Integrand size = 20, antiderivative size = 126 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {175 (5+6 x) \sqrt {2+5 x+3 x^2}}{82944}-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{165888 \sqrt {3}} \]
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Time = 0.03 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {654, 626, 635, 212} \[ \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {175 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{165888 \sqrt {3}}-\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}+\frac {35}{216} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {175 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{10368}+\frac {175 (6 x+5) \sqrt {3 x^2+5 x+2}}{82944} \]
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Rule 212
Rule 626
Rule 635
Rule 654
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}+\frac {35}{6} \int \left (2+5 x+3 x^2\right )^{5/2} \, dx \\ & = \frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175}{432} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx \\ & = -\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}+\frac {175 \int \sqrt {2+5 x+3 x^2} \, dx}{6912} \\ & = \frac {175 (5+6 x) \sqrt {2+5 x+3 x^2}}{82944}-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{165888} \\ & = \frac {175 (5+6 x) \sqrt {2+5 x+3 x^2}}{82944}-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175 \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{82944} \\ & = \frac {175 (5+6 x) \sqrt {2+5 x+3 x^2}}{82944}-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{165888 \sqrt {3}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.64 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-1568541-9651790 x-23110872 x^2-26388720 x^3-13454208 x^4-1347840 x^5+746496 x^6\right )-1225 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{1741824} \]
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Time = 0.33 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.60
method | result | size |
risch | \(-\frac {\left (746496 x^{6}-1347840 x^{5}-13454208 x^{4}-26388720 x^{3}-23110872 x^{2}-9651790 x -1568541\right ) \sqrt {3 x^{2}+5 x +2}}{580608}-\frac {175 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{497664}\) | \(75\) |
trager | \(\left (-\frac {9}{7} x^{6}+\frac {65}{28} x^{5}+\frac {3893}{168} x^{4}+\frac {61085}{1344} x^{3}+\frac {962953}{24192} x^{2}+\frac {4825895}{290304} x +\frac {522847}{193536}\right ) \sqrt {3 x^{2}+5 x +2}+\frac {175 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{497664}\) | \(86\) |
default | \(\frac {35 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{216}-\frac {175 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{10368}+\frac {175 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{82944}-\frac {175 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{497664}-\frac {\left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{21}\) | \(102\) |
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Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.66 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{580608} \, {\left (746496 \, x^{6} - 1347840 \, x^{5} - 13454208 \, x^{4} - 26388720 \, x^{3} - 23110872 \, x^{2} - 9651790 \, x - 1568541\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {175}{995328} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
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Time = 0.59 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.71 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {9 x^{6}}{7} + \frac {65 x^{5}}{28} + \frac {3893 x^{4}}{168} + \frac {61085 x^{3}}{1344} + \frac {962953 x^{2}}{24192} + \frac {4825895 x}{290304} + \frac {522847}{193536}\right ) - \frac {175 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{497664} \]
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Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.03 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{21} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {35}{36} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {175}{216} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {175}{1728} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {875}{10368} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {175}{13824} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {175}{497664} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {875}{82944} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
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Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.63 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{580608} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (6 \, {\left (36 \, x - 65\right )} x - 3893\right )} x - 61085\right )} x - 962953\right )} x - 4825895\right )} x - 1568541\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {175}{497664} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
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Timed out. \[ \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=\int -\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \]
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