Integrand size = 23, antiderivative size = 20 \[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=\sqrt {6} x+2 \sqrt {\frac {2}{3}} x^3 \]
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Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {22} \[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=2 \sqrt {\frac {2}{3}} x^3+\sqrt {6} x \]
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Rule 22
Rubi steps \begin{align*} \text {integral}& = \sqrt {\frac {2}{3}} \int \left (3+6 x^2\right ) \, dx \\ & = \sqrt {6} x+2 \sqrt {\frac {2}{3}} x^3 \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=\sqrt {6} \left (x+\frac {2 x^3}{3}\right ) \]
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Result contains higher order function than in optimal. Order 2 vs. order 1.
Time = 2.36 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90
method | result | size |
gosper | \(\frac {x \left (2 x^{2}+3\right ) \sqrt {4 x^{2}+2}}{\sqrt {6 x^{2}+3}}\) | \(38\) |
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Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (14) = 28\).
Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90 \[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=\frac {{\left (2 \, x^{3} + 3 \, x\right )} \sqrt {6 \, x^{2} + 3} \sqrt {4 \, x^{2} + 2}}{3 \, {\left (2 \, x^{2} + 1\right )}} \]
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Time = 1.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=\frac {2 \sqrt {6} x^{3}}{3} + \sqrt {6} x \]
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\[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=\int { \sqrt {6 \, x^{2} + 3} \sqrt {4 \, x^{2} + 2} \,d x } \]
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none
Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=\frac {1}{3} \, \sqrt {3} \sqrt {2} {\left (2 \, x^{3} + 3 \, x\right )} \]
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Timed out. \[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=\int \sqrt {4\,x^2+2}\,\sqrt {6\,x^2+3} \,d x \]
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