Optimal. Leaf size=38 \[ \frac {\sqrt [3]{x^6-1} \left (4 x^{12}+7 x^9-8 x^6-7 x^3+4\right )}{28 x^7} \]
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Rubi [A] time = 0.13, antiderivative size = 49, normalized size of antiderivative = 1.29, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1835, 1586, 1584, 449} \begin {gather*} \frac {\left (x^6-1\right )^{4/3}}{7 x}-\frac {\left (x^6-1\right )^{4/3}}{7 x^7}+\frac {\left (x^6-1\right )^{4/3}}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rule 1584
Rule 1586
Rule 1835
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right ) \left (-1+x^3+x^6\right )}{x^8} \, dx &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {1}{14} \int \frac {\sqrt [3]{-1+x^6} \left (14 x^2+2 x^5+14 x^8+14 x^{11}\right )}{x^7} \, dx\\ &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {1}{14} \int \frac {\sqrt [3]{-1+x^6} \left (14 x+2 x^4+14 x^7+14 x^{10}\right )}{x^6} \, dx\\ &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {1}{14} \int \frac {\sqrt [3]{-1+x^6} \left (14+2 x^3+14 x^6+14 x^9\right )}{x^5} \, dx\\ &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {\left (-1+x^6\right )^{4/3}}{4 x^4}+\frac {1}{112} \int \frac {\sqrt [3]{-1+x^6} \left (16 x^2+112 x^8\right )}{x^4} \, dx\\ &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {\left (-1+x^6\right )^{4/3}}{4 x^4}+\frac {1}{112} \int \frac {\sqrt [3]{-1+x^6} \left (16+112 x^6\right )}{x^2} \, dx\\ &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {\left (-1+x^6\right )^{4/3}}{4 x^4}+\frac {\left (-1+x^6\right )^{4/3}}{7 x}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 100, normalized size = 2.63 \begin {gather*} \frac {\sqrt [3]{x^6-1} \left (20 \, _2F_1\left (-\frac {7}{6},-\frac {1}{3};-\frac {1}{6};x^6\right )+7 x^3 \left (10 x^6 \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {4}{3};x^6\right )-5 \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {1}{3};x^6\right )+4 x^9 \, _2F_1\left (-\frac {1}{3},\frac {5}{6};\frac {11}{6};x^6\right )\right )\right )}{140 x^7 \sqrt [3]{1-x^6}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.84, size = 28, normalized size = 0.74 \begin {gather*} \frac {\left (x^6-1\right )^{4/3} \left (4 x^6+7 x^3-4\right )}{28 x^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 34, normalized size = 0.89 \begin {gather*} \frac {{\left (4 \, x^{12} + 7 \, x^{9} - 8 \, x^{6} - 7 \, x^{3} + 4\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{28 \, x^{7}} \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 (x^{6} + x^{3} - 1\right )} {\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{8}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 45, normalized size = 1.18 \begin {gather*} \frac {\left (x^{6}-1\right )^{\frac {1}{3}} \left (4 x^{6}+7 x^{3}-4\right ) \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{28 x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 55, normalized size = 1.45 \begin {gather*} \frac {{\left (4 \, x^{12} + 7 \, x^{9} - 8 \, x^{6} - 7 \, x^{3} + 4\right )} {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{\frac {1}{3}}}{28 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 56, normalized size = 1.47 \begin {gather*} {\left (x^6-1\right )}^{1/3}\,\left (\frac {x^5}{7}+\frac {x^2}{4}\right )-\frac {2\,{\left (x^6-1\right )}^{1/3}}{7\,x}-\frac {{\left (x^6-1\right )}^{1/3}}{4\,x^4}+\frac {{\left (x^6-1\right )}^{1/3}}{7\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.08, size = 148, normalized size = 3.89 \begin {gather*} \frac {x^{5} e^{\frac {i \pi }{3}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {11}{6}\right )} + \frac {x^{2} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {4}{3}\right )} - \frac {e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{4} \Gamma \left (\frac {1}{3}\right )} + \frac {e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, - \frac {1}{3} \\ - \frac {1}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{7} \Gamma \left (- \frac {1}{6}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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