Optimal. Leaf size=59 \[ -\text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\text {$\#$1} \log \left (\sqrt [3]{-x^4+x^3+x}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{2 \text {$\#$1}^3-1}\& \right ] \]
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Rubi [F] time = 1.58, 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 {\left (2+x^3\right ) \sqrt [3]{x+x^3-x^4}}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (2+x^3\right ) \sqrt [3]{x+x^3-x^4}}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx &=\frac {\sqrt [3]{x+x^3-x^4} \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2-x^3} \left (2+x^3\right )}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^3}}\\ &=\frac {\left (3 \sqrt [3]{x+x^3-x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6-x^9} \left (2+x^9\right )}{1+x^6-2 x^9+x^{12}-x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^3}}\\ &=\frac {\left (3 \sqrt [3]{x+x^3-x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {2 x^3 \sqrt [3]{1+x^6-x^9}}{1+x^6-2 x^9+x^{12}-x^{15}+x^{18}}+\frac {x^{12} \sqrt [3]{1+x^6-x^9}}{1+x^6-2 x^9+x^{12}-x^{15}+x^{18}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^3}}\\ &=\frac {\left (3 \sqrt [3]{x+x^3-x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{12} \sqrt [3]{1+x^6-x^9}}{1+x^6-2 x^9+x^{12}-x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^3}}+\frac {\left (6 \sqrt [3]{x+x^3-x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6-x^9}}{1+x^6-2 x^9+x^{12}-x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^3}}\\ \end {align*}
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Mathematica [F] time = 0.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2+x^3\right ) \sqrt [3]{x+x^3-x^4}}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.13, size = 59, normalized size = 1.00 \begin {gather*} -\text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x+x^3-x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^3}\&\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^{4} + x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} + 2\right )}}{x^{6} - x^{5} + x^{4} - 2 \, x^{3} + x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 13.59, size = 1808, normalized size = 30.64
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1808\) |
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^{4} + x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} + 2\right )}}{x^{6} - x^{5} + x^{4} - 2 \, x^{3} + x^{2} + 1}\,{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 = -0.02 \begin {gather*} \int \frac {\left (x^3+2\right )\,{\left (-x^4+x^3+x\right )}^{1/3}}{x^6-x^5+x^4-2\,x^3+x^2+1} \,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^{3} - x^{2} - 1\right )} \left (x^{3} + 2\right )}{x^{6} - x^{5} + x^{4} - 2 x^{3} + x^{2} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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