Integrand size = 16, antiderivative size = 44 \[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{5/2}} \, dx=-\frac {1}{12 \left (9+12 x+4 x^2\right )^{3/2}}+\frac {3}{16 (3+2 x) \left (9+12 x+4 x^2\right )^{3/2}} \]
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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} \[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{5/2}} \, dx=\frac {3}{16 (2 x+3) \left (4 x^2+12 x+9\right )^{3/2}}-\frac {1}{12 \left (4 x^2+12 x+9\right )^{3/2}} \]
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Rule 621
Rule 654
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{12 \left (9+12 x+4 x^2\right )^{3/2}}-\frac {3}{2} \int \frac {1}{\left (9+12 x+4 x^2\right )^{5/2}} \, dx \\ & = -\frac {1}{12 \left (9+12 x+4 x^2\right )^{3/2}}+\frac {3}{16 (3+2 x) \left (9+12 x+4 x^2\right )^{3/2}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.61 \[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{5/2}} \, dx=\frac {-3-8 x}{48 (3+2 x)^3 \sqrt {(3+2 x)^2}} \]
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Time = 1.95 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.50
method | result | size |
gosper | \(-\frac {\left (2 x +3\right ) \left (8 x +3\right )}{48 \left (\left (2 x +3\right )^{2}\right )^{\frac {5}{2}}}\) | \(22\) |
default | \(-\frac {\left (2 x +3\right ) \left (8 x +3\right )}{48 \left (\left (2 x +3\right )^{2}\right )^{\frac {5}{2}}}\) | \(22\) |
meijerg | \(\frac {x^{2} \left (\frac {4}{9} x^{2}+\frac {8}{3} x +6\right )}{2916 \left (1+\frac {2 x}{3}\right )^{4}}\) | \(23\) |
risch | \(\frac {16 \sqrt {\left (2 x +3\right )^{2}}\, \left (-\frac {x}{96}-\frac {1}{256}\right )}{\left (2 x +3\right )^{5}}\) | \(24\) |
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Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.66 \[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{5/2}} \, dx=-\frac {8 \, x + 3}{48 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]
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\[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{5/2}} \, dx=\int \frac {x}{\left (\left (2 x + 3\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.55 \[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{5/2}} \, dx=-\frac {1}{12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {3}{2}}} + \frac {3}{16 \, {\left (2 \, x + 3\right )}^{4}} \]
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Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.50 \[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{5/2}} \, dx=-\frac {8 \, x + 3}{48 \, {\left (2 \, x + 3\right )}^{4} \mathrm {sgn}\left (2 \, x + 3\right )} \]
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Time = 9.55 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.59 \[ \int \frac {x}{\left (9+12 x+4 x^2\right )^{5/2}} \, dx=-\frac {\left (8\,x+3\right )\,\sqrt {4\,x^2+12\,x+9}}{48\,{\left (2\,x+3\right )}^5} \]
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