Optimal. Leaf size=34 \[ \frac {x^{a+1} \left (2 x^{2 a}+3 x^a+6\right )^{\frac {1}{a}+1}}{6 (a+1)} \]
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Rubi [A] time = 0.04, 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 = {1594, 1747} \[ \frac {x^{a+1} \left (3 x^a+2 x^{2 a}+6\right )^{\frac {1}{a}+1}}{6 (a+1)} \]
Antiderivative was successfully verified.
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Rule 1594
Rule 1747
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 = 0.13, size = 33, normalized size = 0.97 \[ \frac {x^{a+1} \left (2 x^{2 a}+3 x^a+6\right )^{\frac {1}{a}+1}}{6 a+6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 48, normalized size = 1.41 \[ \frac {{\left (2 \, x x^{3 \, a} + 3 \, x x^{2 \, a} + 6 \, x x^{a}\right )} {\left (2 \, x^{2 \, a} + 3 \, x^{a} + 6\right )}^{\left (\frac {1}{a}\right )}}{6 \, {\left (a + 1\right )}} \]
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 \[ \int {\left (2 \, x^{2 \, a} + 3 \, x^{a} + 6\right )}^{\left (\frac {1}{a}\right )} {\left (x^{3 \, a} + x^{2 \, a} + x^{a}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 44, normalized size = 1.29 \[ \frac {\left (3 x^{a}+2 x^{2 a}+6\right ) x \,x^{a} \left (3 x^{a}+2 x^{2 a}+6\right )^{\frac {1}{a}}}{6 a +6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 48, normalized size = 1.41 \[ \frac {{\left (2 \, x x^{3 \, a} + 3 \, x x^{2 \, a} + 6 \, x x^{a}\right )} {\left (2 \, x^{2 \, a} + 3 \, x^{a} + 6\right )}^{\left (\frac {1}{a}\right )}}{6 \, {\left (a + 1\right )}} \]
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 \[ \int \left (x^a+x^{2\,a}+x^{3\,a}\right )\,{\left (3\,x^a+2\,x^{2\,a}+6\right )}^{1/a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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