Integrand size = 33, antiderivative size = 34 \[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\frac {x^{1+a} \left (6+3 x^a+2 x^{2 a}\right )^{1+\frac {1}{a}}}{6 (1+a)} \]
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Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1608, 1761} \[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\frac {x^{a+1} \left (2 x^{2 a}+3 x^a+6\right )^{\frac {1}{a}+1}}{6 (a+1)} \]
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Rule 1608
Rule 1761
Rubi steps \begin{align*} \text {integral}& = \int x^a \left (1+x^a+x^{2 a}\right ) \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \, dx \\ & = \frac {x^{1+a} \left (6+3 x^a+2 x^{2 a}\right )^{1+\frac {1}{a}}}{6 (1+a)} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\frac {x^{1+a} \left (6+3 x^a+2 x^{2 a}\right )^{1+\frac {1}{a}}}{6+6 a} \]
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Time = 0.72 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29
method | result | size |
risch | \(\frac {x \,x^{a} \left (6+3 x^{a}+2 x^{2 a}\right ) \left (6+3 x^{a}+2 x^{2 a}\right )^{\frac {1}{a}}}{6+6 a}\) | \(44\) |
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Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41 \[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\frac {{\left (2 \, x x^{3 \, a} + 3 \, x x^{2 \, a} + 6 \, x x^{a}\right )} {\left (2 \, x^{2 \, a} + 3 \, x^{a} + 6\right )}^{\left (\frac {1}{a}\right )}}{6 \, {\left (a + 1\right )}} \]
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Timed out. \[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\text {Timed out} \]
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none
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41 \[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\frac {{\left (2 \, x x^{3 \, a} + 3 \, x x^{2 \, a} + 6 \, x x^{a}\right )} {\left (2 \, x^{2 \, a} + 3 \, x^{a} + 6\right )}^{\left (\frac {1}{a}\right )}}{6 \, {\left (a + 1\right )}} \]
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\[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\int { {\left (2 \, x^{2 \, a} + 3 \, x^{a} + 6\right )}^{\left (\frac {1}{a}\right )} {\left (x^{3 \, a} + x^{2 \, a} + x^{a}\right )} \,d x } \]
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Timed out. \[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\int \left (x^a+x^{2\,a}+x^{3\,a}\right )\,{\left (3\,x^a+2\,x^{2\,a}+6\right )}^{1/a} \,d x \]
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