Integrand size = 25, antiderivative size = 131 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {1865 (5+6 x) \sqrt {2+5 x+3 x^2}}{663552}-\frac {1865 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{82944}+\frac {373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac {1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1865 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1327104 \sqrt {3}} \]
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Time = 0.03 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {793, 626, 635, 212} \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1865 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1327104 \sqrt {3}}+\frac {1}{168} (71-14 x) \left (3 x^2+5 x+2\right )^{7/2}+\frac {373 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{1728}-\frac {1865 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{82944}+\frac {1865 (6 x+5) \sqrt {3 x^2+5 x+2}}{663552} \]
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Rule 212
Rule 626
Rule 635
Rule 793
Rubi steps \begin{align*} \text {integral}& = \frac {1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}+\frac {373}{48} \int \left (2+5 x+3 x^2\right )^{5/2} \, dx \\ & = \frac {373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac {1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1865 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{3456} \\ & = -\frac {1865 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{82944}+\frac {373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac {1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}+\frac {1865 \int \sqrt {2+5 x+3 x^2} \, dx}{55296} \\ & = \frac {1865 (5+6 x) \sqrt {2+5 x+3 x^2}}{663552}-\frac {1865 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{82944}+\frac {373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac {1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1865 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{1327104} \\ & = \frac {1865 (5+6 x) \sqrt {2+5 x+3 x^2}}{663552}-\frac {1865 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{82944}+\frac {373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac {1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1865 \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{663552} \\ & = \frac {1865 (5+6 x) \sqrt {2+5 x+3 x^2}}{663552}-\frac {1865 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{82944}+\frac {373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac {1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1865 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1327104 \sqrt {3}} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.66 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-34777419-235223330 x-642995688 x^2-897818256 x^3-655212672 x^4-211154688 x^5-746496 x^6+10450944 x^7\right )-13055 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{13934592} \]
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Time = 0.33 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.61
method | result | size |
risch | \(-\frac {\left (10450944 x^{7}-746496 x^{6}-211154688 x^{5}-655212672 x^{4}-897818256 x^{3}-642995688 x^{2}-235223330 x -34777419\right ) \sqrt {3 x^{2}+5 x +2}}{4644864}-\frac {1865 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{3981312}\) | \(80\) |
trager | \(\left (-\frac {9}{4} x^{7}+\frac {9}{56} x^{6}+\frac {10183}{224} x^{5}+\frac {189587}{1344} x^{4}+\frac {692761}{3584} x^{3}+\frac {26791487}{193536} x^{2}+\frac {117611665}{2322432} x +\frac {11592473}{1548288}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {1865 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{3981312}\) | \(91\) |
default | \(\frac {373 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{1728}-\frac {1865 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{82944}+\frac {1865 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{663552}-\frac {1865 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{3981312}+\frac {71 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{168}-\frac {x \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{12}\) | \(117\) |
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Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.67 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{4644864} \, {\left (10450944 \, x^{7} - 746496 \, x^{6} - 211154688 \, x^{5} - 655212672 \, x^{4} - 897818256 \, x^{3} - 642995688 \, x^{2} - 235223330 \, x - 34777419\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {1865}{7962624} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
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Time = 0.76 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.74 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {9 x^{7}}{4} + \frac {9 x^{6}}{56} + \frac {10183 x^{5}}{224} + \frac {189587 x^{4}}{1344} + \frac {692761 x^{3}}{3584} + \frac {26791487 x^{2}}{193536} + \frac {117611665 x}{2322432} + \frac {11592473}{1548288}\right ) - \frac {1865 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{3981312} \]
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Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.11 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{12} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {71}{168} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {373}{288} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {1865}{1728} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {1865}{13824} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {9325}{82944} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {1865}{110592} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {1865}{3981312} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {9325}{663552} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
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Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.64 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{4644864} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (14 \, x - 1\right )} x - 10183\right )} x - 189587\right )} x - 2078283\right )} x - 26791487\right )} x - 117611665\right )} x - 34777419\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {1865}{3981312} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
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Timed out. \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\int \left (2\,x+3\right )\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \]
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