Integrand size = 20, antiderivative size = 50 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^{3/2} \, dx=\frac {1}{16} (2 d-3 e) (3+2 x) \left (9+12 x+4 x^2\right )^{3/2}+\frac {1}{20} e \left (9+12 x+4 x^2\right )^{5/2} \]
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Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {654, 623} \[ \int (d+e x) \left (9+12 x+4 x^2\right )^{3/2} \, dx=\frac {1}{16} (2 x+3) \left (4 x^2+12 x+9\right )^{3/2} (2 d-3 e)+\frac {1}{20} e \left (4 x^2+12 x+9\right )^{5/2} \]
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Rule 623
Rule 654
Rubi steps \begin{align*} \text {integral}& = \frac {1}{20} e \left (9+12 x+4 x^2\right )^{5/2}+\frac {1}{2} (2 d-3 e) \int \left (9+12 x+4 x^2\right )^{3/2} \, dx \\ & = \frac {1}{16} (2 d-3 e) (3+2 x) \left (9+12 x+4 x^2\right )^{3/2}+\frac {1}{20} e \left (9+12 x+4 x^2\right )^{5/2} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.66 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^{3/2} \, dx=\frac {1}{80} (3+2 x)^3 \sqrt {(3+2 x)^2} (10 d+e (-3+8 x)) \]
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Time = 2.17 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.24
method | result | size |
gosper | \(\frac {x \left (16 e \,x^{4}+20 d \,x^{3}+90 e \,x^{3}+120 d \,x^{2}+180 e \,x^{2}+270 d x +135 e x +270 d \right ) \left (\left (2 x +3\right )^{2}\right )^{\frac {3}{2}}}{10 \left (2 x +3\right )^{3}}\) | \(62\) |
default | \(\frac {x \left (16 e \,x^{4}+20 d \,x^{3}+90 e \,x^{3}+120 d \,x^{2}+180 e \,x^{2}+270 d x +135 e x +270 d \right ) \left (\left (2 x +3\right )^{2}\right )^{\frac {3}{2}}}{10 \left (2 x +3\right )^{3}}\) | \(62\) |
risch | \(\frac {8 \sqrt {\left (2 x +3\right )^{2}}\, e \,x^{5}}{5 \left (2 x +3\right )}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (8 d +36 e \right ) x^{4}}{8 x +12}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (36 d +54 e \right ) x^{3}}{9+6 x}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (54 d +27 e \right ) x^{2}}{4 x +6}+\frac {27 \sqrt {\left (2 x +3\right )^{2}}\, d x}{2 x +3}\) | \(128\) |
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Time = 0.36 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^{3/2} \, dx=\frac {8}{5} \, e x^{5} + {\left (2 \, d + 9 \, e\right )} x^{4} + 6 \, {\left (2 \, d + 3 \, e\right )} x^{3} + \frac {27}{2} \, {\left (2 \, d + e\right )} x^{2} + 27 \, d x \]
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Time = 0.61 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.32 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^{3/2} \, dx=\sqrt {4 x^{2} + 12 x + 9} \cdot \left (\frac {27 d}{8} + \frac {4 e x^{4}}{5} - \frac {81 e}{80} + x^{3} \left (d + \frac {33 e}{10}\right ) + x^{2} \cdot \left (\frac {9 d}{2} + \frac {81 e}{20}\right ) + x \left (\frac {27 d}{4} + \frac {27 e}{40}\right )\right ) \]
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Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.56 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^{3/2} \, dx=\frac {1}{20} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {5}{2}} e + \frac {1}{4} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {3}{2}} d x - \frac {3}{8} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {3}{2}} e x + \frac {3}{8} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {3}{2}} d - \frac {9}{16} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {3}{2}} e \]
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Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (42) = 84\).
Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.20 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^{3/2} \, dx=\frac {8}{5} \, e x^{5} \mathrm {sgn}\left (2 \, x + 3\right ) + 2 \, d x^{4} \mathrm {sgn}\left (2 \, x + 3\right ) + 9 \, e x^{4} \mathrm {sgn}\left (2 \, x + 3\right ) + 12 \, d x^{3} \mathrm {sgn}\left (2 \, x + 3\right ) + 18 \, e x^{3} \mathrm {sgn}\left (2 \, x + 3\right ) + 27 \, d x^{2} \mathrm {sgn}\left (2 \, x + 3\right ) + \frac {27}{2} \, e x^{2} \mathrm {sgn}\left (2 \, x + 3\right ) + 27 \, d x \mathrm {sgn}\left (2 \, x + 3\right ) + \frac {81}{80} \, {\left (10 \, d - 3 \, e\right )} \mathrm {sgn}\left (2 \, x + 3\right ) \]
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Time = 9.60 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.34 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^{3/2} \, dx=\frac {e\,{\left (4\,x^2+12\,x+9\right )}^{5/2}}{20}-\frac {9\,e\,{\left (4\,x^2+12\,x+9\right )}^{3/2}}{16}-\frac {3\,e\,x\,{\left (4\,x^2+12\,x+9\right )}^{3/2}}{8}+\frac {d\,\left (2\,x+3\right )\,{\left (4\,x^2+12\,x+9\right )}^{3/2}}{8} \]
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