Optimal. Leaf size=114 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^6-4 \text {$\#$1}^3+5\& ,\frac {-\text {$\#$1}^3 \log \left (\sqrt [3]{2 x^3-x}-\text {$\#$1} x\right )+\text {$\#$1}^3 \log (x)+5 \log \left (\sqrt [3]{2 x^3-x}-\text {$\#$1} x\right )-5 \log (x)}{\text {$\#$1}^5-2 \text {$\#$1}^2}\& \right ]-\frac {3 \sqrt [3]{2 x^3-x}}{2 x} \]
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Rubi [C] time = 0.66, antiderivative size = 153, normalized size of antiderivative = 1.34, number of steps used = 11, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2056, 6725, 466, 465, 511, 510} \begin {gather*} -\frac {\left (\frac {3}{4}-\frac {3 i}{4}\right ) \sqrt [3]{1-i x^2} \sqrt [3]{2 x^3-x} \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};\frac {(2-i) x^2}{1-i x^2}\right )}{x \sqrt [3]{1-2 x^2}}-\frac {\left (\frac {3}{4}+\frac {3 i}{4}\right ) \sqrt [3]{1+i x^2} \sqrt [3]{2 x^3-x} \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};\frac {(2+i) x^2}{i x^2+1}\right )}{x \sqrt [3]{1-2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 465
Rule 466
Rule 510
Rule 511
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx &=\frac {\sqrt [3]{-x+2 x^3} \int \frac {\left (1+x^2\right ) \sqrt [3]{-1+2 x^2}}{x^{5/3} \left (1+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}\\ &=\frac {\sqrt [3]{-x+2 x^3} \int \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [3]{-1+2 x^2}}{x^{5/3} \left (i-x^2\right )}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [3]{-1+2 x^2}}{x^{5/3} \left (i+x^2\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}\\ &=-\frac {\left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [3]{-x+2 x^3}\right ) \int \frac {\sqrt [3]{-1+2 x^2}}{x^{5/3} \left (i-x^2\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}+\frac {\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [3]{-x+2 x^3}\right ) \int \frac {\sqrt [3]{-1+2 x^2}}{x^{5/3} \left (i+x^2\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}\\ &=-\frac {\left (\left (\frac {3}{2}-\frac {3 i}{2}\right ) \sqrt [3]{-x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+2 x^6}}{x^3 \left (i-x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}+\frac {\left (\left (\frac {3}{2}+\frac {3 i}{2}\right ) \sqrt [3]{-x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+2 x^6}}{x^3 \left (i+x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}\\ &=-\frac {\left (\left (\frac {3}{4}-\frac {3 i}{4}\right ) \sqrt [3]{-x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+2 x^3}}{x^2 \left (i-x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}+\frac {\left (\left (\frac {3}{4}+\frac {3 i}{4}\right ) \sqrt [3]{-x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+2 x^3}}{x^2 \left (i+x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}\\ &=-\frac {\left (\left (\frac {3}{4}-\frac {3 i}{4}\right ) \sqrt [3]{-x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-2 x^3}}{x^2 \left (i-x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-2 x^2}}+\frac {\left (\left (\frac {3}{4}+\frac {3 i}{4}\right ) \sqrt [3]{-x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-2 x^3}}{x^2 \left (i+x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-2 x^2}}\\ &=-\frac {\left (\frac {3}{4}-\frac {3 i}{4}\right ) \sqrt [3]{1-i x^2} \sqrt [3]{-x+2 x^3} \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};\frac {(2-i) x^2}{1-i x^2}\right )}{x \sqrt [3]{1-2 x^2}}-\frac {\left (\frac {3}{4}+\frac {3 i}{4}\right ) \sqrt [3]{1+i x^2} \sqrt [3]{-x+2 x^3} \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};\frac {(2+i) x^2}{1+i x^2}\right )}{x \sqrt [3]{1-2 x^2}}\\ \end {align*}
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Mathematica [F] time = 2.95, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.26, size = 114, normalized size = 1.00 \begin {gather*} -\frac {3 \sqrt [3]{-x+2 x^3}}{2 x}+\frac {1}{4} \text {RootSum}\left [5-4 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{-x+2 x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-x+2 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}^2+\text {$\#$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*} \mathit {undef} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 234.18, size = 10723, normalized size = 94.06
method | result | size |
trager | \(\text {Expression too large to display}\) | \(10723\) |
risch | \(\text {Expression too large to display}\) | \(19101\) |
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 (2 \, x^{3} - x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{{\left (x^{4} + 1\right )} x^{2}}\,{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 (2\,x^3-x\right )}^{1/3}\,\left (x^2+1\right )}{x^2\,\left (x^4+1\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 (2 x^{2} - 1\right )} \left (x^{2} + 1\right )}{x^{2} \left (x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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