Optimal. Leaf size=34 \[ \frac {x^{1+a} \left (6+3 x^a+2 x^{2 a}\right )^{1+\frac {1}{a}}}{6 (1+a)} \]
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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1608, 1761}
\begin {gather*} \frac {x^{a+1} \left (2 x^{2 a}+3 x^a+6\right )^{\frac {1}{a}+1}}{6 (a+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 1608
Rule 1761
Rubi steps
\begin {align*} \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 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*}
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Mathematica [A]
time = 0.10, size = 33, normalized size = 0.97 \begin {gather*} \frac {x^{1+a} \left (6+3 x^a+2 x^{2 a}\right )^{1+\frac {1}{a}}}{6+6 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 44, normalized size = 1.29
method | result | size |
risch | \(\frac {x \,x^{a} \left (6+3 x^{a}+2 x^{2 a}\right ) \left (6+3 x^{a}+2 x^{2 a}\right )^{\frac {1}{a}}}{6+6 a}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.61, size = 48, normalized size = 1.41 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.72, size = 48, normalized size = 1.41 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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*} \text {could not integrate} \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 = -0.03 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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