Integrand size = 21, antiderivative size = 13 \[ \int \frac {x+3 x^2}{\sqrt {x^2+2 x^3}} \, dx=\sqrt {x^2+2 x^3} \]
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Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1602} \[ \int \frac {x+3 x^2}{\sqrt {x^2+2 x^3}} \, dx=\sqrt {2 x^3+x^2} \]
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Rule 1602
Rubi steps \begin{align*} \text {integral}& = \sqrt {x^2+2 x^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {x+3 x^2}{\sqrt {x^2+2 x^3}} \, dx=\sqrt {x^2 (1+2 x)} \]
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Time = 1.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92
method | result | size |
trager | \(\sqrt {2 x^{3}+x^{2}}\) | \(12\) |
pseudoelliptic | \(\sqrt {x^{2} \left (1+2 x \right )}\) | \(12\) |
gosper | \(\frac {x^{2} \left (1+2 x \right )}{\sqrt {2 x^{3}+x^{2}}}\) | \(21\) |
default | \(\frac {x^{2} \left (1+2 x \right )}{\sqrt {2 x^{3}+x^{2}}}\) | \(21\) |
risch | \(\frac {x^{2} \left (1+2 x \right )}{\sqrt {x^{2} \left (1+2 x \right )}}\) | \(21\) |
meijerg | \(\frac {\sqrt {\pi }-\frac {\sqrt {\pi }\, \left (-8 x +8\right ) \sqrt {1+2 x}}{8}}{\sqrt {\pi }}+\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1+2 x}}{2 \sqrt {\pi }}\) | \(53\) |
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none
Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {x+3 x^2}{\sqrt {x^2+2 x^3}} \, dx=\sqrt {2 \, x^{3} + x^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {x+3 x^2}{\sqrt {x^2+2 x^3}} \, dx=\sqrt {2 x^{3} + x^{2}} \]
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none
Time = 0.18 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {x+3 x^2}{\sqrt {x^2+2 x^3}} \, dx=\sqrt {2 \, x^{3} + x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (11) = 22\).
Time = 0.37 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.77 \[ \int \frac {x+3 x^2}{\sqrt {x^2+2 x^3}} \, dx=\frac {{\left (2 \, x + 1\right )}^{\frac {3}{2}} - \sqrt {2 \, x + 1}}{2 \, \mathrm {sgn}\left (x\right )} \]
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Time = 17.10 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {x+3 x^2}{\sqrt {x^2+2 x^3}} \, dx=\left |x\right |\,\sqrt {2\,x+1} \]
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