Integrand size = 40, antiderivative size = 518 \[ \int \frac {\sqrt {\left (-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right )^3}}{-1+x} \, dx=\frac {\left (-1245336401+4423205098 x-508033624 x^2-2700564848 x^3+2304529024 x^4-910869760 x^5+199009280 x^6-23296000 x^7+1146880 x^8\right ) \sqrt {-531441+531441 x+2480058 x^2-4704237 x^3-885735 x^4+9880866 x^5-10219851 x^6+592677 x^7+8764767 x^8-10819710 x^9+7498953 x^{10}-3554163 x^{11}+1221371 x^{12}-309774 x^{13}+57735 x^{14}-7717 x^{15}+702 x^{16}-39 x^{17}+x^{18}}}{10321920 (-3+x)^6 \left (-1-x+x^2\right )}+128 \arctan \left (\frac {\left (-1-x+x^2\right ) \left (729-1458 x+1215 x^2-540 x^3+135 x^4-18 x^5+x^6\right )}{729-1458 x-243 x^2+3105 x^3-3753 x^4+2277 x^5-809 x^6+171 x^7-20 x^8+x^9-\sqrt {-531441+531441 x+2480058 x^2-4704237 x^3-885735 x^4+9880866 x^5-10219851 x^6+592677 x^7+8764767 x^8-10819710 x^9+7498953 x^{10}-3554163 x^{11}+1221371 x^{12}-309774 x^{13}+57735 x^{14}-7717 x^{15}+702 x^{16}-39 x^{17}+x^{18}}}\right )-\frac {19451047 \log \left (-729+729 x+972 x^2-2133 x^3+1620 x^4-657 x^5+152 x^6-19 x^7+x^8\right )}{65536}+\frac {19451047 \log \left (-729+2187 x-486 x^2-4077 x^3+5886 x^4-3897 x^5+1466 x^6-323 x^7+39 x^8-2 x^9+2 \sqrt {-531441+531441 x+2480058 x^2-4704237 x^3-885735 x^4+9880866 x^5-10219851 x^6+592677 x^7+8764767 x^8-10819710 x^9+7498953 x^{10}-3554163 x^{11}+1221371 x^{12}-309774 x^{13}+57735 x^{14}-7717 x^{15}+702 x^{16}-39 x^{17}+x^{18}}\right )}{65536} \]
1/10321920*(1146880*x^8-23296000*x^7+199009280*x^6-910869760*x^5+230452902 4*x^4-2700564848*x^3-508033624*x^2+4423205098*x-1245336401)*(x^18-39*x^17+ 702*x^16-7717*x^15+57735*x^14-309774*x^13+1221371*x^12-3554163*x^11+749895 3*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)^(1/2)/(-3+x)^6/(x^2-x-1)+128 *arctan((x^2-x-1)*(x^6-18*x^5+135*x^4-540*x^3+1215*x^2-1458*x+729)/(729-14 58*x-243*x^2+3105*x^3-3753*x^4+2277*x^5-809*x^6+171*x^7-20*x^8+x^9-(x^18-3 9*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)^(1/2)))-19451047/6553 6*ln(x^8-19*x^7+152*x^6-657*x^5+1620*x^4-2133*x^3+972*x^2+729*x-729)+19451 047/65536*ln(-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*(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-5314 41)^(1/2))
Time = 0.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt {\left (-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right )^3}}{-1+x} \, dx=\frac {(-3+x)^6 \left (-1-x+x^2\right )^{3/2} \left (2 \sqrt {-1-x+x^2} \left (-1245336401+4423205098 x-508033624 x^2-2700564848 x^3+2304529024 x^4-910869760 x^5+199009280 x^6-23296000 x^7+1146880 x^8\right )+2642411520 \arctan \left (1-x+\sqrt {-1-x+x^2}\right )+6127079805 \log \left (1-2 x+2 \sqrt {-1-x+x^2}\right )\right )}{20643840 \sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3}} \]
((-3 + x)^6*(-1 - x + x^2)^(3/2)*(2*Sqrt[-1 - x + x^2]*(-1245336401 + 4423 205098*x - 508033624*x^2 - 2700564848*x^3 + 2304529024*x^4 - 910869760*x^5 + 199009280*x^6 - 23296000*x^7 + 1146880*x^8) + 2642411520*ArcTan[1 - x + Sqrt[-1 - x + x^2]] + 6127079805*Log[1 - 2*x + 2*Sqrt[-1 - x + x^2]]))/(2 0643840*Sqrt[(-3 + x)^12*(-1 - x + x^2)^3])
Time = 1.39 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.53, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {7239, 7270, 25, 1267, 27, 2184, 27, 2184, 27, 2184, 27, 2184, 27, 1231, 27, 1231, 27, 1269, 1092, 219, 1154, 217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {\left (x^6-13 x^5+65 x^4-150 x^3+135 x^2+27 x-81\right )^3}}{x-1} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {(x-3)^{12} \left (x^2-x-1\right )^3}}{x-1}dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \int -\frac {(3-x)^6 \left (x^2-x-1\right )^{3/2}}{1-x}dx}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \int \frac {(3-x)^6 \left (x^2-x-1\right )^{3/2}}{1-x}dx}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 1267 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{9} \int \frac {\left (x^2-x-1\right )^{3/2} \left (-229 x^5+2233 x^4-9522 x^3+21778 x^2-26233 x+13125\right )}{2 (1-x)}dx-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \int \frac {\left (x^2-x-1\right )^{3/2} \left (-229 x^5+2233 x^4-9522 x^3+21778 x^2-26233 x+13125\right )}{1-x}dx-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (-\frac {1}{8} \int -\frac {\left (x^2-x-1\right )^{3/2} \left (19927 x^4-127162 x^3+331044 x^2-415606 x+210229\right )}{2 (1-x)}dx-\frac {229}{8} \left (x^2-x-1\right )^{5/2} (1-x)^3\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \int \frac {\left (x^2-x-1\right )^{3/2} \left (19927 x^4-127162 x^3+331044 x^2-415606 x+210229\right )}{1-x}dx-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \left (\frac {1}{7} \int \frac {\left (x^2-x-1\right )^{3/2} \left (-843699 x^3+3578485 x^2-5400017 x+2923279\right )}{2 (1-x)}dx-\frac {19927}{7} (1-x)^2 \left (x^2-x-1\right )^{5/2}\right )-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \left (\frac {1}{14} \int \frac {\left (x^2-x-1\right )^{3/2} \left (-843699 x^3+3578485 x^2-5400017 x+2923279\right )}{1-x}dx-\frac {19927}{7} (1-x)^2 \left (x^2-x-1\right )^{5/2}\right )-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \left (\frac {1}{14} \left (-\frac {1}{6} \int -\frac {3 \left (x^2-x-1\right )^{3/2} \left (6158183 x^2-15975408 x+10849417\right )}{2 (1-x)}dx-\frac {281233}{2} (1-x) \left (x^2-x-1\right )^{5/2}\right )-\frac {19927}{7} (1-x)^2 \left (x^2-x-1\right )^{5/2}\right )-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \left (\frac {1}{14} \left (\frac {1}{4} \int \frac {\left (x^2-x-1\right )^{3/2} \left (6158183 x^2-15975408 x+10849417\right )}{1-x}dx-\frac {281233}{2} (1-x) \left (x^2-x-1\right )^{5/2}\right )-\frac {19927}{7} (1-x)^2 \left (x^2-x-1\right )^{5/2}\right )-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \left (\frac {1}{14} \left (\frac {1}{4} \left (\frac {1}{5} \int \frac {315 (246677-213909 x) \left (x^2-x-1\right )^{3/2}}{2 (1-x)}dx-\frac {6158183}{5} \left (x^2-x-1\right )^{5/2}\right )-\frac {281233}{2} (1-x) \left (x^2-x-1\right )^{5/2}\right )-\frac {19927}{7} (1-x)^2 \left (x^2-x-1\right )^{5/2}\right )-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \left (\frac {1}{14} \left (\frac {1}{4} \left (\frac {63}{2} \int \frac {(246677-213909 x) \left (x^2-x-1\right )^{3/2}}{1-x}dx-\frac {6158183}{5} \left (x^2-x-1\right )^{5/2}\right )-\frac {281233}{2} (1-x) \left (x^2-x-1\right )^{5/2}\right )-\frac {19927}{7} (1-x)^2 \left (x^2-x-1\right )^{5/2}\right )-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \left (\frac {1}{14} \left (\frac {1}{4} \left (\frac {63}{2} \left (-\frac {1}{8} \int \frac {(3995067-3470779 x) \sqrt {x^2-x-1}}{2 (1-x)}dx-\frac {1}{24} (903871-1283454 x) \left (x^2-x-1\right )^{3/2}\right )-\frac {6158183}{5} \left (x^2-x-1\right )^{5/2}\right )-\frac {281233}{2} (1-x) \left (x^2-x-1\right )^{5/2}\right )-\frac {19927}{7} (1-x)^2 \left (x^2-x-1\right )^{5/2}\right )-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \left (\frac {1}{14} \left (\frac {1}{4} \left (\frac {63}{2} \left (-\frac {1}{16} \int \frac {(3995067-3470779 x) \sqrt {x^2-x-1}}{1-x}dx-\frac {1}{24} (903871-1283454 x) \left (x^2-x-1\right )^{3/2}\right )-\frac {6158183}{5} \left (x^2-x-1\right )^{5/2}\right )-\frac {281233}{2} (1-x) \left (x^2-x-1\right )^{5/2}\right )-\frac {19927}{7} (1-x)^2 \left (x^2-x-1\right )^{5/2}\right )-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \left (\frac {1}{14} \left (\frac {1}{4} \left (\frac {63}{2} \left (\frac {1}{16} \left (\frac {1}{4} \int \frac {23645351-19451047 x}{2 (1-x) \sqrt {x^2-x-1}}dx+\frac {1}{4} \sqrt {x^2-x-1} (5567931-6941558 x)\right )-\frac {1}{24} (903871-1283454 x) \left (x^2-x-1\right )^{3/2}\right )-\frac {6158183}{5} \left (x^2-x-1\right )^{5/2}\right )-\frac {281233}{2} (1-x) \left (x^2-x-1\right )^{5/2}\right )-\frac {19927}{7} (1-x)^2 \left (x^2-x-1\right )^{5/2}\right )-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \left (\frac {1}{14} \left (\frac {1}{4} \left (\frac {63}{2} \left (\frac {1}{16} \left (\frac {1}{8} \int \frac {23645351-19451047 x}{(1-x) \sqrt {x^2-x-1}}dx+\frac {1}{4} \sqrt {x^2-x-1} (5567931-6941558 x)\right )-\frac {1}{24} (903871-1283454 x) \left (x^2-x-1\right )^{3/2}\right )-\frac {6158183}{5} \left (x^2-x-1\right )^{5/2}\right )-\frac {281233}{2} (1-x) \left (x^2-x-1\right )^{5/2}\right )-\frac {19927}{7} (1-x)^2 \left (x^2-x-1\right )^{5/2}\right )-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \left (\frac {1}{14} \left (\frac {1}{4} \left (\frac {63}{2} \left (\frac {1}{16} \left (\frac {1}{8} \left (19451047 \int \frac {1}{\sqrt {x^2-x-1}}dx+4194304 \int \frac {1}{(1-x) \sqrt {x^2-x-1}}dx\right )+\frac {1}{4} \sqrt {x^2-x-1} (5567931-6941558 x)\right )-\frac {1}{24} (903871-1283454 x) \left (x^2-x-1\right )^{3/2}\right )-\frac {6158183}{5} \left (x^2-x-1\right )^{5/2}\right )-\frac {281233}{2} (1-x) \left (x^2-x-1\right )^{5/2}\right )-\frac {19927}{7} (1-x)^2 \left (x^2-x-1\right )^{5/2}\right )-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \left (\frac {1}{14} \left (\frac {1}{4} \left (\frac {63}{2} \left (\frac {1}{16} \left (\frac {1}{8} \left (4194304 \int \frac {1}{(1-x) \sqrt {x^2-x-1}}dx+38902094 \int \frac {1}{4-\frac {(1-2 x)^2}{x^2-x-1}}d\left (-\frac {1-2 x}{\sqrt {x^2-x-1}}\right )\right )+\frac {1}{4} \sqrt {x^2-x-1} (5567931-6941558 x)\right )-\frac {1}{24} (903871-1283454 x) \left (x^2-x-1\right )^{3/2}\right )-\frac {6158183}{5} \left (x^2-x-1\right )^{5/2}\right )-\frac {281233}{2} (1-x) \left (x^2-x-1\right )^{5/2}\right )-\frac {19927}{7} (1-x)^2 \left (x^2-x-1\right )^{5/2}\right )-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \left (\frac {1}{14} \left (\frac {1}{4} \left (\frac {63}{2} \left (\frac {1}{16} \left (\frac {1}{8} \left (4194304 \int \frac {1}{(1-x) \sqrt {x^2-x-1}}dx-19451047 \text {arctanh}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )\right )+\frac {1}{4} \sqrt {x^2-x-1} (5567931-6941558 x)\right )-\frac {1}{24} (903871-1283454 x) \left (x^2-x-1\right )^{3/2}\right )-\frac {6158183}{5} \left (x^2-x-1\right )^{5/2}\right )-\frac {281233}{2} (1-x) \left (x^2-x-1\right )^{5/2}\right )-\frac {19927}{7} (1-x)^2 \left (x^2-x-1\right )^{5/2}\right )-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \left (\frac {1}{14} \left (\frac {1}{4} \left (\frac {63}{2} \left (\frac {1}{16} \left (\frac {1}{8} \left (-8388608 \int \frac {1}{-\frac {(3-x)^2}{x^2-x-1}-4}d\frac {3-x}{\sqrt {x^2-x-1}}-19451047 \text {arctanh}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )\right )+\frac {1}{4} \sqrt {x^2-x-1} (5567931-6941558 x)\right )-\frac {1}{24} (903871-1283454 x) \left (x^2-x-1\right )^{3/2}\right )-\frac {6158183}{5} \left (x^2-x-1\right )^{5/2}\right )-\frac {281233}{2} (1-x) \left (x^2-x-1\right )^{5/2}\right )-\frac {19927}{7} (1-x)^2 \left (x^2-x-1\right )^{5/2}\right )-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \left (\frac {1}{18} \left (\frac {1}{16} \left (\frac {1}{14} \left (\frac {1}{4} \left (\frac {63}{2} \left (\frac {1}{16} \left (\frac {1}{8} \left (4194304 \arctan \left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )-19451047 \text {arctanh}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )\right )+\frac {1}{4} \sqrt {x^2-x-1} (5567931-6941558 x)\right )-\frac {1}{24} (903871-1283454 x) \left (x^2-x-1\right )^{3/2}\right )-\frac {6158183}{5} \left (x^2-x-1\right )^{5/2}\right )-\frac {281233}{2} (1-x) \left (x^2-x-1\right )^{5/2}\right )-\frac {19927}{7} (1-x)^2 \left (x^2-x-1\right )^{5/2}\right )-\frac {229}{8} (1-x)^3 \left (x^2-x-1\right )^{5/2}\right )-\frac {1}{9} (1-x)^4 \left (x^2-x-1\right )^{5/2}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}\) |
-((Sqrt[-((3 - x)^12*(1 + x - x^2)^3)]*(-1/9*((1 - x)^4*(-1 - x + x^2)^(5/ 2)) + ((-229*(1 - x)^3*(-1 - x + x^2)^(5/2))/8 + ((-19927*(1 - x)^2*(-1 - x + x^2)^(5/2))/7 + ((-281233*(1 - x)*(-1 - x + x^2)^(5/2))/2 + ((-6158183 *(-1 - x + x^2)^(5/2))/5 + (63*(-1/24*((903871 - 1283454*x)*(-1 - x + x^2) ^(3/2)) + (((5567931 - 6941558*x)*Sqrt[-1 - x + x^2])/4 + (4194304*ArcTan[ (3 - x)/(2*Sqrt[-1 - x + x^2])] - 19451047*ArcTanh[(1 - 2*x)/(2*Sqrt[-1 - x + x^2])])/8)/16))/2)/4)/14)/16)/18))/((3 - x)^6*(-1 - x + x^2)^(3/2)))
3.31.84.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b *x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m + n + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d + e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g , m, p}, x] && IGtQ[n, 1] && IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c *d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGt Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
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 | \(\frac {\left (1146880 x^{8}-23296000 x^{7}+199009280 x^{6}-910869760 x^{5}+2304529024 x^{4}-2700564848 x^{3}-508033624 x^{2}+4423205098 x -1245336401\right ) \sqrt {x^{18}-39 x^{17}+702 x^{16}-7717 x^{15}+57735 x^{14}-309774 x^{13}+1221371 x^{12}-3554163 x^{11}+7498953 x^{10}-10819710 x^{9}+8764767 x^{8}+592677 x^{7}-10219851 x^{6}+9880866 x^{5}-885735 x^{4}-4704237 x^{3}+2480058 x^{2}+531441 x -531441}}{10321920 \left (-3+x \right )^{6} \left (x^{2}-x -1\right )}+\frac {19451047 \ln \left (-\frac {-729+2187 x -486 x^{2}-4077 x^{3}+5886 x^{4}-3897 x^{5}+1466 x^{6}-323 x^{7}+39 x^{8}-2 x^{9}+2 \sqrt {x^{18}-39 x^{17}+702 x^{16}-7717 x^{15}+57735 x^{14}-309774 x^{13}+1221371 x^{12}-3554163 x^{11}+7498953 x^{10}-10819710 x^{9}+8764767 x^{8}+592677 x^{7}-10219851 x^{6}+9880866 x^{5}-885735 x^{4}-4704237 x^{3}+2480058 x^{2}+531441 x -531441}}{\left (x^{2}-x -1\right ) \left (-3+x \right )^{6}}\right )}{65536}+64 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{9}+22 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{8}-209 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{7}+1113 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-3591 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}+6993 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-7371 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2187 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2916 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +2 \sqrt {x^{18}-39 x^{17}+702 x^{16}-7717 x^{15}+57735 x^{14}-309774 x^{13}+1221371 x^{12}-3554163 x^{11}+7498953 x^{10}-10819710 x^{9}+8764767 x^{8}+592677 x^{7}-10219851 x^{6}+9880866 x^{5}-885735 x^{4}-4704237 x^{3}+2480058 x^{2}+531441 x -531441}-2187 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right ) \left (x^{2}-x -1\right ) \left (-3+x \right )^{6}}\right )\) | \(534\) |
1/10321920/(x^2-x-1)*(1146880*x^8-23296000*x^7+199009280*x^6-910869760*x^5 +2304529024*x^4-2700564848*x^3-508033624*x^2+4423205098*x-1245336401)*((x^ 2-x-1)^3*(-3+x)^12)^(1/2)/(-3+x)^6+(-19451047/65536*ln(-1/2+x+(x^2-x-1)^(1 /2))+64*arctan(1/2*(-3+x)/((-1+x)^2-2+x)^(1/2)))*((x^2-x-1)^3*(-3+x)^12)^( 1/2)/(x^2-x-1)^(3/2)/(-3+x)^6
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 )}} \]
1/330301440*(4819349233*x^8 - 91567635427*x^7 + 732541083416*x^6 - 3166312 446081*x^5 + 7807345757460*x^4 - 10279671913989*x^3 + 4684407454476*x^2 + 42278584320*(x^8 - 19*x^7 + 152*x^6 - 657*x^5 + 1620*x^4 - 2133*x^3 + 972* x^2 + 729*x - 729)*arctan(-(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 + 74989 53*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)) + 98033276880*(x^8 - 19*x^7 + 152*x^6 - 657*x^5 + 1620*x^4 - 2133* x^3 + 972*x^2 + 729*x - 729)*log(-(2*x^9 - 39*x^8 + 323*x^7 - 1466*x^6 + 3 897*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 - 531 441) + 729)/(x^8 - 19*x^7 + 152*x^6 - 657*x^5 + 1620*x^4 - 2133*x^3 + 972* x^2 + 729*x - 729)) + 32*sqrt(x^18 - 39*x^17 + 702*x^16 - 7717*x^15 + 5773 5*x^14 - 309774*x^13 + 1221371*x^12 - 3554163*x^11 + 7498953*x^10 - 108197 10*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)*(1146880*x^8 - 23296...
\[ \int \frac {\sqrt {\left (-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right )^3}}{-1+x} \, dx=\int \frac {\sqrt {\left (x - 3\right )^{12} \left (x^{2} - x - 1\right )^{3}}}{x - 1}\, dx \]
\[ \int \frac {\sqrt {\left (-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right )^3}}{-1+x} \, dx=\int { \frac {\sqrt {{\left (x^{6} - 13 \, x^{5} + 65 \, x^{4} - 150 \, x^{3} + 135 \, x^{2} + 27 \, x - 81\right )}^{3}}}{x - 1} \,d x } \]
Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {\left (-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right )^3}}{-1+x} \, dx=\frac {1}{10321920} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, {\left (14 \, {\left (16 \, x - 325\right )} x + 38869\right )} x - 711617\right )} x + 18004133\right )} x - 168785303\right )} x - 63504203\right )} x + 2211602549\right )} x - 1245336401\right )} \sqrt {x^{2} - x - 1} + 128 \, \arctan \left (-x + \sqrt {x^{2} - x - 1} + 1\right ) + \frac {19451047}{65536} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1 \right |}\right ) \]
1/10321920*(2*(4*(2*(8*(10*(4*(14*(16*x - 325)*x + 38869)*x - 711617)*x + 18004133)*x - 168785303)*x - 63504203)*x + 2211602549)*x - 1245336401)*sqr t(x^2 - x - 1) + 128*arctan(-x + sqrt(x^2 - x - 1) + 1) + 19451047/65536*l og(abs(-2*x + 2*sqrt(x^2 - x - 1) + 1))
Timed out. \[ \int \frac {\sqrt {\left (-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right )^3}}{-1+x} \, dx=\int \frac {\sqrt {{\left (x^6-13\,x^5+65\,x^4-150\,x^3+135\,x^2+27\,x-81\right )}^3}}{x-1} \,d x \]