Integrand size = 23, antiderivative size = 378 \[ \int \frac {1}{\left (3-2 x+x^2\right )^{11/2} \left (1+x+2 x^2\right )^5} \, dx=-\frac {3450497-2004270 x}{123480000 \left (3-2 x+x^2\right )^{9/2}}-\frac {4878869-2578034 x}{411600000 \left (3-2 x+x^2\right )^{7/2}}-\frac {30316369-15043110 x}{6860000000 \left (3-2 x+x^2\right )^{5/2}}-\frac {63043297-29625922 x}{41160000000 \left (3-2 x+x^2\right )^{3/2}}-\frac {31 (7434109-3088870 x)}{411600000000 \sqrt {3-2 x+x^2}}-\frac {1-10 x}{280 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^4}+\frac {28+67 x}{1050 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^3}+\frac {5485+8878 x}{117600 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^2}+\frac {3 (8822+8233 x)}{343000 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )}+\frac {\sqrt {\frac {1}{70} \left (151363871237318045+110320475741093888 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {5}{7 \left (151363871237318045+110320475741093888 \sqrt {2}\right )}} \left (308108167+312239803 \sqrt {2}+\left (932587773+620347970 \sqrt {2}\right ) x\right )}{\sqrt {3-2 x+x^2}}\right )}{137200000000}-\frac {\sqrt {\frac {1}{70} \left (-151363871237318045+110320475741093888 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {5}{7 \left (-151363871237318045+110320475741093888 \sqrt {2}\right )}} \left (308108167-312239803 \sqrt {2}+\left (932587773-620347970 \sqrt {2}\right ) x\right )}{\sqrt {3-2 x+x^2}}\right )}{137200000000} \] Output:
1/123480000*(-3450497+2004270*x)/(x^2-2*x+3)^(9/2)+1/411600000*(-4878869+2 578034*x)/(x^2-2*x+3)^(7/2)+1/6860000000*(-30316369+15043110*x)/(x^2-2*x+3 )^(5/2)+1/41160000000*(-63043297+29625922*x)/(x^2-2*x+3)^(3/2)+1/280*(-1+1 0*x)/(x^2-2*x+3)^(9/2)/(2*x^2+x+1)^4+1/1050*(28+67*x)/(x^2-2*x+3)^(9/2)/(2 *x^2+x+1)^3+1/117600*(5485+8878*x)/(x^2-2*x+3)^(9/2)/(2*x^2+x+1)^2+3/34300 0*(8822+8233*x)/(x^2-2*x+3)^(9/2)/(2*x^2+x+1)-31/411600000000*(7434109-308 8870*x)/(x^2-2*x+3)^(1/2)-1/9604000000000*arctanh(1/7*(308108167+x*(932587 773-620347970*2^(1/2))-312239803*2^(1/2))*35^(1/2)/(-151363871237318045+11 0320475741093888*2^(1/2))^(1/2)/(x^2-2*x+3)^(1/2))*(-10595470986612263150+ 7722433301876572160*2^(1/2))^(1/2)+1/9604000000000*arctan(1/7*(308108167+3 12239803*2^(1/2)+x*(932587773+620347970*2^(1/2)))*35^(1/2)/(15136387123731 8045+110320475741093888*2^(1/2))^(1/2)/(x^2-2*x+3)^(1/2))*(105954709866122 63150+7722433301876572160*2^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 2.58 (sec) , antiderivative size = 733, normalized size of antiderivative = 1.94 \[ \int \frac {1}{\left (3-2 x+x^2\right )^{11/2} \left (1+x+2 x^2\right )^5} \, dx =\text {Too large to display} \] Input:
Integrate[1/((3 - 2*x + x^2)^(11/2)*(1 + x + 2*x^2)^5),x]
Output:
((-53205422447 + 261702502714*x - 266966654968*x^2 + 1002897791524*x^3 - 1 409335257371*x^4 + 2503427226914*x^5 - 3359813871472*x^6 + 4591320676952*x ^7 - 5134334619701*x^8 + 5380603084494*x^9 - 4915797913008*x^10 + 39996561 32532*x^11 - 2679143870481*x^12 + 1459208021718*x^13 - 606785954952*x^14 + 188603773872*x^15 - 38639385552*x^16 + 4596238560*x^17)/((3 - 2*x + x^2)^ (9/2)*(1 + x + 2*x^2)^4) - 49392*RootSum[14 + 7*#1 - 5*#1^2 - #1^3 + #1^4 & , (-6014*Log[-x + Sqrt[3 - 2*x + x^2] - #1] - 10727*Log[-x + Sqrt[3 - 2* x + x^2] - #1]*#1 + 3229*Log[-x + Sqrt[3 - 2*x + x^2] - #1]*#1^2)/(7 - 10* #1 - 3*#1^2 + 4*#1^3) & ] - 56448*RootSum[14 + 7*#1 - 5*#1^2 - #1^3 + #1^4 & , (73781*Log[-x + Sqrt[3 - 2*x + x^2] - #1] - 60407*Log[-x + Sqrt[3 - 2 *x + x^2] - #1]*#1 + 13104*Log[-x + Sqrt[3 - 2*x + x^2] - #1]*#1^2)/(7 - 1 0*#1 - 3*#1^2 + 4*#1^3) & ] - 504*RootSum[14 + 7*#1 - 5*#1^2 - #1^3 + #1^4 & , (275935046*Log[-x + Sqrt[3 - 2*x + x^2] - #1] - 208696097*Log[-x + Sq rt[3 - 2*x + x^2] - #1]*#1 + 50007219*Log[-x + Sqrt[3 - 2*x + x^2] - #1]*# 1^2)/(7 - 10*#1 - 3*#1^2 + 4*#1^3) & ] + 1440*RootSum[14 + 7*#1 - 5*#1^2 - #1^3 + #1^4 & , (3276009822*Log[-x + Sqrt[3 - 2*x + x^2] - #1] - 24478316 21*Log[-x + Sqrt[3 - 2*x + x^2] - #1]*#1 + 590084719*Log[-x + Sqrt[3 - 2*x + x^2] - #1]*#1^2)/(7 - 10*#1 - 3*#1^2 + 4*#1^3) & ] - 18*RootSum[14 + 7* #1 - 5*#1^2 - #1^3 + #1^4 & , (254137663854*Log[-x + Sqrt[3 - 2*x + x^2] - #1] - 189631531133*Log[-x + Sqrt[3 - 2*x + x^2] - #1]*#1 + 45801521671...
Time = 1.33 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.12, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {1305, 27, 2135, 27, 2135, 27, 2135, 27, 2135, 27, 2135, 27, 2135, 27, 2135, 27, 2135, 27, 1368, 27, 1362, 217, 219}
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.
| \(\displaystyle \int \frac {1}{\left (x^2-2 x+3\right )^{11/2} \left (2 x^2+x+1\right )^5} \, dx\) |
| \(\Big \downarrow \) 1305 |
| \(\displaystyle -\frac {\int -\frac {5 \left (160 x^2-267 x+247\right )}{\left (x^2-2 x+3\right )^{11/2} \left (2 x^2+x+1\right )^4}dx}{1400}-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{280} \int \frac {160 x^2-267 x+247}{\left (x^2-2 x+3\right )^{11/2} \left (2 x^2+x+1\right )^4}dx-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{280} \left (\frac {\int \frac {70 \left (3752 x^2-3814 x+2901\right )}{\left (x^2-2 x+3\right )^{11/2} \left (2 x^2+x+1\right )^3}dx}{1050}+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \int \frac {3752 x^2-3814 x+2901}{\left (x^2-2 x+3\right )^{11/2} \left (2 x^2+x+1\right )^3}dx+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {1}{700} \int \frac {75 \left (35512 x^2-23581 x+12713\right )}{\left (x^2-2 x+3\right )^{11/2} \left (2 x^2+x+1\right )^2}dx+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \int \frac {35512 x^2-23581 x+12713}{\left (x^2-2 x+3\right )^{11/2} \left (2 x^2+x+1\right )^2}dx+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{350} \int -\frac {10 \left (-987960 x^2-28350 x+486617\right )}{\left (x^2-2 x+3\right )^{11/2} \left (2 x^2+x+1\right )}dx+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}-\frac {1}{35} \int \frac {-987960 x^2-28350 x+486617}{\left (x^2-2 x+3\right )^{11/2} \left (2 x^2+x+1\right )}dx\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{35} \left (-\frac {\int \frac {60 \left (-10689440 x^2+3332642 x+2083763\right )}{\left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}dx}{1800}-\frac {3450497-2004270 x}{90 \left (x^2-2 x+3\right )^{9/2}}\right )+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{35} \left (-\frac {1}{30} \int \frac {-10689440 x^2+3332642 x+2083763}{\left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}dx-\frac {3450497-2004270 x}{90 \left (x^2-2 x+3\right )^{9/2}}\right )+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{35} \left (\frac {1}{30} \left (-\frac {\int -\frac {420 \left (10312136 x^2-5581162 x+620039\right )}{\left (x^2-2 x+3\right )^{7/2} \left (2 x^2+x+1\right )}dx}{1400}-\frac {4878869-2578034 x}{10 \left (x^2-2 x+3\right )^{7/2}}\right )-\frac {3450497-2004270 x}{90 \left (x^2-2 x+3\right )^{9/2}}\right )+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{35} \left (\frac {1}{30} \left (\frac {3}{10} \int \frac {10312136 x^2-5581162 x+620039}{\left (x^2-2 x+3\right )^{7/2} \left (2 x^2+x+1\right )}dx-\frac {4878869-2578034 x}{10 \left (x^2-2 x+3\right )^{7/2}}\right )-\frac {3450497-2004270 x}{90 \left (x^2-2 x+3\right )^{9/2}}\right )+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{35} \left (\frac {1}{30} \left (\frac {3}{10} \left (\frac {\int \frac {100 \left (24068976 x^2-18512030 x+9163631\right )}{\left (x^2-2 x+3\right )^{5/2} \left (2 x^2+x+1\right )}dx}{1000}-\frac {30316369-15043110 x}{50 \left (x^2-2 x+3\right )^{5/2}}\right )-\frac {4878869-2578034 x}{10 \left (x^2-2 x+3\right )^{7/2}}\right )-\frac {3450497-2004270 x}{90 \left (x^2-2 x+3\right )^{9/2}}\right )+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{35} \left (\frac {1}{30} \left (\frac {3}{10} \left (\frac {1}{10} \int \frac {24068976 x^2-18512030 x+9163631}{\left (x^2-2 x+3\right )^{5/2} \left (2 x^2+x+1\right )}dx-\frac {30316369-15043110 x}{50 \left (x^2-2 x+3\right )^{5/2}}\right )-\frac {4878869-2578034 x}{10 \left (x^2-2 x+3\right )^{7/2}}\right )-\frac {3450497-2004270 x}{90 \left (x^2-2 x+3\right )^{9/2}}\right )+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{35} \left (\frac {1}{30} \left (\frac {3}{10} \left (\frac {1}{10} \left (\frac {1}{600} \int \frac {20 \left (118503688 x^2-141252406 x+125053685\right )}{\left (x^2-2 x+3\right )^{3/2} \left (2 x^2+x+1\right )}dx-\frac {63043297-29625922 x}{30 \left (x^2-2 x+3\right )^{3/2}}\right )-\frac {30316369-15043110 x}{50 \left (x^2-2 x+3\right )^{5/2}}\right )-\frac {4878869-2578034 x}{10 \left (x^2-2 x+3\right )^{7/2}}\right )-\frac {3450497-2004270 x}{90 \left (x^2-2 x+3\right )^{9/2}}\right )+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{35} \left (\frac {1}{30} \left (\frac {3}{10} \left (\frac {1}{10} \left (\frac {1}{30} \int \frac {118503688 x^2-141252406 x+125053685}{\left (x^2-2 x+3\right )^{3/2} \left (2 x^2+x+1\right )}dx-\frac {63043297-29625922 x}{30 \left (x^2-2 x+3\right )^{3/2}}\right )-\frac {30316369-15043110 x}{50 \left (x^2-2 x+3\right )^{5/2}}\right )-\frac {4878869-2578034 x}{10 \left (x^2-2 x+3\right )^{7/2}}\right )-\frac {3450497-2004270 x}{90 \left (x^2-2 x+3\right )^{9/2}}\right )+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{35} \left (\frac {1}{30} \left (\frac {3}{10} \left (\frac {1}{10} \left (\frac {1}{30} \left (\frac {1}{200} \int \frac {60 (132636591-89801606 x)}{\sqrt {x^2-2 x+3} \left (2 x^2+x+1\right )}dx-\frac {31 (7434109-3088870 x)}{10 \sqrt {x^2-2 x+3}}\right )-\frac {63043297-29625922 x}{30 \left (x^2-2 x+3\right )^{3/2}}\right )-\frac {30316369-15043110 x}{50 \left (x^2-2 x+3\right )^{5/2}}\right )-\frac {4878869-2578034 x}{10 \left (x^2-2 x+3\right )^{7/2}}\right )-\frac {3450497-2004270 x}{90 \left (x^2-2 x+3\right )^{9/2}}\right )+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{35} \left (\frac {1}{30} \left (\frac {3}{10} \left (\frac {1}{10} \left (\frac {1}{30} \left (\frac {3}{10} \int \frac {132636591-89801606 x}{\sqrt {x^2-2 x+3} \left (2 x^2+x+1\right )}dx-\frac {31 (7434109-3088870 x)}{10 \sqrt {x^2-2 x+3}}\right )-\frac {63043297-29625922 x}{30 \left (x^2-2 x+3\right )^{3/2}}\right )-\frac {30316369-15043110 x}{50 \left (x^2-2 x+3\right )^{5/2}}\right )-\frac {4878869-2578034 x}{10 \left (x^2-2 x+3\right )^{7/2}}\right )-\frac {3450497-2004270 x}{90 \left (x^2-2 x+3\right )^{9/2}}\right )+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{35} \left (\frac {1}{30} \left (\frac {3}{10} \left (\frac {1}{10} \left (\frac {1}{30} \left (\frac {3}{10} \left (\frac {\int -\frac {5 \left (-\left (\left (42834985-89801606 \sqrt {2}\right ) x\right )-132636591 \sqrt {2}+222438197\right )}{\sqrt {x^2-2 x+3} \left (2 x^2+x+1\right )}dx}{10 \sqrt {2}}-\frac {\int -\frac {5 \left (-\left (\left (42834985+89801606 \sqrt {2}\right ) x\right )+132636591 \sqrt {2}+222438197\right )}{\sqrt {x^2-2 x+3} \left (2 x^2+x+1\right )}dx}{10 \sqrt {2}}\right )-\frac {31 (7434109-3088870 x)}{10 \sqrt {x^2-2 x+3}}\right )-\frac {63043297-29625922 x}{30 \left (x^2-2 x+3\right )^{3/2}}\right )-\frac {30316369-15043110 x}{50 \left (x^2-2 x+3\right )^{5/2}}\right )-\frac {4878869-2578034 x}{10 \left (x^2-2 x+3\right )^{7/2}}\right )-\frac {3450497-2004270 x}{90 \left (x^2-2 x+3\right )^{9/2}}\right )+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{35} \left (\frac {1}{30} \left (\frac {3}{10} \left (\frac {1}{10} \left (\frac {1}{30} \left (\frac {3}{10} \left (\frac {\int \frac {-\left (\left (42834985+89801606 \sqrt {2}\right ) x\right )+132636591 \sqrt {2}+222438197}{\sqrt {x^2-2 x+3} \left (2 x^2+x+1\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {-\left (\left (42834985-89801606 \sqrt {2}\right ) x\right )-132636591 \sqrt {2}+222438197}{\sqrt {x^2-2 x+3} \left (2 x^2+x+1\right )}dx}{2 \sqrt {2}}\right )-\frac {31 (7434109-3088870 x)}{10 \sqrt {x^2-2 x+3}}\right )-\frac {63043297-29625922 x}{30 \left (x^2-2 x+3\right )^{3/2}}\right )-\frac {30316369-15043110 x}{50 \left (x^2-2 x+3\right )^{5/2}}\right )-\frac {4878869-2578034 x}{10 \left (x^2-2 x+3\right )^{7/2}}\right )-\frac {3450497-2004270 x}{90 \left (x^2-2 x+3\right )^{9/2}}\right )+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{35} \left (\frac {1}{30} \left (\frac {3}{10} \left (\frac {1}{10} \left (\frac {1}{30} \left (\frac {3}{10} \left (\frac {\left (151363871237318045-110320475741093888 \sqrt {2}\right ) \int \frac {1}{-\frac {5 \left (\left (932587773-620347970 \sqrt {2}\right ) x-312239803 \sqrt {2}+308108167\right )^2}{x^2-2 x+3}-7 \left (151363871237318045-110320475741093888 \sqrt {2}\right )}d\frac {\left (932587773-620347970 \sqrt {2}\right ) x-312239803 \sqrt {2}+308108167}{\sqrt {x^2-2 x+3}}}{\sqrt {2}}-\frac {\left (151363871237318045+110320475741093888 \sqrt {2}\right ) \int \frac {1}{-\frac {5 \left (\left (932587773+620347970 \sqrt {2}\right ) x+312239803 \sqrt {2}+308108167\right )^2}{x^2-2 x+3}-7 \left (151363871237318045+110320475741093888 \sqrt {2}\right )}d\frac {\left (932587773+620347970 \sqrt {2}\right ) x+312239803 \sqrt {2}+308108167}{\sqrt {x^2-2 x+3}}}{\sqrt {2}}\right )-\frac {31 (7434109-3088870 x)}{10 \sqrt {x^2-2 x+3}}\right )-\frac {63043297-29625922 x}{30 \left (x^2-2 x+3\right )^{3/2}}\right )-\frac {30316369-15043110 x}{50 \left (x^2-2 x+3\right )^{5/2}}\right )-\frac {4878869-2578034 x}{10 \left (x^2-2 x+3\right )^{7/2}}\right )-\frac {3450497-2004270 x}{90 \left (x^2-2 x+3\right )^{9/2}}\right )+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{35} \left (\frac {1}{30} \left (\frac {3}{10} \left (\frac {1}{10} \left (\frac {1}{30} \left (\frac {3}{10} \left (\frac {\left (151363871237318045-110320475741093888 \sqrt {2}\right ) \int \frac {1}{-\frac {5 \left (\left (932587773-620347970 \sqrt {2}\right ) x-312239803 \sqrt {2}+308108167\right )^2}{x^2-2 x+3}-7 \left (151363871237318045-110320475741093888 \sqrt {2}\right )}d\frac {\left (932587773-620347970 \sqrt {2}\right ) x-312239803 \sqrt {2}+308108167}{\sqrt {x^2-2 x+3}}}{\sqrt {2}}+\sqrt {\frac {1}{70} \left (151363871237318045+110320475741093888 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {5}{7 \left (151363871237318045+110320475741093888 \sqrt {2}\right )}} \left (\left (932587773+620347970 \sqrt {2}\right ) x+312239803 \sqrt {2}+308108167\right )}{\sqrt {x^2-2 x+3}}\right )\right )-\frac {31 (7434109-3088870 x)}{10 \sqrt {x^2-2 x+3}}\right )-\frac {63043297-29625922 x}{30 \left (x^2-2 x+3\right )^{3/2}}\right )-\frac {30316369-15043110 x}{50 \left (x^2-2 x+3\right )^{5/2}}\right )-\frac {4878869-2578034 x}{10 \left (x^2-2 x+3\right )^{7/2}}\right )-\frac {3450497-2004270 x}{90 \left (x^2-2 x+3\right )^{9/2}}\right )+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{280} \left (\frac {1}{15} \left (\frac {3}{28} \left (\frac {1}{35} \left (\frac {1}{30} \left (\frac {3}{10} \left (\frac {1}{10} \left (\frac {1}{30} \left (\frac {3}{10} \left (\sqrt {\frac {1}{70} \left (151363871237318045+110320475741093888 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {5}{7 \left (151363871237318045+110320475741093888 \sqrt {2}\right )}} \left (\left (932587773+620347970 \sqrt {2}\right ) x+312239803 \sqrt {2}+308108167\right )}{\sqrt {x^2-2 x+3}}\right )+\frac {\left (151363871237318045-110320475741093888 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {5}{7 \left (110320475741093888 \sqrt {2}-151363871237318045\right )}} \left (\left (932587773-620347970 \sqrt {2}\right ) x-312239803 \sqrt {2}+308108167\right )}{\sqrt {x^2-2 x+3}}\right )}{\sqrt {70 \left (110320475741093888 \sqrt {2}-151363871237318045\right )}}\right )-\frac {31 (7434109-3088870 x)}{10 \sqrt {x^2-2 x+3}}\right )-\frac {63043297-29625922 x}{30 \left (x^2-2 x+3\right )^{3/2}}\right )-\frac {30316369-15043110 x}{50 \left (x^2-2 x+3\right )^{5/2}}\right )-\frac {4878869-2578034 x}{10 \left (x^2-2 x+3\right )^{7/2}}\right )-\frac {3450497-2004270 x}{90 \left (x^2-2 x+3\right )^{9/2}}\right )+\frac {12 (8233 x+8822)}{35 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {8878 x+5485}{28 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {4 (67 x+28)}{15 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}\right )-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}\) |
Input:
Int[1/((3 - 2*x + x^2)^(11/2)*(1 + x + 2*x^2)^5),x]
Output:
-1/280*(1 - 10*x)/((3 - 2*x + x^2)^(9/2)*(1 + x + 2*x^2)^4) + ((4*(28 + 67 *x))/(15*(3 - 2*x + x^2)^(9/2)*(1 + x + 2*x^2)^3) + ((5485 + 8878*x)/(28*( 3 - 2*x + x^2)^(9/2)*(1 + x + 2*x^2)^2) + (3*((12*(8822 + 8233*x))/(35*(3 - 2*x + x^2)^(9/2)*(1 + x + 2*x^2)) + (-1/90*(3450497 - 2004270*x)/(3 - 2* x + x^2)^(9/2) + (-1/10*(4878869 - 2578034*x)/(3 - 2*x + x^2)^(7/2) + (3*( -1/50*(30316369 - 15043110*x)/(3 - 2*x + x^2)^(5/2) + (-1/30*(63043297 - 2 9625922*x)/(3 - 2*x + x^2)^(3/2) + ((-31*(7434109 - 3088870*x))/(10*Sqrt[3 - 2*x + x^2]) + (3*(Sqrt[(151363871237318045 + 110320475741093888*Sqrt[2] )/70]*ArcTan[(Sqrt[5/(7*(151363871237318045 + 110320475741093888*Sqrt[2])) ]*(308108167 + 312239803*Sqrt[2] + (932587773 + 620347970*Sqrt[2])*x))/Sqr t[3 - 2*x + x^2]] + ((151363871237318045 - 110320475741093888*Sqrt[2])*Arc Tanh[(Sqrt[5/(7*(-151363871237318045 + 110320475741093888*Sqrt[2]))]*(3081 08167 - 312239803*Sqrt[2] + (932587773 - 620347970*Sqrt[2])*x))/Sqrt[3 - 2 *x + x^2]])/Sqrt[70*(-151363871237318045 + 110320475741093888*Sqrt[2])]))/ 10)/30)/10))/10)/30)/35))/28)/15)/280
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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x _)^2)^(q_), x_Symbol] :> Simp[(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a *f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*(( d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*( c*e - b*f))*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Si mp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f *(p + 1) - c*e*(2*p + q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))* (2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b ^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q , 0]
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[-2*g*(g*b - 2*a*h) Subst[I nt[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, Simp[ g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b , c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && Ne Q[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(c*e - b*f ), 0]
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Simp[1/(2*q) Int[Simp[h*(b*d - a*e) - g*(c*d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqr t[d + e*x + f*x^2]), x], x] - Simp[1/(2*q) Int[Simp[h*(b*d - a*e) - g*(c* d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x + c*x^2) *Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]
Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. )*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] , C = Coeff[Px, x, 2]}, Simp[(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^( q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)))*( (A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b *e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b*c*d - 2*a*c* e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x), x] + Simp[1/((b^2 - 4 *a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)) Int[(a + b*x + c *x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B )*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a *c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*( c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4))) *x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; F reeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
Time = 2.50 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.20
| method | result | size |
| risch | \(\frac {4596238560 x^{17}-38639385552 x^{16}+188603773872 x^{15}-606785954952 x^{14}+1459208021718 x^{13}-2679143870481 x^{12}+3999656132532 x^{11}-4915797913008 x^{10}+5380603084494 x^{9}-5134334619701 x^{8}+4591320676952 x^{7}-3359813871472 x^{6}+2503427226914 x^{5}-1409335257371 x^{4}+1002897791524 x^{3}-266966654968 x^{2}+261702502714 x -53205422447}{1234800000000 \left (x^{2}-2 x +3\right )^{\frac {9}{2}} \left (2 x^{2}+x +1\right )^{4}}+\frac {\sqrt {4}\, \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}\, \sqrt {2}\, \left (9625722625 \sqrt {-6050+4280 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-6050+4280 \sqrt {2}}\, \left (57+40 \sqrt {2}\right ) \left (\sqrt {2}-1+x \right )}{49 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}\, \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-350+280 \sqrt {2}}\, \sqrt {2}+13664181884 \sqrt {-6050+4280 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-6050+4280 \sqrt {2}}\, \left (57+40 \sqrt {2}\right ) \left (\sqrt {2}-1+x \right )}{49 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}\, \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-350+280 \sqrt {2}}+456968008770 \sqrt {2}\, \operatorname {arctanh}\left (\frac {7 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}}{\sqrt {-350+280 \sqrt {2}}}\right )-607941010600 \,\operatorname {arctanh}\left (\frac {7 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}}{\sqrt {-350+280 \sqrt {2}}}\right )\right )}{268912000000000 \sqrt {\frac {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}{\left (\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}+1\right )^{2}}}\, \left (\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}+1\right ) \sqrt {-350+280 \sqrt {2}}}\) | \(452\) |
| trager | \(\text {Expression too large to display}\) | \(541\) |
| default | \(\text {Expression too large to display}\) | \(21028\) |
Input:
int(1/(x^2-2*x+3)^(11/2)/(2*x^2+x+1)^5,x,method=_RETURNVERBOSE)
Output:
1/1234800000000*(4596238560*x^17-38639385552*x^16+188603773872*x^15-606785 954952*x^14+1459208021718*x^13-2679143870481*x^12+3999656132532*x^11-49157 97913008*x^10+5380603084494*x^9-5134334619701*x^8+4591320676952*x^7-335981 3871472*x^6+2503427226914*x^5-1409335257371*x^4+1002897791524*x^3-26696665 4968*x^2+261702502714*x-53205422447)/(x^2-2*x+3)^(9/2)/(2*x^2+x+1)^4+1/268 912000000000*4^(1/2)*((2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+1)^(1/2)*2^(1/2)*(96 25722625*(-6050+4280*2^(1/2))^(1/2)*arctan(1/49*(-6050+4280*2^(1/2))^(1/2) /((2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+1)^(1/2)*(57+40*2^(1/2))*(2^(1/2)-1+x)/( 2^(1/2)+1-x))*(-350+280*2^(1/2))^(1/2)*2^(1/2)+13664181884*(-6050+4280*2^( 1/2))^(1/2)*arctan(1/49*(-6050+4280*2^(1/2))^(1/2)/((2^(1/2)-1+x)^2/(2^(1/ 2)+1-x)^2+1)^(1/2)*(57+40*2^(1/2))*(2^(1/2)-1+x)/(2^(1/2)+1-x))*(-350+280* 2^(1/2))^(1/2)+456968008770*2^(1/2)*arctanh(7*((2^(1/2)-1+x)^2/(2^(1/2)+1- x)^2+1)^(1/2)/(-350+280*2^(1/2))^(1/2))-607941010600*arctanh(7*((2^(1/2)-1 +x)^2/(2^(1/2)+1-x)^2+1)^(1/2)/(-350+280*2^(1/2))^(1/2)))/(((2^(1/2)-1+x)^ 2/(2^(1/2)+1-x)^2+1)/((2^(1/2)-1+x)/(2^(1/2)+1-x)+1)^2)^(1/2)/((2^(1/2)-1+ x)/(2^(1/2)+1-x)+1)/(-350+280*2^(1/2))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 1014 vs. \(2 (298) = 596\).
Time = 0.10 (sec) , antiderivative size = 1014, normalized size of antiderivative = 2.68 \[ \int \frac {1}{\left (3-2 x+x^2\right )^{11/2} \left (1+x+2 x^2\right )^5} \, dx=\text {Too large to display} \] Input:
integrate(1/(x^2-2*x+3)^(11/2)/(2*x^2+x+1)^5,x, algorithm="fricas")
Output:
1/2469600000000*(9192477120*x^18 - 73539816960*x^17 + 353910369120*x^16 - 1116885970080*x^15 + 2670989133180*x^14 - 4857075098280*x^13 + 72879107667 00*x^12 - 8932789641360*x^11 + 9990499039980*x^10 - 9478592970360*x^9 + 88 80507427740*x^8 - 6269269395840*x^7 + 5282801694900*x^6 - 2524484029080*x^ 5 + 2531952916740*x^4 - 227513808720*x^3 + 18*(16*x^18 - 128*x^17 + 616*x^ 16 - 1944*x^15 + 4649*x^14 - 8454*x^13 + 12685*x^12 - 15548*x^11 + 17389*x ^10 - 16498*x^9 + 15457*x^8 - 10912*x^7 + 9195*x^6 - 4394*x^5 + 4407*x^4 - 396*x^3 + 1647*x^2 + 162*x + 243)*sqrt(55160237870546944/35*sqrt(2) + 302 72774247463609/14)*arctan(-5/14293820940408247*(sqrt(2)*(89801606*x - 4696 6621) + (5*sqrt(2)*(x - 1) - sqrt(x^2 - 2*x + 3)*(5*sqrt(2) + 8) + 8*x - 8 )*sqrt(55160237870546944/35*sqrt(2) - 30272774247463609/14) - sqrt(x^2 - 2 *x + 3)*(89801606*sqrt(2) - 42834985) - 42834985*x - 136768227)*sqrt(55160 237870546944/35*sqrt(2) + 30272774247463609/14)) - 18*(16*x^18 - 128*x^17 + 616*x^16 - 1944*x^15 + 4649*x^14 - 8454*x^13 + 12685*x^12 - 15548*x^11 + 17389*x^10 - 16498*x^9 + 15457*x^8 - 10912*x^7 + 9195*x^6 - 4394*x^5 + 44 07*x^4 - 396*x^3 + 1647*x^2 + 162*x + 243)*sqrt(55160237870546944/35*sqrt( 2) + 30272774247463609/14)*arctan(5/14293820940408247*(sqrt(2)*(89801606*x - 46966621) - (5*sqrt(2)*(x - 1) - sqrt(x^2 - 2*x + 3)*(5*sqrt(2) + 8) + 8*x - 8)*sqrt(55160237870546944/35*sqrt(2) - 30272774247463609/14) - sqrt( x^2 - 2*x + 3)*(89801606*sqrt(2) - 42834985) - 42834985*x - 136768227)*...
\[ \int \frac {1}{\left (3-2 x+x^2\right )^{11/2} \left (1+x+2 x^2\right )^5} \, dx=\int \frac {1}{\left (x^{2} - 2 x + 3\right )^{\frac {11}{2}} \left (2 x^{2} + x + 1\right )^{5}}\, dx \] Input:
integrate(1/(x**2-2*x+3)**(11/2)/(2*x**2+x+1)**5,x)
Output:
Integral(1/((x**2 - 2*x + 3)**(11/2)*(2*x**2 + x + 1)**5), x)
\[ \int \frac {1}{\left (3-2 x+x^2\right )^{11/2} \left (1+x+2 x^2\right )^5} \, dx=\int { \frac {1}{{\left (2 \, x^{2} + x + 1\right )}^{5} {\left (x^{2} - 2 \, x + 3\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate(1/(x^2-2*x+3)^(11/2)/(2*x^2+x+1)^5,x, algorithm="maxima")
Output:
integrate(1/((2*x^2 + x + 1)^5*(x^2 - 2*x + 3)^(11/2)), x)
Result contains complex when optimal does not.
Time = 34.72 (sec) , antiderivative size = 2509, normalized size of antiderivative = 6.64 \[ \int \frac {1}{\left (3-2 x+x^2\right )^{11/2} \left (1+x+2 x^2\right )^5} \, dx=\text {Too large to display} \] Input:
integrate(1/(x^2-2*x+3)^(11/2)/(2*x^2+x+1)^5,x, algorithm="giac")
Output:
1/19208000000000*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)* log(3136*(2474301535988301451359142266380914651280177790712513272161012361 81293485559300330785024470114864584026604284622700*sqrt(7)*sqrt(2)*sqrt(77 22433301876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt (2) - 151363871237318045)^2 + 14433425626598425132928329887222002132467703 77915632742093923877724211999095918596245976075670043406821858326965750*sq rt(7)*(110320475741093888*sqrt(2) - 151363871237318045)^3 + 28866851253196 85026585665977444400426493540755831265484187847755448423998191837192491952 151340086813643716653931500*sqrt(2)*(110320475741093888*sqrt(2) - 15136387 1237318045)^3 + 2061917946656917876132618555317428876066814825593761060134 17696817744571299416942320853725095720486688836903852250*sqrt(772243330187 6572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 1513 63871237318045)^3 - 104913854112296962573522080729623041436733404265622592 321084093289259251415027575686933144355006438004151420024881229481000*sqrt (7)*sqrt(2)*(110320475741093888*sqrt(2) - 151363871237318045)^2 - 10491385 41122969625735220807296230414367334042656225923210840932892592514150275756 8693314435500643800415142002488122948100*sqrt(7)*sqrt(7722433301876572160* sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 151363871237 318045)^2 - 20982770822459392514704416145924608287346680853124518464216818 657851850283005515137386628871001287600830284004976245896200*sqrt(2)*sq...
Timed out. \[ \int \frac {1}{\left (3-2 x+x^2\right )^{11/2} \left (1+x+2 x^2\right )^5} \, dx=\int \frac {1}{{\left (2\,x^2+x+1\right )}^5\,{\left (x^2-2\,x+3\right )}^{11/2}} \,d x \] Input:
int(1/((x + 2*x^2 + 1)^5*(x^2 - 2*x + 3)^(11/2)),x)
Output:
int(1/((x + 2*x^2 + 1)^5*(x^2 - 2*x + 3)^(11/2)), x)
\[ \int \frac {1}{\left (3-2 x+x^2\right )^{11/2} \left (1+x+2 x^2\right )^5} \, dx=\int \frac {\sqrt {x^{2}-2 x +3}}{32 x^{22}-304 x^{21}+1696 x^{20}-6360 x^{19}+18130 x^{18}-40343 x^{17}+73881 x^{16}-112604 x^{15}+150340 x^{14}-175580 x^{13}+189860 x^{12}-179940 x^{11}+164080 x^{10}-124970 x^{9}+103430 x^{8}-59524 x^{7}+49708 x^{6}-15372 x^{5}+21060 x^{4}+540 x^{3}+6318 x^{2}+729 x +729}d x \] Input:
int(1/(x^2-2*x+3)^(11/2)/(2*x^2+x+1)^5,x)
Output:
int(sqrt(x**2 - 2*x + 3)/(32*x**22 - 304*x**21 + 1696*x**20 - 6360*x**19 + 18130*x**18 - 40343*x**17 + 73881*x**16 - 112604*x**15 + 150340*x**14 - 1 75580*x**13 + 189860*x**12 - 179940*x**11 + 164080*x**10 - 124970*x**9 + 1 03430*x**8 - 59524*x**7 + 49708*x**6 - 15372*x**5 + 21060*x**4 + 540*x**3 + 6318*x**2 + 729*x + 729),x)