Optimal. Leaf size=20 \[ \sqrt {6} x+2 \sqrt {\frac {2}{3}} x^3 \]
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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {22}
\begin {gather*} 2 \sqrt {\frac {2}{3}} x^3+\sqrt {6} x \end {gather*}
Antiderivative was successfully verified.
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Rule 22
Rubi steps
\begin {align*} \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx &=\sqrt {\frac {2}{3}} \int \left (3+6 x^2\right ) \, dx\\ &=\sqrt {6} x+2 \sqrt {\frac {2}{3}} x^3\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 0.75 \begin {gather*} \sqrt {6} \left (x+\frac {2 x^3}{3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 2 vs. order
1.
time = 0.06, size = 38, normalized size = 1.90
method | result | size |
gosper | \(\frac {x \left (2 x^{2}+3\right ) \sqrt {4 x^{2}+2}}{\sqrt {6 x^{2}+3}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 38 vs.
\(2 (14) = 28\).
time = 0.63, size = 38, normalized size = 1.90 \begin {gather*} \frac {{\left (2 \, x^{3} + 3 \, x\right )} \sqrt {6 \, x^{2} + 3} \sqrt {4 \, x^{2} + 2}}{3 \, {\left (2 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.43, size = 17, normalized size = 0.85 \begin {gather*} \frac {2 \sqrt {6} x^{3}}{3} + \sqrt {6} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.13, size = 17, normalized size = 0.85 \begin {gather*} \frac {1}{3} \, \sqrt {3} \sqrt {2} {\left (2 \, x^{3} + 3 \, x\right )} \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 = -0.05 \begin {gather*} \int \sqrt {4\,x^2+2}\,\sqrt {6\,x^2+3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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