Integrand size = 16, antiderivative size = 44 \[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=-\frac {1}{4 \sqrt {9+12 x+4 x^2}}+\frac {3}{8 (3+2 x) \sqrt {9+12 x+4 x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {654, 621} \[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=\frac {3}{8 (2 x+3) \sqrt {4 x^2+12 x+9}}-\frac {1}{4 \sqrt {4 x^2+12 x+9}} \]
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Rule 621
Rule 654
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4 \sqrt {9+12 x+4 x^2}}-\frac {3}{2} \int \frac {1}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx \\ & = -\frac {1}{4 \sqrt {9+12 x+4 x^2}}+\frac {3}{8 (3+2 x) \sqrt {9+12 x+4 x^2}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.57 \[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=\frac {x^2 \sqrt {(3+2 x)^2}}{6 (3+2 x)^3} \]
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Time = 2.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.30
method | result | size |
meijerg | \(\frac {x^{2}}{54 \left (1+\frac {2 x}{3}\right )^{2}}\) | \(13\) |
gosper | \(-\frac {\left (2 x +3\right ) \left (4 x +3\right )}{8 \left (\left (2 x +3\right )^{2}\right )^{\frac {3}{2}}}\) | \(22\) |
default | \(-\frac {\left (2 x +3\right ) \left (4 x +3\right )}{8 \left (\left (2 x +3\right )^{2}\right )^{\frac {3}{2}}}\) | \(22\) |
risch | \(\frac {4 \sqrt {\left (2 x +3\right )^{2}}\, \left (-\frac {x}{8}-\frac {3}{32}\right )}{\left (2 x +3\right )^{3}}\) | \(24\) |
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Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.43 \[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=-\frac {4 \, x + 3}{8 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \]
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\[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=\int \frac {x}{\left (\left (2 x + 3\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.55 \[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=-\frac {1}{4 \, \sqrt {4 \, x^{2} + 12 \, x + 9}} + \frac {3}{8 \, {\left (2 \, x + 3\right )}^{2}} \]
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Time = 0.35 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.50 \[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=-\frac {4 \, x + 3}{8 \, {\left (2 \, x + 3\right )}^{2} \mathrm {sgn}\left (2 \, x + 3\right )} \]
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Time = 9.52 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.59 \[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=-\frac {\left (4\,x+3\right )\,\sqrt {4\,x^2+12\,x+9}}{8\,{\left (2\,x+3\right )}^3} \]
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