Optimal. Leaf size=16 \[ \frac {3 \left (x^4+x\right )^{7/3}}{7 x^7} \]
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Rubi [B] time = 0.19, antiderivative size = 47, normalized size of antiderivative = 2.94, number of steps used = 12, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2052, 2004, 2032, 364, 2020, 2025} \begin {gather*} \frac {3}{7} \sqrt [3]{x^4+x} x+\frac {3 \sqrt [3]{x^4+x}}{7 x^5}+\frac {6 \sqrt [3]{x^4+x}}{7 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 364
Rule 2004
Rule 2020
Rule 2025
Rule 2032
Rule 2052
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{x+x^4} \left (-2-x^3+x^6\right )}{x^6} \, dx &=\int \left (\sqrt [3]{x+x^4}-\frac {2 \sqrt [3]{x+x^4}}{x^6}-\frac {\sqrt [3]{x+x^4}}{x^3}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt [3]{x+x^4}}{x^6} \, dx\right )+\int \sqrt [3]{x+x^4} \, dx-\int \frac {\sqrt [3]{x+x^4}}{x^3} \, dx\\ &=\frac {3 \sqrt [3]{x+x^4}}{7 x^5}+\frac {3 \sqrt [3]{x+x^4}}{5 x^2}+\frac {3}{7} x \sqrt [3]{x+x^4}-\frac {3}{7} \int \frac {1}{x^2 \left (x+x^4\right )^{2/3}} \, dx+\frac {3}{7} \int \frac {x}{\left (x+x^4\right )^{2/3}} \, dx-\frac {3}{5} \int \frac {x}{\left (x+x^4\right )^{2/3}} \, dx\\ &=\frac {3 \sqrt [3]{x+x^4}}{7 x^5}+\frac {6 \sqrt [3]{x+x^4}}{7 x^2}+\frac {3}{7} x \sqrt [3]{x+x^4}+\frac {6}{35} \int \frac {x}{\left (x+x^4\right )^{2/3}} \, dx+\frac {\left (3 x^{2/3} \left (1+x^3\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^3\right )^{2/3}} \, dx}{7 \left (x+x^4\right )^{2/3}}-\frac {\left (3 x^{2/3} \left (1+x^3\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^3\right )^{2/3}} \, dx}{5 \left (x+x^4\right )^{2/3}}\\ &=\frac {3 \sqrt [3]{x+x^4}}{7 x^5}+\frac {6 \sqrt [3]{x+x^4}}{7 x^2}+\frac {3}{7} x \sqrt [3]{x+x^4}-\frac {9 x^2 \left (1+x^3\right )^{2/3} \, _2F_1\left (\frac {4}{9},\frac {2}{3};\frac {13}{9};-x^3\right )}{70 \left (x+x^4\right )^{2/3}}+\frac {\left (6 x^{2/3} \left (1+x^3\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^3\right )^{2/3}} \, dx}{35 \left (x+x^4\right )^{2/3}}\\ &=\frac {3 \sqrt [3]{x+x^4}}{7 x^5}+\frac {6 \sqrt [3]{x+x^4}}{7 x^2}+\frac {3}{7} x \sqrt [3]{x+x^4}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 83, normalized size = 5.19 \begin {gather*} \frac {3 \sqrt [3]{x^4+x} \left (28 x^3 \, _2F_1\left (-\frac {5}{9},-\frac {1}{3};\frac {4}{9};-x^3\right )+20 \, _2F_1\left (-\frac {14}{9},-\frac {1}{3};-\frac {5}{9};-x^3\right )+35 x^6 \, _2F_1\left (-\frac {1}{3},\frac {4}{9};\frac {13}{9};-x^3\right )\right )}{140 x^5 \sqrt [3]{x^3+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 16, normalized size = 1.00 \begin {gather*} \frac {3 \left (x+x^4\right )^{7/3}}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 22, normalized size = 1.38 \begin {gather*} \frac {3 \, {\left (x^{6} + 2 \, x^{3} + 1\right )} {\left (x^{4} + x\right )}^{\frac {1}{3}}}{7 \, x^{5}} \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} - 2\right )} {\left (x^{4} + x\right )}^{\frac {1}{3}}}{x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 23, normalized size = 1.44
method | result | size |
trager | \(\frac {3 \left (x^{6}+2 x^{3}+1\right ) \left (x^{4}+x \right )^{\frac {1}{3}}}{7 x^{5}}\) | \(23\) |
gosper | \(\frac {3 \left (x^{3}+1\right ) \left (x^{4}+x \right )^{\frac {1}{3}} \left (1+x \right ) \left (x^{2}-x +1\right )}{7 x^{5}}\) | \(29\) |
risch | \(\frac {3 \left (x \left (x^{3}+1\right )\right )^{\frac {1}{3}} \left (x^{9}+3 x^{6}+3 x^{3}+1\right )}{7 x^{5} \left (x^{3}+1\right )}\) | \(37\) |
meijerg | \(\frac {3 \hypergeom \left (\left [-\frac {1}{3}, \frac {4}{9}\right ], \left [\frac {13}{9}\right ], -x^{3}\right ) x^{\frac {4}{3}}}{4}+\frac {3 \hypergeom \left (\left [-\frac {5}{9}, -\frac {1}{3}\right ], \left [\frac {4}{9}\right ], -x^{3}\right )}{5 x^{\frac {5}{3}}}+\frac {3 \hypergeom \left (\left [-\frac {14}{9}, -\frac {1}{3}\right ], \left [-\frac {5}{9}\right ], -x^{3}\right )}{7 x^{\frac {14}{3}}}\) | \(50\) |
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^{6} - x^{3} - 2\right )} {\left (x^{4} + x\right )}^{\frac {1}{3}}}{x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 19, normalized size = 1.19 \begin {gather*} \frac {3\,{\left (x^3+1\right )}^2\,{\left (x^4+x\right )}^{1/3}}{7\,x^5} \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]{x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 1\right ) \left (x^{3} - 2\right ) \left (x^{2} - x + 1\right )}{x^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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