Optimal. Leaf size=20 \[ \frac {1}{6} \left (2 x^4+2 x^2-1\right )^{3/2} \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1247, 629} \begin {gather*} \frac {1}{6} \left (2 x^4+2 x^2-1\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 629
Rule 1247
Rubi steps
\begin {align*} \int x \left (1+2 x^2\right ) \sqrt {-1+2 x^2+2 x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (1+2 x) \sqrt {-1+2 x+2 x^2} \, dx,x,x^2\right )\\ &=\frac {1}{6} \left (-1+2 x^2+2 x^4\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {1}{6} \left (2 x^4+2 x^2-1\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 20, normalized size = 1.00 \begin {gather*} \frac {1}{6} \left (2 x^4+2 x^2-1\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, {\left (2 \, x^{4} + 2 \, x^{2} - 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, {\left (2 \, x^{4} + 2 \, x^{2} - 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 0.85 \begin {gather*} \frac {\left (2 x^{4}+2 x^{2}-1\right )^{\frac {3}{2}}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, {\left (2 \, x^{4} + 2 \, x^{2} - 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 16, normalized size = 0.80 \begin {gather*} \frac {{\left (2\,x^4+2\,x^2-1\right )}^{3/2}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.19, size = 56, normalized size = 2.80 \begin {gather*} \frac {x^{4} \sqrt {2 x^{4} + 2 x^{2} - 1}}{3} + \frac {x^{2} \sqrt {2 x^{4} + 2 x^{2} - 1}}{3} - \frac {\sqrt {2 x^{4} + 2 x^{2} - 1}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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