Integrand size = 27, antiderivative size = 135 \[ \int (5-x) (3+2 x)^3 \sqrt {2+5 x+3 x^2} \, dx=\frac {6221 (5+6 x) \sqrt {2+5 x+3 x^2}}{5184}+\frac {11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(27487+11538 x) \left (2+5 x+3 x^2\right )^{3/2}}{3240}-\frac {6221 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{10368 \sqrt {3}} \]
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Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {846, 793, 626, 635, 212} \[ \int (5-x) (3+2 x)^3 \sqrt {2+5 x+3 x^2} \, dx=-\frac {6221 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{10368 \sqrt {3}}-\frac {1}{18} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3+\frac {11}{15} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac {(11538 x+27487) \left (3 x^2+5 x+2\right )^{3/2}}{3240}+\frac {6221 (6 x+5) \sqrt {3 x^2+5 x+2}}{5184} \]
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Rule 212
Rule 626
Rule 635
Rule 793
Rule 846
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac {1}{18} \int (3+2 x)^2 \left (\frac {609}{2}+198 x\right ) \sqrt {2+5 x+3 x^2} \, dx \\ & = \frac {11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac {1}{270} \int (3+2 x) \left (\frac {15327}{2}+5769 x\right ) \sqrt {2+5 x+3 x^2} \, dx \\ & = \frac {11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(27487+11538 x) \left (2+5 x+3 x^2\right )^{3/2}}{3240}+\frac {6221}{432} \int \sqrt {2+5 x+3 x^2} \, dx \\ & = \frac {6221 (5+6 x) \sqrt {2+5 x+3 x^2}}{5184}+\frac {11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(27487+11538 x) \left (2+5 x+3 x^2\right )^{3/2}}{3240}-\frac {6221 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{10368} \\ & = \frac {6221 (5+6 x) \sqrt {2+5 x+3 x^2}}{5184}+\frac {11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(27487+11538 x) \left (2+5 x+3 x^2\right )^{3/2}}{3240}-\frac {6221 \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{5184} \\ & = \frac {6221 (5+6 x) \sqrt {2+5 x+3 x^2}}{5184}+\frac {11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(27487+11538 x) \left (2+5 x+3 x^2\right )^{3/2}}{3240}-\frac {6221 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{10368 \sqrt {3}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.56 \[ \int (5-x) (3+2 x)^3 \sqrt {2+5 x+3 x^2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-859701-2432350 x-2317848 x^2-825840 x^3-14976 x^4+34560 x^5\right )-31105 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{77760} \]
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Time = 0.33 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.52
method | result | size |
risch | \(-\frac {\left (34560 x^{5}-14976 x^{4}-825840 x^{3}-2317848 x^{2}-2432350 x -859701\right ) \sqrt {3 x^{2}+5 x +2}}{25920}-\frac {6221 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{31104}\) | \(70\) |
trager | \(\left (-\frac {4}{3} x^{5}+\frac {26}{45} x^{4}+\frac {1147}{36} x^{3}+\frac {96577}{1080} x^{2}+\frac {243235}{2592} x +\frac {286567}{8640}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {6221 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{31104}\) | \(81\) |
default | \(\frac {6221 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{5184}-\frac {6221 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{31104}+\frac {44011 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{3240}-\frac {4 x^{3} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{9}+\frac {14 x^{2} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{15}+\frac {337 x \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{36}\) | \(113\) |
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Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.58 \[ \int (5-x) (3+2 x)^3 \sqrt {2+5 x+3 x^2} \, dx=-\frac {1}{25920} \, {\left (34560 \, x^{5} - 14976 \, x^{4} - 825840 \, x^{3} - 2317848 \, x^{2} - 2432350 \, x - 859701\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {6221}{62208} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
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Time = 0.54 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.61 \[ \int (5-x) (3+2 x)^3 \sqrt {2+5 x+3 x^2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {4 x^{5}}{3} + \frac {26 x^{4}}{45} + \frac {1147 x^{3}}{36} + \frac {96577 x^{2}}{1080} + \frac {243235 x}{2592} + \frac {286567}{8640}\right ) - \frac {6221 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{31104} \]
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Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.90 \[ \int (5-x) (3+2 x)^3 \sqrt {2+5 x+3 x^2} \, dx=-\frac {4}{9} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{3} + \frac {14}{15} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{2} + \frac {337}{36} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {44011}{3240} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {6221}{864} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {6221}{31104} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {31105}{5184} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
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Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.55 \[ \int (5-x) (3+2 x)^3 \sqrt {2+5 x+3 x^2} \, dx=-\frac {1}{25920} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (30 \, x - 13\right )} x - 5735\right )} x - 96577\right )} x - 1216175\right )} x - 859701\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {6221}{31104} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
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Time = 12.47 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.13 \[ \int (5-x) (3+2 x)^3 \sqrt {2+5 x+3 x^2} \, dx=\frac {14\,x^2\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{15}-\frac {4\,x^3\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{9}-\frac {2093\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{1296}+\frac {2093\,\left (\frac {x}{2}+\frac {5}{12}\right )\,\sqrt {3\,x^2+5\,x+2}}{18}+\frac {44011\,\sqrt {3\,x^2+5\,x+2}\,\left (72\,x^2+30\,x-27\right )}{77760}+\frac {337\,x\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{36}+\frac {44011\,\sqrt {3}\,\ln \left (2\,\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (6\,x+5\right )}{3}\right )}{31104} \]
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