Integrand size = 34, antiderivative size = 59 \[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=-\frac {1}{4} \text {RootSum}\left [1-4 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{-1-2 x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ] \]
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\[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=\int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 x \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6}+\frac {4 x^2 \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6}\right ) \, dx \\ & = 3 \int \frac {x \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx+4 \int \frac {x^2 \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=-\frac {1}{4} \text {RootSum}\left [1-4 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{-1-2 x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ] \]
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Time = 30.76 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-4 \textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +\left (x^{3}-2 x -1\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{3}-1}\right )}{4}\) | \(48\) |
trager | \(\text {Expression too large to display}\) | \(10661\) |
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Exception generated. \[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 1.92 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58 \[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=\int \frac {x \sqrt [3]{\left (x + 1\right ) \left (x^{2} - x - 1\right )} \left (4 x + 3\right )}{x^{6} - 8 x^{2} - 8 x - 2}\, dx \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58 \[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=\int { \frac {{\left (x^{3} - 2 \, x - 1\right )}^{\frac {1}{3}} {\left (4 \, x + 3\right )} x}{x^{6} - 8 \, x^{2} - 8 \, x - 2} \,d x } \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58 \[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=\int { \frac {{\left (x^{3} - 2 \, x - 1\right )}^{\frac {1}{3}} {\left (4 \, x + 3\right )} x}{x^{6} - 8 \, x^{2} - 8 \, x - 2} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.63 \[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=\int -\frac {x\,\left (4\,x+3\right )\,{\left (x^3-2\,x-1\right )}^{1/3}}{-x^6+8\,x^2+8\,x+2} \,d x \]
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