Integrand size = 34, antiderivative size = 139 \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^3 \left (30+31 x-12 x^2\right )^3} \, dx=-\frac {275+388 x}{294 (10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}}+\frac {738029+1042556 x}{8232 (10-3 x)^2 \sqrt {6+17 x+12 x^2}}-\frac {50555899 \sqrt {6+17 x+12 x^2}}{19361664 (10-3 x)^2}-\frac {1634466587 \sqrt {6+17 x+12 x^2}}{7589772288 (10-3 x)}+\frac {40325 \text {arctanh}\left (\frac {206+291 x}{84 \sqrt {6+17 x+12 x^2}}\right )}{637540872192} \]
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Time = 0.07 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {1016, 754, 836, 848, 820, 738, 212} \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^3 \left (30+31 x-12 x^2\right )^3} \, dx=\frac {40325 \text {arctanh}\left (\frac {291 x+206}{84 \sqrt {12 x^2+17 x+6}}\right )}{637540872192}-\frac {388 x+275}{294 (10-3 x)^2 \left (12 x^2+17 x+6\right )^{3/2}}-\frac {1634466587 \sqrt {12 x^2+17 x+6}}{7589772288 (10-3 x)}-\frac {50555899 \sqrt {12 x^2+17 x+6}}{19361664 (10-3 x)^2}+\frac {1042556 x+738029}{8232 (10-3 x)^2 \sqrt {12 x^2+17 x+6}} \]
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Rule 212
Rule 738
Rule 754
Rule 820
Rule 836
Rule 848
Rule 1016
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(10-3 x)^3 \left (6+17 x+12 x^2\right )^{5/2}} \, dx \\ & = -\frac {275+388 x}{294 (10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}}-\frac {\int \frac {\frac {109953}{2}-41904 x}{(10-3 x)^3 \left (6+17 x+12 x^2\right )^{3/2}} \, dx}{2646} \\ & = -\frac {275+388 x}{294 (10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}}+\frac {738029+1042556 x}{8232 (10-3 x)^2 \sqrt {6+17 x+12 x^2}}+\frac {\int \frac {-\frac {5020024653}{4}-1773387756 x}{(10-3 x)^3 \sqrt {6+17 x+12 x^2}} \, dx}{2333772} \\ & = -\frac {275+388 x}{294 (10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}}+\frac {738029+1042556 x}{8232 (10-3 x)^2 \sqrt {6+17 x+12 x^2}}-\frac {50555899 \sqrt {6+17 x+12 x^2}}{19361664 (10-3 x)^2}-\frac {\int \frac {\frac {1461036257541}{8}+257986752597 x}{(10-3 x)^2 \sqrt {6+17 x+12 x^2}} \, dx}{8233547616} \\ & = -\frac {275+388 x}{294 (10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}}+\frac {738029+1042556 x}{8232 (10-3 x)^2 \sqrt {6+17 x+12 x^2}}-\frac {50555899 \sqrt {6+17 x+12 x^2}}{19361664 (10-3 x)^2}-\frac {1634466587 \sqrt {6+17 x+12 x^2}}{7589772288 (10-3 x)}+\frac {40325 \int \frac {1}{(10-3 x) \sqrt {6+17 x+12 x^2}} \, dx}{15179544576} \\ & = -\frac {275+388 x}{294 (10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}}+\frac {738029+1042556 x}{8232 (10-3 x)^2 \sqrt {6+17 x+12 x^2}}-\frac {50555899 \sqrt {6+17 x+12 x^2}}{19361664 (10-3 x)^2}-\frac {1634466587 \sqrt {6+17 x+12 x^2}}{7589772288 (10-3 x)}-\frac {40325 \text {Subst}\left (\int \frac {1}{7056-x^2} \, dx,x,\frac {-206-291 x}{\sqrt {6+17 x+12 x^2}}\right )}{7589772288} \\ & = -\frac {275+388 x}{294 (10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}}+\frac {738029+1042556 x}{8232 (10-3 x)^2 \sqrt {6+17 x+12 x^2}}-\frac {50555899 \sqrt {6+17 x+12 x^2}}{19361664 (10-3 x)^2}-\frac {1634466587 \sqrt {6+17 x+12 x^2}}{7589772288 (10-3 x)}+\frac {40325 \tanh ^{-1}\left (\frac {206+291 x}{84 \sqrt {6+17 x+12 x^2}}\right )}{637540872192} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^3 \left (30+31 x-12 x^2\right )^3} \, dx=\frac {\sqrt {6+17 x+12 x^2} \left (2773753482408+10124325497244 x+9848047480070 x^2-1096520427663 x^3-3206824169544 x^4+706089565584 x^5\right )}{7589772288 (-10+3 x)^2 (2+3 x)^2 (3+4 x)^2}+\frac {40325 \text {arctanh}\left (\frac {6 \sqrt {6+17 x+12 x^2}}{7 (2+3 x)}\right )}{318770436096} \]
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Time = 0.81 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {706089565584 x^{5}-3206824169544 x^{4}-1096520427663 x^{3}+9848047480070 x^{2}+10124325497244 x +2773753482408}{7589772288 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}} \left (3 x -10\right )^{2}}+\frac {40325 \,\operatorname {arctanh}\left (\frac {\frac {206}{3}+97 x}{28 \sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}}\right )}{637540872192}\) | \(72\) |
trager | \(\frac {\left (706089565584 x^{5}-3206824169544 x^{4}-1096520427663 x^{3}+9848047480070 x^{2}+10124325497244 x +2773753482408\right ) \sqrt {12 x^{2}+17 x +6}}{7589772288 \left (36 x^{3}-69 x^{2}-152 x -60\right )^{2}}-\frac {40325 \ln \left (-\frac {84 \sqrt {12 x^{2}+17 x +6}-206-291 x}{3 x -10}\right )}{637540872192}\) | \(89\) |
default | \(-\frac {\left (12 \left (x +\frac {2}{3}\right )^{2}+x +\frac {2}{3}\right )^{\frac {3}{2}}}{2592 \left (x +\frac {2}{3}\right )^{3}}+\frac {47 \left (12 \left (x +\frac {2}{3}\right )^{2}+x +\frac {2}{3}\right )^{\frac {3}{2}}}{1152 \left (x +\frac {2}{3}\right )^{2}}-\frac {23 \sqrt {12 \left (x +\frac {2}{3}\right )^{2}+x +\frac {2}{3}}}{4608}-\frac {23 \ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 \left (x +\frac {2}{3}\right )^{2}+x +\frac {2}{3}}\right ) \sqrt {12}}{110592}+\frac {\left (12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}\right )^{\frac {3}{2}}}{79692609024 \left (x -\frac {10}{3}\right )^{2}}-\frac {1261 \left (12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}\right )^{\frac {3}{2}}}{31239502737408 \left (x -\frac {10}{3}\right )}-\frac {40325 \sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}}{8925572210688}+\frac {40325 \,\operatorname {arctanh}\left (\frac {\frac {206}{3}+97 x}{28 \sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}}\right )}{637540872192}+\frac {1261 \left (17+24 x \right ) \sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}}{62479005474816}-\frac {128 \left (12 \left (x +\frac {3}{4}\right )^{2}-x -\frac {3}{4}\right )^{\frac {3}{2}}}{352947 \left (x +\frac {3}{4}\right )^{3}}-\frac {570457 \ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}\right ) \sqrt {12}}{31239502737408}-\frac {230400 \left (12 \left (x +\frac {3}{4}\right )^{2}-x -\frac {3}{4}\right )^{\frac {3}{2}}}{5764801 \left (x +\frac {3}{4}\right )^{2}}-\frac {1410048 \sqrt {12 \left (x +\frac {3}{4}\right )^{2}-x -\frac {3}{4}}}{282475249}+\frac {58752 \ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 \left (x +\frac {3}{4}\right )^{2}-x -\frac {3}{4}}\right ) \sqrt {12}}{282475249}\) | \(306\) |
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Time = 0.27 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^3 \left (30+31 x-12 x^2\right )^3} \, dx=\frac {40325 \, {\left (1296 \, x^{6} - 4968 \, x^{5} - 6183 \, x^{4} + 16656 \, x^{3} + 31384 \, x^{2} + 18240 \, x + 3600\right )} \log \left (\frac {291 \, x + 84 \, \sqrt {12 \, x^{2} + 17 \, x + 6} + 206}{x}\right ) - 40325 \, {\left (1296 \, x^{6} - 4968 \, x^{5} - 6183 \, x^{4} + 16656 \, x^{3} + 31384 \, x^{2} + 18240 \, x + 3600\right )} \log \left (\frac {291 \, x - 84 \, \sqrt {12 \, x^{2} + 17 \, x + 6} + 206}{x}\right ) + 168 \, {\left (706089565584 \, x^{5} - 3206824169544 \, x^{4} - 1096520427663 \, x^{3} + 9848047480070 \, x^{2} + 10124325497244 \, x + 2773753482408\right )} \sqrt {12 \, x^{2} + 17 \, x + 6}}{1275081744384 \, {\left (1296 \, x^{6} - 4968 \, x^{5} - 6183 \, x^{4} + 16656 \, x^{3} + 31384 \, x^{2} + 18240 \, x + 3600\right )}} \]
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\[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^3 \left (30+31 x-12 x^2\right )^3} \, dx=- \int \frac {\sqrt {12 x^{2} + 17 x + 6}}{46656 x^{9} - 268272 x^{8} - 76788 x^{7} + 1703619 x^{6} + 1218456 x^{5} - 3669588 x^{4} - 6898688 x^{3} - 4903920 x^{2} - 1641600 x - 216000}\, dx \]
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\[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^3 \left (30+31 x-12 x^2\right )^3} \, dx=\int { -\frac {\sqrt {12 \, x^{2} + 17 \, x + 6}}{{\left (12 \, x^{2} - 31 \, x - 30\right )}^{3} {\left (3 \, x + 2\right )}^{3}} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.67 \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^3 \left (30+31 x-12 x^2\right )^3} \, dx=\frac {\sqrt {3} {\left (282273 \, \sqrt {3} {\left (2 \, \sqrt {3} x - \sqrt {12 \, x^{2} + 17 \, x + 6}\right )}^{3} - 11460924 \, {\left (2 \, \sqrt {3} x - \sqrt {12 \, x^{2} + 17 \, x + 6}\right )}^{2} - 37551180 \, \sqrt {3} {\left (2 \, \sqrt {3} x - \sqrt {12 \, x^{2} + 17 \, x + 6}\right )} - 83365264\right )}}{159385218048 \, {\left (3 \, {\left (2 \, \sqrt {3} x - \sqrt {12 \, x^{2} + 17 \, x + 6}\right )}^{2} - 40 \, \sqrt {3} {\left (2 \, \sqrt {3} x - \sqrt {12 \, x^{2} + 17 \, x + 6}\right )} - 188\right )}^{2}} + \frac {{\left (8 \, {\left (2860316794 \, x + 6078171227\right )} x + 34383350229\right )} x + 8090114146}{2213683584 \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}}} + \frac {40325}{637540872192} \, \log \left ({\left | -6 \, \sqrt {3} x + 20 \, \sqrt {3} + 3 \, \sqrt {12 \, x^{2} + 17 \, x + 6} + 42 \right |}\right ) - \frac {40325}{637540872192} \, \log \left ({\left | -6 \, \sqrt {3} x + 20 \, \sqrt {3} + 3 \, \sqrt {12 \, x^{2} + 17 \, x + 6} - 42 \right |}\right ) \]
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Timed out. \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^3 \left (30+31 x-12 x^2\right )^3} \, dx=\int \frac {\sqrt {12\,x^2+17\,x+6}}{{\left (3\,x+2\right )}^3\,{\left (-12\,x^2+31\,x+30\right )}^3} \,d x \]
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