Optimal. Leaf size=116 \[ \frac {1}{16} \text {RootSum}\left [\text {$\#$1}^6-4 \text {$\#$1}^3+2\& ,\frac {-11 \text {$\#$1}^3 \log \left (\sqrt [3]{x^3-2 x}-\text {$\#$1} x\right )+11 \text {$\#$1}^3 \log (x)+6 \log \left (\sqrt [3]{x^3-2 x}-\text {$\#$1} x\right )-6 \log (x)}{\text {$\#$1}^5-2 \text {$\#$1}^2}\& \right ]-\frac {3 \left (7 x^2-2\right ) \sqrt [3]{x^3-2 x}}{16 x^3} \]
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Rubi [B] time = 1.30, antiderivative size = 357, normalized size of antiderivative = 3.08, number of steps used = 11, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2056, 6728, 466, 465, 511, 510} \begin {gather*} \frac {3 \left (4-\sqrt {2}\right ) \sqrt [3]{x^3-2 x} \left (-3 \left (\left (4-3 \sqrt {2}\right ) x^2+2 \left (2-\sqrt {2}\right )\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {\left (2-\sqrt {2}\right ) x^2}{2-x^2}\right )+\left (2 \left (2-\sqrt {2}\right ) x^2-3 \left (4-3 \sqrt {2}\right ) x^4\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {\left (2-\sqrt {2}\right ) x^2}{2-x^2}\right )+2 \left (2-x^2\right ) \left (2-3 \left (1-\sqrt {2}\right ) x^2\right )\right )}{256 x^3 \left (2-x^2\right )}+\frac {3 \left (4+\sqrt {2}\right ) \sqrt [3]{x^3-2 x} \left (-3 \left (\left (4+3 \sqrt {2}\right ) x^2+2 \left (2+\sqrt {2}\right )\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {\left (2+\sqrt {2}\right ) x^2}{2-x^2}\right )+\left (2 \left (2+\sqrt {2}\right ) x^2-3 \left (4+3 \sqrt {2}\right ) x^4\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {\left (2+\sqrt {2}\right ) x^2}{2-x^2}\right )+2 \left (2-x^2\right ) \left (2-3 \left (1+\sqrt {2}\right ) x^2\right )\right )}{256 x^3 \left (2-x^2\right )} \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 {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx &=\frac {\sqrt [3]{-2 x+x^3} \int \frac {\sqrt [3]{-2+x^2} \left (4+x^2\right )}{x^{11/3} \left (-4-4 x^2+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+x^2}}\\ &=\frac {\sqrt [3]{-2 x+x^3} \int \left (\frac {\left (1+\frac {3}{\sqrt {2}}\right ) \sqrt [3]{-2+x^2}}{x^{11/3} \left (-4-4 \sqrt {2}+2 x^2\right )}+\frac {\left (1-\frac {3}{\sqrt {2}}\right ) \sqrt [3]{-2+x^2}}{x^{11/3} \left (-4+4 \sqrt {2}+2 x^2\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-2+x^2}}\\ &=\frac {\left (\left (2-3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \int \frac {\sqrt [3]{-2+x^2}}{x^{11/3} \left (-4+4 \sqrt {2}+2 x^2\right )} \, dx}{2 \sqrt [3]{x} \sqrt [3]{-2+x^2}}+\frac {\left (\left (2+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \int \frac {\sqrt [3]{-2+x^2}}{x^{11/3} \left (-4-4 \sqrt {2}+2 x^2\right )} \, dx}{2 \sqrt [3]{x} \sqrt [3]{-2+x^2}}\\ &=\frac {\left (3 \left (2-3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-2+x^6}}{x^9 \left (-4+4 \sqrt {2}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{-2+x^2}}+\frac {\left (3 \left (2+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-2+x^6}}{x^9 \left (-4-4 \sqrt {2}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{-2+x^2}}\\ &=\frac {\left (3 \left (2-3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-2+x^3}}{x^5 \left (-4+4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{x} \sqrt [3]{-2+x^2}}+\frac {\left (3 \left (2+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-2+x^3}}{x^5 \left (-4-4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{x} \sqrt [3]{-2+x^2}}\\ &=\frac {\left (3 \left (2-3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-\frac {x^3}{2}}}{x^5 \left (-4+4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2\ 2^{2/3} \sqrt [3]{x} \sqrt [3]{2-x^2}}+\frac {\left (3 \left (2+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-\frac {x^3}{2}}}{x^5 \left (-4-4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2\ 2^{2/3} \sqrt [3]{x} \sqrt [3]{2-x^2}}\\ &=\frac {3 \left (4-\sqrt {2}\right ) \sqrt [3]{-2 x+x^3} \left (2 \left (2-x^2\right ) \left (2-3 \left (1-\sqrt {2}\right ) x^2\right )+\left (2 \left (2-\sqrt {2}\right ) x^2-3 \left (4-3 \sqrt {2}\right ) x^4\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {\left (2-\sqrt {2}\right ) x^2}{2-x^2}\right )-3 x^2 \left (2 \left (2-\sqrt {2}\right )+\left (4-3 \sqrt {2}\right ) x^2\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {\left (2-\sqrt {2}\right ) x^2}{2-x^2}\right )\right )}{256 x^3 \left (2-x^2\right )}+\frac {3 \left (4+\sqrt {2}\right ) \sqrt [3]{-2 x+x^3} \left (2 \left (2-x^2\right ) \left (2-3 \left (1+\sqrt {2}\right ) x^2\right )+\left (2 \left (2+\sqrt {2}\right ) x^2-3 \left (4+3 \sqrt {2}\right ) x^4\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {\left (2+\sqrt {2}\right ) x^2}{2-x^2}\right )-3 x^2 \left (2 \left (2+\sqrt {2}\right )+\left (4+3 \sqrt {2}\right ) x^2\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {\left (2+\sqrt {2}\right ) x^2}{2-x^2}\right )\right )}{256 x^3 \left (2-x^2\right )}\\ \end {align*}
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Mathematica [B] time = 0.20, size = 318, normalized size = 2.74 \begin {gather*} \frac {3 \left (-\frac {\left (3+\sqrt {2}\right ) \left (\left (3 \left (3 \sqrt {2}-4\right ) x^2-2 \sqrt {2}+4\right ) x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {\left (-2+\sqrt {2}\right ) x^2}{x^2-2}\right )+3 \left (\left (3 \sqrt {2}-4\right ) x^2+2 \left (\sqrt {2}-2\right )\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {\left (-2+\sqrt {2}\right ) x^2}{x^2-2}\right )-2 \left (x^2-2\right ) \left (3 \left (\sqrt {2}-1\right ) x^2+2\right )\right )}{2+\sqrt {2}}-\left (1+\frac {1}{\sqrt {2}}\right ) \left (3-\sqrt {2}\right ) \left (-3 \left (\left (4+3 \sqrt {2}\right ) x^2+2 \left (2+\sqrt {2}\right )\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {\left (2+\sqrt {2}\right ) x^2}{x^2-2}\right )+\left (2 \left (2+\sqrt {2}\right ) x^2-3 \left (4+3 \sqrt {2}\right ) x^4\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\left (2+\sqrt {2}\right ) x^2}{x^2-2}\right )+2 \left (2-x^2\right ) \left (2-3 \left (1+\sqrt {2}\right ) x^2\right )\right )\right )}{128 x^2 \left (x \left (x^2-2\right )\right )^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.00, size = 116, normalized size = 1.00 \begin {gather*} -\frac {3 \left (-2+7 x^2\right ) \sqrt [3]{-2 x+x^3}}{16 x^3}+\frac {1}{16} \text {RootSum}\left [2-4 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-6 \log (x)+6 \log \left (\sqrt [3]{-2 x+x^3}-x \text {$\#$1}\right )+11 \log (x) \text {$\#$1}^3-11 \log \left (\sqrt [3]{-2 x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - 2 \, x\right )}^{\frac {1}{3}} {\left (x^{2} + 4\right )}}{{\left (x^{4} - 4 \, x^{2} - 4\right )} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 238.14, size = 9346, normalized size = 80.57
method | result | size |
trager | \(\text {Expression too large to display}\) | \(9346\) |
risch | \(\text {Expression too large to display}\) | \(16927\) |
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*} \frac {3 \, {\left (36 \, x^{7} - 36 \, x^{5} + 5 \, {\left (3 \, x^{5} + 2 \, x^{3} - 16 \, x\right )} x^{2} - 8 \, x^{3} - 128 \, x\right )} {\left (x^{2} - 2\right )}^{\frac {1}{3}}}{1120 \, {\left (x^{\frac {23}{3}} - 4 \, x^{\frac {17}{3}} - 4 \, x^{\frac {11}{3}}\right )}} + \int \frac {3 \, {\left (36 \, x^{6} - 6 \, x^{4} + {\left (18 \, x^{6} + 27 \, x^{4} + 26 \, x^{2} - 304\right )} x^{2} + 12 \, x^{2} - 288\right )} {\left (x^{2} - 2\right )}^{\frac {1}{3}}}{70 \, {\left (x^{\frac {35}{3}} - 8 \, x^{\frac {29}{3}} + 8 \, x^{\frac {23}{3}} + 32 \, x^{\frac {17}{3}} + 16 \, x^{\frac {11}{3}}\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 {\left (x^2+4\right )\,{\left (x^3-2\,x\right )}^{1/3}}{x^4\,\left (-x^4+4\,x^2+4\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 {\sqrt [3]{x \left (x^{2} - 2\right )} \left (x^{2} + 4\right )}{x^{4} \left (x^{4} - 4 x^{2} - 4\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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