Integrand size = 27, antiderivative size = 160 \[ \int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx=\frac {25969 (5+6 x) \sqrt {2+5 x+3 x^2}}{15552}+\frac {478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(874301+378774 x) \left (2+5 x+3 x^2\right )^{3/2}}{68040}-\frac {25969 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{31104 \sqrt {3}} \]
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Time = 0.06 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {846, 793, 626, 635, 212} \[ \int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx=-\frac {25969 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{31104 \sqrt {3}}-\frac {1}{21} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^4+\frac {229}{378} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3+\frac {478}{315} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac {(378774 x+874301) \left (3 x^2+5 x+2\right )^{3/2}}{68040}+\frac {25969 (6 x+5) \sqrt {3 x^2+5 x+2}}{15552} \]
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Rule 212
Rule 626
Rule 635
Rule 793
Rule 846
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {1}{21} \int (3+2 x)^3 \left (\frac {707}{2}+229 x\right ) \sqrt {2+5 x+3 x^2} \, dx \\ & = \frac {229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {1}{378} \int (3+2 x)^2 \left (\frac {22377}{2}+8604 x\right ) \sqrt {2+5 x+3 x^2} \, dx \\ & = \frac {478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {\int (3+2 x) \left (\frac {482121}{2}+189387 x\right ) \sqrt {2+5 x+3 x^2} \, dx}{5670} \\ & = \frac {478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(874301+378774 x) \left (2+5 x+3 x^2\right )^{3/2}}{68040}+\frac {25969 \int \sqrt {2+5 x+3 x^2} \, dx}{1296} \\ & = \frac {25969 (5+6 x) \sqrt {2+5 x+3 x^2}}{15552}+\frac {478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(874301+378774 x) \left (2+5 x+3 x^2\right )^{3/2}}{68040}-\frac {25969 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{31104} \\ & = \frac {25969 (5+6 x) \sqrt {2+5 x+3 x^2}}{15552}+\frac {478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(874301+378774 x) \left (2+5 x+3 x^2\right )^{3/2}}{68040}-\frac {25969 \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{15552} \\ & = \frac {25969 (5+6 x) \sqrt {2+5 x+3 x^2}}{15552}+\frac {478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(874301+378774 x) \left (2+5 x+3 x^2\right )^{3/2}}{68040}-\frac {25969 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{31104 \sqrt {3}} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.51 \[ \int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-47009103-161915450 x-208601544 x^2-123633360 x^3-28649088 x^4+1624320 x^5+1244160 x^6\right )-908915 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{1632960} \]
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Time = 0.36 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.47
method | result | size |
risch | \(-\frac {\left (1244160 x^{6}+1624320 x^{5}-28649088 x^{4}-123633360 x^{3}-208601544 x^{2}-161915450 x -47009103\right ) \sqrt {3 x^{2}+5 x +2}}{544320}-\frac {25969 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{93312}\) | \(75\) |
trager | \(\left (-\frac {16}{7} x^{6}-\frac {188}{63} x^{5}+\frac {49738}{945} x^{4}+\frac {171713}{756} x^{3}+\frac {8691731}{22680} x^{2}+\frac {16191545}{54432} x +\frac {15669701}{181440}\right ) \sqrt {3 x^{2}+5 x +2}+\frac {25969 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{93312}\) | \(86\) |
default | \(\frac {25969 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{15552}-\frac {25969 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{93312}+\frac {2654033 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{68040}-\frac {16 x^{4} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{21}+\frac {52 x^{3} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{189}+\frac {5542 x^{2} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{315}+\frac {34931 x \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{756}\) | \(130\) |
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Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.52 \[ \int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx=-\frac {1}{544320} \, {\left (1244160 \, x^{6} + 1624320 \, x^{5} - 28649088 \, x^{4} - 123633360 \, x^{3} - 208601544 \, x^{2} - 161915450 \, x - 47009103\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {25969}{186624} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
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Time = 0.56 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.56 \[ \int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {16 x^{6}}{7} - \frac {188 x^{5}}{63} + \frac {49738 x^{4}}{945} + \frac {171713 x^{3}}{756} + \frac {8691731 x^{2}}{22680} + \frac {16191545 x}{54432} + \frac {15669701}{181440}\right ) - \frac {25969 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{93312} \]
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Time = 0.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.86 \[ \int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx=-\frac {16}{21} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{4} + \frac {52}{189} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{3} + \frac {5542}{315} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{2} + \frac {34931}{756} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {2654033}{68040} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {25969}{2592} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {25969}{93312} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {129845}{15552} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
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Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.49 \[ \int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx=-\frac {1}{544320} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (30 \, {\left (36 \, x + 47\right )} x - 24869\right )} x - 858565\right )} x - 8691731\right )} x - 80957725\right )} x - 47009103\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {25969}{93312} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
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Time = 12.63 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.06 \[ \int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx=\frac {5542\,x^2\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{315}+\frac {52\,x^3\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{189}-\frac {16\,x^4\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{21}-\frac {118159\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{27216}+\frac {118159\,\left (\frac {x}{2}+\frac {5}{12}\right )\,\sqrt {3\,x^2+5\,x+2}}{378}+\frac {2654033\,\sqrt {3\,x^2+5\,x+2}\,\left (72\,x^2+30\,x-27\right )}{1632960}+\frac {34931\,x\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{756}+\frac {2654033\,\sqrt {3}\,\ln \left (2\,\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (6\,x+5\right )}{3}\right )}{653184} \]
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