Integrand size = 27, antiderivative size = 181 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {182917 (5+6 x) \sqrt {2+5 x+3 x^2}}{35831808}-\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}-\frac {182917 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{71663616 \sqrt {3}} \]
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Time = 0.06 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {846, 793, 626, 635, 212} \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {182917 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{71663616 \sqrt {3}}-\frac {1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}+\frac {169}{405} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}+\frac {(213878 x+477101) \left (3 x^2+5 x+2\right )^{7/2}}{136080}+\frac {182917 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{466560}-\frac {182917 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{4478976}+\frac {182917 (6 x+5) \sqrt {3 x^2+5 x+2}}{35831808} \]
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Rule 212
Rule 626
Rule 635
Rule 793
Rule 846
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {1}{30} \int (3+2 x)^2 \left (\frac {1029}{2}+338 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx \\ & = \frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {1}{810} \int (3+2 x) \left (\frac {42451}{2}+15277 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx \\ & = \frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}+\frac {182917 \int \left (2+5 x+3 x^2\right )^{5/2} \, dx}{12960} \\ & = \frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}-\frac {182917 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{186624} \\ & = -\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}+\frac {182917 \int \sqrt {2+5 x+3 x^2} \, dx}{2985984} \\ & = \frac {182917 (5+6 x) \sqrt {2+5 x+3 x^2}}{35831808}-\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}-\frac {182917 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{71663616} \\ & = \frac {182917 (5+6 x) \sqrt {2+5 x+3 x^2}}{35831808}-\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}-\frac {182917 \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{35831808} \\ & = \frac {182917 (5+6 x) \sqrt {2+5 x+3 x^2}}{35831808}-\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}-\frac {182917 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{71663616 \sqrt {3}} \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.53 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-73178684475-585749416130 x-1995914277480 x^2-3762746217360 x^3-4253933381760 x^4-2893044950784 x^5-1086687912960 x^6-147947046912 x^7+29262643200 x^8+9029615616 x^9\right )-6402095 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{3762339840} \]
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Time = 0.42 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.50
method | result | size |
risch | \(-\frac {\left (9029615616 x^{9}+29262643200 x^{8}-147947046912 x^{7}-1086687912960 x^{6}-2893044950784 x^{5}-4253933381760 x^{4}-3762746217360 x^{3}-1995914277480 x^{2}-585749416130 x -73178684475\right ) \sqrt {3 x^{2}+5 x +2}}{1254113280}-\frac {182917 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{214990848}\) | \(90\) |
trager | \(\left (-\frac {36}{5} x^{9}-\frac {70}{3} x^{8}+\frac {42469}{360} x^{7}+\frac {873431}{1008} x^{6}+\frac {418553957}{181440} x^{5}+\frac {738530101}{217728} x^{4}+\frac {5226036413}{1741824} x^{3}+\frac {16632618979}{10450944} x^{2}+\frac {58574941613}{125411328} x +\frac {4878578965}{83607552}\right ) \sqrt {3 x^{2}+5 x +2}+\frac {182917 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{214990848}\) | \(101\) |
default | \(\frac {182917 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{466560}-\frac {182917 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{4478976}+\frac {182917 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{35831808}-\frac {182917 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{214990848}+\frac {173137 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{27216}-\frac {4 x^{3} \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{15}+\frac {38 x^{2} \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{81}+\frac {46453 x \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{9720}\) | \(151\) |
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Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.54 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{1254113280} \, {\left (9029615616 \, x^{9} + 29262643200 \, x^{8} - 147947046912 \, x^{7} - 1086687912960 \, x^{6} - 2893044950784 \, x^{5} - 4253933381760 \, x^{4} - 3762746217360 \, x^{3} - 1995914277480 \, x^{2} - 585749416130 \, x - 73178684475\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {182917}{429981696} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
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Time = 0.74 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.61 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {36 x^{9}}{5} - \frac {70 x^{8}}{3} + \frac {42469 x^{7}}{360} + \frac {873431 x^{6}}{1008} + \frac {418553957 x^{5}}{181440} + \frac {738530101 x^{4}}{217728} + \frac {5226036413 x^{3}}{1741824} + \frac {16632618979 x^{2}}{10450944} + \frac {58574941613 x}{125411328} + \frac {4878578965}{83607552}\right ) - \frac {182917 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{214990848} \]
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Time = 0.28 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.99 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {4}{15} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{3} + \frac {38}{81} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{2} + \frac {46453}{9720} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {173137}{27216} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {182917}{77760} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {182917}{93312} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {182917}{746496} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {914585}{4478976} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {182917}{5971968} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {182917}{214990848} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {914585}{35831808} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
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Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.52 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{1254113280} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (14 \, {\left (48 \, {\left (54 \, x + 175\right )} x - 42469\right )} x - 4367155\right )} x - 418553957\right )} x - 3692650505\right )} x - 26130182065\right )} x - 83163094895\right )} x - 292874708065\right )} x - 73178684475\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {182917}{214990848} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
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Timed out. \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\int {\left (2\,x+3\right )}^3\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \]
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