Integrand size = 30, antiderivative size = 103 \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=-\frac {97 (17+24 x) \sqrt {6+17 x+12 x^2}}{24576}+\frac {97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac {97 \text {arctanh}\left (\frac {17+24 x}{4 \sqrt {3} \sqrt {6+17 x+12 x^2}}\right )}{98304 \sqrt {3}} \]
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Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1016, 654, 626, 635, 212} \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=\frac {97 \text {arctanh}\left (\frac {24 x+17}{4 \sqrt {3} \sqrt {12 x^2+17 x+6}}\right )}{98304 \sqrt {3}}-\frac {1}{20} \left (12 x^2+17 x+6\right )^{5/2}+\frac {97}{768} (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}-\frac {97 (24 x+17) \sqrt {12 x^2+17 x+6}}{24576} \]
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 1016
Rubi steps \begin{align*} \text {integral}& = \int (10-3 x) \left (6+17 x+12 x^2\right )^{3/2} \, dx \\ & = -\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac {97}{8} \int \left (6+17 x+12 x^2\right )^{3/2} \, dx \\ & = \frac {97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}-\frac {97}{512} \int \sqrt {6+17 x+12 x^2} \, dx \\ & = -\frac {97 (17+24 x) \sqrt {6+17 x+12 x^2}}{24576}+\frac {97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac {97 \int \frac {1}{\sqrt {6+17 x+12 x^2}} \, dx}{49152} \\ & = -\frac {97 (17+24 x) \sqrt {6+17 x+12 x^2}}{24576}+\frac {97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac {97 \text {Subst}\left (\int \frac {1}{48-x^2} \, dx,x,\frac {17+24 x}{\sqrt {6+17 x+12 x^2}}\right )}{24576} \\ & = -\frac {97 (17+24 x) \sqrt {6+17 x+12 x^2}}{24576}+\frac {97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac {97 \tanh ^{-1}\left (\frac {17+24 x}{4 \sqrt {3} \sqrt {6+17 x+12 x^2}}\right )}{98304 \sqrt {3}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.72 \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=\frac {6 \sqrt {6+17 x+12 x^2} \left (1353611+5455144 x+6837888 x^2+1963008 x^3-884736 x^4\right )+485 \sqrt {3} \text {arctanh}\left (\frac {2 \sqrt {2+\frac {17 x}{3}+4 x^2}}{3+4 x}\right )}{737280} \]
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Time = 0.53 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.63
method | result | size |
risch | \(-\frac {\left (884736 x^{4}-1963008 x^{3}-6837888 x^{2}-5455144 x -1353611\right ) \sqrt {12 x^{2}+17 x +6}}{122880}+\frac {97 \ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 x^{2}+17 x +6}\right ) \sqrt {12}}{589824}\) | \(65\) |
trager | \(\left (-\frac {36}{5} x^{4}+\frac {639}{40} x^{3}+\frac {17807}{320} x^{2}+\frac {681893}{15360} x +\frac {1353611}{122880}\right ) \sqrt {12 x^{2}+17 x +6}-\frac {97 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-24 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -17 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+12 \sqrt {12 x^{2}+17 x +6}\right )}{294912}\) | \(76\) |
default | \(-\frac {97 \left (17+24 x \right ) \sqrt {12 x^{2}+17 x +6}}{24576}+\frac {97 \ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 x^{2}+17 x +6}\right ) \sqrt {12}}{589824}+\frac {7093 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{3840}-\frac {3 x^{2} \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{5}+\frac {349 x \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{160}\) | \(96\) |
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Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.71 \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=-\frac {1}{122880} \, {\left (884736 \, x^{4} - 1963008 \, x^{3} - 6837888 \, x^{2} - 5455144 \, x - 1353611\right )} \sqrt {12 \, x^{2} + 17 \, x + 6} + \frac {97}{589824} \, \sqrt {3} \log \left (8 \, \sqrt {3} \sqrt {12 \, x^{2} + 17 \, x + 6} {\left (24 \, x + 17\right )} + 1152 \, x^{2} + 1632 \, x + 577\right ) \]
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Time = 0.52 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.74 \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=\sqrt {12 x^{2} + 17 x + 6} \left (- \frac {36 x^{4}}{5} + \frac {639 x^{3}}{40} + \frac {17807 x^{2}}{320} + \frac {681893 x}{15360} + \frac {1353611}{122880}\right ) + \frac {97 \sqrt {3} \log {\left (24 x + 4 \sqrt {3} \sqrt {12 x^{2} + 17 x + 6} + 17 \right )}}{294912} \]
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Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01 \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=-\frac {3}{5} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x^{2} + \frac {349}{160} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x + \frac {7093}{3840} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} - \frac {97}{1024} \, \sqrt {12 \, x^{2} + 17 \, x + 6} x + \frac {97}{294912} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {12 \, x^{2} + 17 \, x + 6} + 24 \, x + 17\right ) - \frac {1649}{24576} \, \sqrt {12 \, x^{2} + 17 \, x + 6} \]
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Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.68 \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=-\frac {1}{122880} \, {\left (8 \, {\left (48 \, {\left (72 \, {\left (32 \, x - 71\right )} x - 17807\right )} x - 681893\right )} x - 1353611\right )} \sqrt {12 \, x^{2} + 17 \, x + 6} - \frac {97}{294912} \, \sqrt {3} \log \left ({\left | -4 \, \sqrt {3} {\left (2 \, \sqrt {3} x - \sqrt {12 \, x^{2} + 17 \, x + 6}\right )} - 17 \right |}\right ) \]
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Time = 13.33 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.32 \[ \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx=\frac {3753\,\left (\frac {x}{2}+\frac {17}{48}\right )\,\sqrt {12\,x^2+17\,x+6}}{80}-\frac {417\,\sqrt {12}\,\ln \left (\sqrt {12\,x^2+17\,x+6}+\frac {\sqrt {12}\,\left (12\,x+\frac {17}{2}\right )}{12}\right )}{10240}-\frac {3\,x^2\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{5}+\frac {7093\,\sqrt {12\,x^2+17\,x+6}\,\left (1152\,x^2+408\,x-291\right )}{368640}+\frac {349\,x\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{160}+\frac {120581\,\sqrt {12}\,\ln \left (2\,\sqrt {12\,x^2+17\,x+6}+\frac {\sqrt {12}\,\left (24\,x+17\right )}{12}\right )}{2949120} \]
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