Integrand size = 34, antiderivative size = 149 \[ \int (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx=\frac {125455 (17+24 x) \sqrt {6+17 x+12 x^2}}{150994944}-\frac {125455 (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}}{4718592}+\frac {25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac {873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}-\frac {125455 \text {arctanh}\left (\frac {17+24 x}{4 \sqrt {3} \sqrt {6+17 x+12 x^2}}\right )}{603979776 \sqrt {3}} \]
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Time = 0.06 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1016, 756, 654, 626, 635, 212} \[ \int (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx=-\frac {125455 \text {arctanh}\left (\frac {24 x+17}{4 \sqrt {3} \sqrt {12 x^2+17 x+6}}\right )}{603979776 \sqrt {3}}-\frac {1}{32} (10-3 x) \left (12 x^2+17 x+6\right )^{7/2}-\frac {873 \left (12 x^2+17 x+6\right )^{7/2}}{1792}+\frac {25091 (24 x+17) \left (12 x^2+17 x+6\right )^{5/2}}{24576}-\frac {125455 (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}}{4718592}+\frac {125455 (24 x+17) \sqrt {12 x^2+17 x+6}}{150994944} \]
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 756
Rule 1016
Rubi steps \begin{align*} \text {integral}& = \int (10-3 x)^2 \left (6+17 x+12 x^2\right )^{5/2} \, dx \\ & = -\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}+\frac {1}{96} \int \left (11331-\frac {7857 x}{2}\right ) \left (6+17 x+12 x^2\right )^{5/2} \, dx \\ & = -\frac {873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}+\frac {75273}{512} \int \left (6+17 x+12 x^2\right )^{5/2} \, dx \\ & = \frac {25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac {873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}-\frac {125455 \int \left (6+17 x+12 x^2\right )^{3/2} \, dx}{49152} \\ & = -\frac {125455 (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}}{4718592}+\frac {25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac {873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}+\frac {125455 \int \sqrt {6+17 x+12 x^2} \, dx}{3145728} \\ & = \frac {125455 (17+24 x) \sqrt {6+17 x+12 x^2}}{150994944}-\frac {125455 (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}}{4718592}+\frac {25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac {873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}-\frac {125455 \int \frac {1}{\sqrt {6+17 x+12 x^2}} \, dx}{301989888} \\ & = \frac {125455 (17+24 x) \sqrt {6+17 x+12 x^2}}{150994944}-\frac {125455 (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}}{4718592}+\frac {25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac {873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}-\frac {125455 \text {Subst}\left (\int \frac {1}{48-x^2} \, dx,x,\frac {17+24 x}{\sqrt {6+17 x+12 x^2}}\right )}{150994944} \\ & = \frac {125455 (17+24 x) \sqrt {6+17 x+12 x^2}}{150994944}-\frac {125455 (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}}{4718592}+\frac {25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac {873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}-\frac {125455 \tanh ^{-1}\left (\frac {17+24 x}{4 \sqrt {3} \sqrt {6+17 x+12 x^2}}\right )}{603979776 \sqrt {3}} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.60 \[ \int (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx=\frac {6 \sqrt {6+17 x+12 x^2} \left (474999091769+3132157281976 x+7899203409792 x^2+8974844476416 x^3+3438453030912 x^4-1190083166208 x^5-732816211968 x^6+171228266496 x^7\right )-878185 \sqrt {3} \text {arctanh}\left (\frac {2 \sqrt {2+\frac {17 x}{3}+4 x^2}}{3+4 x}\right )}{6341787648} \]
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Time = 0.66 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.54
method | result | size |
risch | \(\frac {\left (171228266496 x^{7}-732816211968 x^{6}-1190083166208 x^{5}+3438453030912 x^{4}+8974844476416 x^{3}+7899203409792 x^{2}+3132157281976 x +474999091769\right ) \sqrt {12 x^{2}+17 x +6}}{1056964608}-\frac {125455 \ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 x^{2}+17 x +6}\right ) \sqrt {12}}{3623878656}\) | \(80\) |
trager | \(\left (162 x^{7}-\frac {19413}{28} x^{6}-\frac {504423}{448} x^{5}+\frac {11659251}{3584} x^{4}+\frac {139118993}{16384} x^{3}+\frac {20570842213}{2752512} x^{2}+\frac {391519660247}{132120576} x +\frac {474999091769}{1056964608}\right ) \sqrt {12 x^{2}+17 x +6}+\frac {125455 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-24 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -17 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+12 \sqrt {12 x^{2}+17 x +6}\right )}{1811939328}\) | \(91\) |
default | \(\frac {125455 \left (17+24 x \right ) \sqrt {12 x^{2}+17 x +6}}{150994944}-\frac {125455 \ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 x^{2}+17 x +6}\right ) \sqrt {12}}{3623878656}+\frac {2473875847 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{33030144}+\frac {27 x^{5} \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{2}-\frac {8613 x^{4} \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{112}+\frac {14991 x^{3} \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{1792}+\frac {4267751 x^{2} \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{14336}+\frac {129220757 x \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{458752}\) | \(147\) |
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Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.59 \[ \int (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx=\frac {1}{1056964608} \, {\left (171228266496 \, x^{7} - 732816211968 \, x^{6} - 1190083166208 \, x^{5} + 3438453030912 \, x^{4} + 8974844476416 \, x^{3} + 7899203409792 \, x^{2} + 3132157281976 \, x + 474999091769\right )} \sqrt {12 \, x^{2} + 17 \, x + 6} + \frac {125455}{3623878656} \, \sqrt {3} \log \left (-8 \, \sqrt {3} \sqrt {12 \, x^{2} + 17 \, x + 6} {\left (24 \, x + 17\right )} + 1152 \, x^{2} + 1632 \, x + 577\right ) \]
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Time = 0.53 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.64 \[ \int (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx=\sqrt {12 x^{2} + 17 x + 6} \cdot \left (162 x^{7} - \frac {19413 x^{6}}{28} - \frac {504423 x^{5}}{448} + \frac {11659251 x^{4}}{3584} + \frac {139118993 x^{3}}{16384} + \frac {20570842213 x^{2}}{2752512} + \frac {391519660247 x}{132120576} + \frac {474999091769}{1056964608}\right ) - \frac {125455 \sqrt {3} \log {\left (24 x + 4 \sqrt {3} \sqrt {12 x^{2} + 17 x + 6} + 17 \right )}}{1811939328} \]
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Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.04 \[ \int (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx=\frac {27}{2} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x^{5} - \frac {8613}{112} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x^{4} + \frac {14991}{1792} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x^{3} + \frac {4267751}{14336} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x^{2} + \frac {129220757}{458752} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x + \frac {2473875847}{33030144} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} + \frac {125455}{6291456} \, \sqrt {12 \, x^{2} + 17 \, x + 6} x - \frac {125455}{1811939328} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {12 \, x^{2} + 17 \, x + 6} + 24 \, x + 17\right ) + \frac {2132735}{150994944} \, \sqrt {12 \, x^{2} + 17 \, x + 6} \]
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Time = 0.30 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.57 \[ \int (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx=\frac {1}{1056964608} \, {\left (8 \, {\left (48 \, {\left (24 \, {\left (96 \, {\left (24 \, {\left (48 \, {\left (168 \, x - 719\right )} x - 56047\right )} x + 3886417\right )} x + 973832951\right )} x + 20570842213\right )} x + 391519660247\right )} x + 474999091769\right )} \sqrt {12 \, x^{2} + 17 \, x + 6} + \frac {125455}{1811939328} \, \sqrt {3} \log \left ({\left | -4 \, \sqrt {3} {\left (2 \, \sqrt {3} x - \sqrt {12 \, x^{2} + 17 \, x + 6}\right )} - 17 \right |}\right ) \]
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Time = 13.92 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.26 \[ \int (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx=\frac {4267751\,x^2\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{14336}+\frac {14991\,x^3\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{1792}-\frac {8613\,x^4\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{112}+\frac {27\,x^5\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{2}-\frac {146030443\,\sqrt {12}\,\ln \left (\sqrt {12\,x^2+17\,x+6}+\frac {\sqrt {12}\,\left (12\,x+\frac {17}{2}\right )}{12}\right )}{88080384}+\frac {438091329\,\left (\frac {x}{2}+\frac {17}{48}\right )\,\sqrt {12\,x^2+17\,x+6}}{229376}+\frac {2473875847\,\sqrt {12\,x^2+17\,x+6}\,\left (1152\,x^2+408\,x-291\right )}{3170893824}+\frac {129220757\,x\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{458752}+\frac {42055889399\,\sqrt {12}\,\ln \left (2\,\sqrt {12\,x^2+17\,x+6}+\frac {\sqrt {12}\,\left (24\,x+17\right )}{12}\right )}{25367150592} \]
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