 2^{\frac {1}{3}} x^{5} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (10 \, x^{4} - x^{2} + 2 \, x\right )} {\left (2 \, x^{3} - x + 2\right )}^{\frac {2}{3}} + 2^{\frac {2}{3}} {\left (76 \, x^{6} - 32 \, x^{4} + 64 \, x^{3} + x^{2} - 4 \, x + 4\right )} + 24 \, {\left (4 \, x^{5} - x^{3} + 2 \, x^{2}\right )} {\left (2 \, x^{3} - x + 2\right )}^{\frac {1}{3}}}{4 \, x^{6} + 4 \, x^{4} - 8 \, x^{3} + x^{2} - 4 \, x + 4}\right ) - 9 \, {\left (7 \, x^{3} - x + 2\right )} {\left (2 \, x^{3} - x + 2\right )}^{\frac {2}{3}}}{30 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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-2\right )\,\left (x-3\right )\,{\left (2\,x^3-x+2\right )}^{2/3}}{x^6\,\left (2\,x^3+x-2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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