Integrand size = 40, antiderivative size = 518 \[ \int \frac {\sqrt {\left (-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right )^3}}{-1+x} \, dx=\frac {\left (-1245336401+4423205098 x-508033624 x^2-2700564848 x^3+2304529024 x^4-910869760 x^5+199009280 x^6-23296000 x^7+1146880 x^8\right ) \sqrt {-531441+531441 x+2480058 x^2-4704237 x^3-885735 x^4+9880866 x^5-10219851 x^6+592677 x^7+8764767 x^8-10819710 x^9+7498953 x^{10}-3554163 x^{11}+1221371 x^{12}-309774 x^{13}+57735 x^{14}-7717 x^{15}+702 x^{16}-39 x^{17}+x^{18}}}{10321920 (-3+x)^6 \left (-1-x+x^2\right )}+128 \arctan \left (\frac {\left (-1-x+x^2\right ) \left (729-1458 x+1215 x^2-540 x^3+135 x^4-18 x^5+x^6\right )}{729-1458 x-243 x^2+3105 x^3-3753 x^4+2277 x^5-809 x^6+171 x^7-20 x^8+x^9-\sqrt {-531441+531441 x+2480058 x^2-4704237 x^3-885735 x^4+9880866 x^5-10219851 x^6+592677 x^7+8764767 x^8-10819710 x^9+7498953 x^{10}-3554163 x^{11}+1221371 x^{12}-309774 x^{13}+57735 x^{14}-7717 x^{15}+702 x^{16}-39 x^{17}+x^{18}}}\right )-\frac {19451047 \log \left (-729+729 x+972 x^2-2133 x^3+1620 x^4-657 x^5+152 x^6-19 x^7+x^8\right )}{65536}+\frac {19451047 \log \left (-729+2187 x-486 x^2-4077 x^3+5886 x^4-3897 x^5+1466 x^6-323 x^7+39 x^8-2 x^9+2 \sqrt {-531441+531441 x+2480058 x^2-4704237 x^3-885735 x^4+9880866 x^5-10219851 x^6+592677 x^7+8764767 x^8-10819710 x^9+7498953 x^{10}-3554163 x^{11}+1221371 x^{12}-309774 x^{13}+57735 x^{14}-7717 x^{15}+702 x^{16}-39 x^{17}+x^{18}}\right )}{65536} \]
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Time = 0.60 (sec) , antiderivative size = 459, normalized size of antiderivative = 0.89, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {6820, 6851, 1667, 828, 857, 635, 212, 738, 210} \[ \int \frac {\sqrt {\left (-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right )^3}}{-1+x} \, dx=-\frac {64 \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \arctan \left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}+\frac {19451047 \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \text {arctanh}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )}{65536 (3-x)^6 \left (x^2-x-1\right )^{3/2}}-\frac {\left (-x^2+x+1\right ) \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} (1-x)^4}{9 (3-x)^6}-\frac {229 \left (-x^2+x+1\right ) \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} (1-x)^3}{144 (3-x)^6}-\frac {19927 \left (-x^2+x+1\right ) \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} (1-x)^2}{2016 (3-x)^6}-\frac {281233 \left (-x^2+x+1\right ) \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} (1-x)}{8064 (3-x)^6}-\frac {6158183 \left (-x^2+x+1\right ) \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3}}{80640 (3-x)^6}+\frac {(5567931-6941558 x) \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3}}{32768 (3-x)^6 \left (-x^2+x+1\right )}+\frac {(903871-1283454 x) \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3}}{12288 (3-x)^6} \]
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Rule 210
Rule 212
Rule 635
Rule 738
Rule 828
Rule 857
Rule 1667
Rule 6820
Rule 6851
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3}}{-1+x} \, dx \\ & = \frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {(-3+x)^6 \left (-1-x+x^2\right )^{3/2}}{-1+x} \, dx}{(-3+x)^6 \left (-1-x+x^2\right )^{3/2}} \\ & = -\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}+\frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {\left (-1-x+x^2\right )^{3/2} \left (\frac {13125}{2}-\frac {26233 x}{2}+10889 x^2-4761 x^3+\frac {2233 x^4}{2}-\frac {229 x^5}{2}\right )}{-1+x} \, dx}{9 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}} \\ & = -\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}+\frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {\left (-1-x+x^2\right )^{3/2} \left (\frac {210229}{4}-\frac {207803 x}{2}+82761 x^2-\frac {63581 x^3}{2}+\frac {19927 x^4}{4}\right )}{-1+x} \, dx}{72 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}} \\ & = -\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}+\frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {\left (-1-x+x^2\right )^{3/2} \left (\frac {2923279}{8}-\frac {5400017 x}{8}+\frac {3578485 x^2}{8}-\frac {843699 x^3}{8}\right )}{-1+x} \, dx}{504 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}} \\ & = -\frac {281233 (1-x) \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{8064 (3-x)^6}-\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}+\frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {\left (-1-x+x^2\right )^{3/2} \left (\frac {32548251}{16}-2995389 x+\frac {18474549 x^2}{16}\right )}{-1+x} \, dx}{3024 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}} \\ & = -\frac {6158183 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{80640 (3-x)^6}-\frac {281233 (1-x) \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{8064 (3-x)^6}-\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}+\frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {\left (\frac {233109765}{32}-\frac {202144005 x}{32}\right ) \left (-1-x+x^2\right )^{3/2}}{-1+x} \, dx}{15120 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}} \\ & = \frac {(903871-1283454 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{12288 (3-x)^6}-\frac {6158183 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{80640 (3-x)^6}-\frac {281233 (1-x) \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{8064 (3-x)^6}-\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}-\frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {\left (\frac {3775338315}{64}-\frac {3279886155 x}{64}\right ) \sqrt {-1-x+x^2}}{-1+x} \, dx}{120960 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}} \\ & = \frac {(903871-1283454 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{12288 (3-x)^6}+\frac {(5567931-6941558 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{32768 (3-x)^6 \left (1+x-x^2\right )}-\frac {6158183 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{80640 (3-x)^6}-\frac {281233 (1-x) \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{8064 (3-x)^6}-\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}+\frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {\frac {22344856695}{128}-\frac {18381239415 x}{128}}{(-1+x) \sqrt {-1-x+x^2}} \, dx}{483840 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}} \\ & = \frac {(903871-1283454 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{12288 (3-x)^6}+\frac {(5567931-6941558 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{32768 (3-x)^6 \left (1+x-x^2\right )}-\frac {6158183 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{80640 (3-x)^6}-\frac {281233 (1-x) \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{8064 (3-x)^6}-\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}+\frac {\left (64 \sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3}\right ) \int \frac {1}{(-1+x) \sqrt {-1-x+x^2}} \, dx}{(-3+x)^6 \left (-1-x+x^2\right )^{3/2}}-\frac {\left (19451047 \sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3}\right ) \int \frac {1}{\sqrt {-1-x+x^2}} \, dx}{65536 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}} \\ & = \frac {(903871-1283454 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{12288 (3-x)^6}+\frac {(5567931-6941558 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{32768 (3-x)^6 \left (1+x-x^2\right )}-\frac {6158183 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{80640 (3-x)^6}-\frac {281233 (1-x) \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{8064 (3-x)^6}-\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}-\frac {\left (128 \sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3}\right ) \text {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {-3+x}{\sqrt {-1-x+x^2}}\right )}{(-3+x)^6 \left (-1-x+x^2\right )^{3/2}}-\frac {\left (19451047 \sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3}\right ) \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x}{\sqrt {-1-x+x^2}}\right )}{32768 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}} \\ & = \frac {(903871-1283454 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{12288 (3-x)^6}+\frac {(5567931-6941558 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{32768 (3-x)^6 \left (1+x-x^2\right )}-\frac {6158183 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{80640 (3-x)^6}-\frac {281233 (1-x) \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{8064 (3-x)^6}-\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}-\frac {64 \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3} \arctan \left (\frac {3-x}{2 \sqrt {-1-x+x^2}}\right )}{(3-x)^6 \left (-1-x+x^2\right )^{3/2}}+\frac {19451047 \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3} \text {arctanh}\left (\frac {1-2 x}{2 \sqrt {-1-x+x^2}}\right )}{65536 (3-x)^6 \left (-1-x+x^2\right )^{3/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt {\left (-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right )^3}}{-1+x} \, dx=\frac {(-3+x)^6 \left (-1-x+x^2\right )^{3/2} \left (2 \sqrt {-1-x+x^2} \left (-1245336401+4423205098 x-508033624 x^2-2700564848 x^3+2304529024 x^4-910869760 x^5+199009280 x^6-23296000 x^7+1146880 x^8\right )+2642411520 \arctan \left (1-x+\sqrt {-1-x+x^2}\right )+6127079805 \log \left (1-2 x+2 \sqrt {-1-x+x^2}\right )\right )}{20643840 \sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3}} \]
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Time = 33.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.28
method | result | size |
risch | \(\frac {\left (1146880 x^{8}-23296000 x^{7}+199009280 x^{6}-910869760 x^{5}+2304529024 x^{4}-2700564848 x^{3}-508033624 x^{2}+4423205098 x -1245336401\right ) \sqrt {\left (x^{2}-x -1\right )^{3} \left (-3+x \right )^{12}}}{10321920 \left (x^{2}-x -1\right ) \left (-3+x \right )^{6}}+\frac {\left (-\frac {19451047 \ln \left (-\frac {1}{2}+x +\sqrt {x^{2}-x -1}\right )}{65536}+64 \arctan \left (\frac {-3+x}{2 \sqrt {\left (-1+x \right )^{2}-2+x}}\right )\right ) \sqrt {\left (x^{2}-x -1\right )^{3} \left (-3+x \right )^{12}}}{\left (x^{2}-x -1\right )^{\frac {3}{2}} \left (-3+x \right )^{6}}\) | \(146\) |
default | \(-\frac {\sqrt {\left (x^{6}-13 x^{5}+65 x^{4}-150 x^{3}+135 x^{2}+27 x -81\right )^{3}}\, \left (-2293760 x^{4} \left (x^{2}-x -1\right )^{\frac {5}{2}}+42004480 x^{3} \left (x^{2}-x -1\right )^{\frac {5}{2}}-316303360 x^{2} \left (x^{2}-x -1\right )^{\frac {5}{2}}+1235724800 x \left (x^{2}-x -1\right )^{\frac {5}{2}}-2535627008 \left (x^{2}-x -1\right )^{\frac {5}{2}}+2156202720 x \left (x^{2}-x -1\right )^{\frac {3}{2}}-1518503280 \left (x^{2}-x -1\right )^{\frac {3}{2}}-4373181540 x \sqrt {x^{2}-x -1}+6127079805 \ln \left (-\frac {1}{2}+x +\sqrt {x^{2}-x -1}\right )-1321205760 \arctan \left (\frac {-3+x}{2 \sqrt {x^{2}-x -1}}\right )+3507796530 \sqrt {x^{2}-x -1}\right )}{20643840 \left (-3+x \right )^{6} \left (x^{2}-x -1\right )^{\frac {3}{2}}}\) | \(205\) |
trager | \(\frac {\left (1146880 x^{8}-23296000 x^{7}+199009280 x^{6}-910869760 x^{5}+2304529024 x^{4}-2700564848 x^{3}-508033624 x^{2}+4423205098 x -1245336401\right ) \sqrt {x^{18}-39 x^{17}+702 x^{16}-7717 x^{15}+57735 x^{14}-309774 x^{13}+1221371 x^{12}-3554163 x^{11}+7498953 x^{10}-10819710 x^{9}+8764767 x^{8}+592677 x^{7}-10219851 x^{6}+9880866 x^{5}-885735 x^{4}-4704237 x^{3}+2480058 x^{2}+531441 x -531441}}{10321920 \left (-3+x \right )^{6} \left (x^{2}-x -1\right )}+\frac {19451047 \ln \left (-\frac {-729+2187 x -486 x^{2}-4077 x^{3}+5886 x^{4}-3897 x^{5}+1466 x^{6}-323 x^{7}+39 x^{8}-2 x^{9}+2 \sqrt {x^{18}-39 x^{17}+702 x^{16}-7717 x^{15}+57735 x^{14}-309774 x^{13}+1221371 x^{12}-3554163 x^{11}+7498953 x^{10}-10819710 x^{9}+8764767 x^{8}+592677 x^{7}-10219851 x^{6}+9880866 x^{5}-885735 x^{4}-4704237 x^{3}+2480058 x^{2}+531441 x -531441}}{\left (x^{2}-x -1\right ) \left (-3+x \right )^{6}}\right )}{65536}+64 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{9}+22 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{8}-209 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{7}+1113 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-3591 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}+6993 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-7371 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2187 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2916 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +2 \sqrt {x^{18}-39 x^{17}+702 x^{16}-7717 x^{15}+57735 x^{14}-309774 x^{13}+1221371 x^{12}-3554163 x^{11}+7498953 x^{10}-10819710 x^{9}+8764767 x^{8}+592677 x^{7}-10219851 x^{6}+9880866 x^{5}-885735 x^{4}-4704237 x^{3}+2480058 x^{2}+531441 x -531441}-2187 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right ) \left (x^{2}-x -1\right ) \left (-3+x \right )^{6}}\right )\) | \(534\) |
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Time = 0.26 (sec) , antiderivative size = 652, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {\left (-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right )^3}}{-1+x} \, dx=\frac {4819349233 \, x^{8} - 91567635427 \, x^{7} + 732541083416 \, x^{6} - 3166312446081 \, x^{5} + 7807345757460 \, x^{4} - 10279671913989 \, x^{3} + 4684407454476 \, x^{2} + 42278584320 \, {\left (x^{8} - 19 \, x^{7} + 152 \, x^{6} - 657 \, x^{5} + 1620 \, x^{4} - 2133 \, x^{3} + 972 \, x^{2} + 729 \, x - 729\right )} \arctan \left (-\frac {x^{9} - 20 \, x^{8} + 171 \, x^{7} - 809 \, x^{6} + 2277 \, x^{5} - 3753 \, x^{4} + 3105 \, x^{3} - 243 \, x^{2} - 1458 \, x - \sqrt {x^{18} - 39 \, x^{17} + 702 \, x^{16} - 7717 \, x^{15} + 57735 \, x^{14} - 309774 \, x^{13} + 1221371 \, x^{12} - 3554163 \, x^{11} + 7498953 \, x^{10} - 10819710 \, x^{9} + 8764767 \, x^{8} + 592677 \, x^{7} - 10219851 \, x^{6} + 9880866 \, x^{5} - 885735 \, x^{4} - 4704237 \, x^{3} + 2480058 \, x^{2} + 531441 \, x - 531441} + 729}{x^{8} - 19 \, x^{7} + 152 \, x^{6} - 657 \, x^{5} + 1620 \, x^{4} - 2133 \, x^{3} + 972 \, x^{2} + 729 \, x - 729}\right ) + 98033276880 \, {\left (x^{8} - 19 \, x^{7} + 152 \, x^{6} - 657 \, x^{5} + 1620 \, x^{4} - 2133 \, x^{3} + 972 \, x^{2} + 729 \, x - 729\right )} \log \left (-\frac {2 \, x^{9} - 39 \, x^{8} + 323 \, x^{7} - 1466 \, x^{6} + 3897 \, x^{5} - 5886 \, x^{4} + 4077 \, x^{3} + 486 \, x^{2} - 2187 \, x - 2 \, \sqrt {x^{18} - 39 \, x^{17} + 702 \, x^{16} - 7717 \, x^{15} + 57735 \, x^{14} - 309774 \, x^{13} + 1221371 \, x^{12} - 3554163 \, x^{11} + 7498953 \, x^{10} - 10819710 \, x^{9} + 8764767 \, x^{8} + 592677 \, x^{7} - 10219851 \, x^{6} + 9880866 \, x^{5} - 885735 \, x^{4} - 4704237 \, x^{3} + 2480058 \, x^{2} + 531441 \, x - 531441} + 729}{x^{8} - 19 \, x^{7} + 152 \, x^{6} - 657 \, x^{5} + 1620 \, x^{4} - 2133 \, x^{3} + 972 \, x^{2} + 729 \, x - 729}\right ) + 32 \, \sqrt {x^{18} - 39 \, x^{17} + 702 \, x^{16} - 7717 \, x^{15} + 57735 \, x^{14} - 309774 \, x^{13} + 1221371 \, x^{12} - 3554163 \, x^{11} + 7498953 \, x^{10} - 10819710 \, x^{9} + 8764767 \, x^{8} + 592677 \, x^{7} - 10219851 \, x^{6} + 9880866 \, x^{5} - 885735 \, x^{4} - 4704237 \, x^{3} + 2480058 \, x^{2} + 531441 \, x - 531441} {\left (1146880 \, x^{8} - 23296000 \, x^{7} + 199009280 \, x^{6} - 910869760 \, x^{5} + 2304529024 \, x^{4} - 2700564848 \, x^{3} - 508033624 \, x^{2} + 4423205098 \, x - 1245336401\right )} + 3513305590857 \, x - 3513305590857}{330301440 \, {\left (x^{8} - 19 \, x^{7} + 152 \, x^{6} - 657 \, x^{5} + 1620 \, x^{4} - 2133 \, x^{3} + 972 \, x^{2} + 729 \, x - 729\right )}} \]
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\[ \int \frac {\sqrt {\left (-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right )^3}}{-1+x} \, dx=\int \frac {\sqrt {\left (x - 3\right )^{12} \left (x^{2} - x - 1\right )^{3}}}{x - 1}\, dx \]
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\[ \int \frac {\sqrt {\left (-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right )^3}}{-1+x} \, dx=\int { \frac {\sqrt {{\left (x^{6} - 13 \, x^{5} + 65 \, x^{4} - 150 \, x^{3} + 135 \, x^{2} + 27 \, x - 81\right )}^{3}}}{x - 1} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {\left (-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right )^3}}{-1+x} \, dx=\frac {1}{10321920} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, {\left (14 \, {\left (16 \, x - 325\right )} x + 38869\right )} x - 711617\right )} x + 18004133\right )} x - 168785303\right )} x - 63504203\right )} x + 2211602549\right )} x - 1245336401\right )} \sqrt {x^{2} - x - 1} + 128 \, \arctan \left (-x + \sqrt {x^{2} - x - 1} + 1\right ) + \frac {19451047}{65536} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1 \right |}\right ) \]
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Timed out. \[ \int \frac {\sqrt {\left (-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right )^3}}{-1+x} \, dx=\int \frac {\sqrt {{\left (x^6-13\,x^5+65\,x^4-150\,x^3+135\,x^2+27\,x-81\right )}^3}}{x-1} \,d x \]
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