Integrand size = 20, antiderivative size = 407 \[ \int \frac {1}{(3-2 x)^{11/2} \left (1+x+2 x^2\right )^5} \, dx=-\frac {19255}{395136 (3-2 x)^{9/2}}-\frac {462025}{30118144 (3-2 x)^{7/2}}-\frac {38491}{8605184 (3-2 x)^{5/2}}-\frac {141045}{120472576 (3-2 x)^{3/2}}-\frac {38225}{240945152 \sqrt {3-2 x}}+\frac {x}{28 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}+\frac {23+73 x}{1176 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^3}+\frac {1387+3049 x}{32928 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^2}+\frac {5 (3049+4377 x)}{153664 (3-2 x)^{9/2} \left (1+x+2 x^2\right )}+\frac {5 \sqrt {\frac {1}{2} \left (149046503977+40815066112 \sqrt {14}\right )} \arctan \left (\frac {\sqrt {7+2 \sqrt {14}}-2 \sqrt {3-2 x}}{\sqrt {-7+2 \sqrt {14}}}\right )}{3373232128}-\frac {5 \sqrt {\frac {1}{2} \left (149046503977+40815066112 \sqrt {14}\right )} \arctan \left (\frac {\sqrt {7+2 \sqrt {14}}+2 \sqrt {3-2 x}}{\sqrt {-7+2 \sqrt {14}}}\right )}{3373232128}+\frac {5 \sqrt {\frac {1}{2} \left (-149046503977+40815066112 \sqrt {14}\right )} \log \left (3+\sqrt {14}-\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}-2 x\right )}{6746464256}-\frac {5 \sqrt {\frac {1}{2} \left (-149046503977+40815066112 \sqrt {14}\right )} \log \left (3+\sqrt {14}+\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}-2 x\right )}{6746464256} \]
-19255/395136/(3-2*x)^(9/2)-462025/30118144/(3-2*x)^(7/2)-38491/8605184/(3 -2*x)^(5/2)-141045/120472576/(3-2*x)^(3/2)+1/28*x/(3-2*x)^(9/2)/(2*x^2+x+1 )^4+1/1176*(23+73*x)/(3-2*x)^(9/2)/(2*x^2+x+1)^3+1/32928*(1387+3049*x)/(3- 2*x)^(9/2)/(2*x^2+x+1)^2+5/153664*(3049+4377*x)/(3-2*x)^(9/2)/(2*x^2+x+1)- 38225/240945152/(3-2*x)^(1/2)+5/13492928512*ln(3-2*x+14^(1/2)-(3-2*x)^(1/2 )*(7+2*14^(1/2))^(1/2))*(-298093007954+81630132224*14^(1/2))^(1/2)-5/13492 928512*ln(3-2*x+14^(1/2)+(3-2*x)^(1/2)*(7+2*14^(1/2))^(1/2))*(-29809300795 4+81630132224*14^(1/2))^(1/2)+5/6746464256*arctan((-2*(3-2*x)^(1/2)+(7+2*1 4^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))*(298093007954+81630132224*14^(1/2)) ^(1/2)-5/6746464256*arctan((2*(3-2*x)^(1/2)+(7+2*14^(1/2))^(1/2))/(-7+2*14 ^(1/2))^(1/2))*(298093007954+81630132224*14^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 7.42 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.44 \[ \int \frac {1}{(3-2 x)^{11/2} \left (1+x+2 x^2\right )^5} \, dx=\frac {-\frac {14 \left (40289347-429812744 x+135202154 x^2-1073855156 x^3+1627773523 x^4-1470758860 x^5+2888625656 x^6-3106712560 x^7+2343370048 x^8-2443779648 x^9+1873554048 x^{10}-677249280 x^{11}+88070400 x^{12}\right )}{(3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}-45 \sqrt {149046503977+12577271771 i \sqrt {7}} \arctan \left (\frac {1}{2} \sqrt {-1-\frac {i}{\sqrt {7}}} \sqrt {3-2 x}\right )-45 \sqrt {149046503977-12577271771 i \sqrt {7}} \arctan \left (\frac {1}{2} \sqrt {-1+\frac {i}{\sqrt {7}}} \sqrt {3-2 x}\right )}{30359089152} \]
((-14*(40289347 - 429812744*x + 135202154*x^2 - 1073855156*x^3 + 162777352 3*x^4 - 1470758860*x^5 + 2888625656*x^6 - 3106712560*x^7 + 2343370048*x^8 - 2443779648*x^9 + 1873554048*x^10 - 677249280*x^11 + 88070400*x^12))/((3 - 2*x)^(9/2)*(1 + x + 2*x^2)^4) - 45*Sqrt[149046503977 + (12577271771*I)*S qrt[7]]*ArcTan[(Sqrt[-1 - I/Sqrt[7]]*Sqrt[3 - 2*x])/2] - 45*Sqrt[149046503 977 - (12577271771*I)*Sqrt[7]]*ArcTan[(Sqrt[-1 + I/Sqrt[7]]*Sqrt[3 - 2*x]) /2])/30359089152
Time = 0.87 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.24, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {1165, 27, 1235, 27, 1235, 27, 1235, 27, 1198, 27, 1198, 27, 1198, 27, 1198, 27, 1198, 27, 1197, 27, 1483, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{(3-2 x)^{11/2} \left (2 x^2+x+1\right )^5} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle \frac {1}{784} \int \frac {28 (25-23 x)}{(3-2 x)^{11/2} \left (2 x^2+x+1\right )^4}dx+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \int \frac {25-23 x}{(3-2 x)^{11/2} \left (2 x^2+x+1\right )^4}dx+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{588} \int \frac {14 (831-1387 x)}{(3-2 x)^{11/2} \left (2 x^2+x+1\right )^3}dx+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \int \frac {831-1387 x}{(3-2 x)^{11/2} \left (2 x^2+x+1\right )^3}dx+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {1}{392} \int \frac {210 (664-3049 x)}{(3-2 x)^{11/2} \left (2 x^2+x+1\right )^2}dx+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \int \frac {664-3049 x}{(3-2 x)^{11/2} \left (2 x^2+x+1\right )^2}dx+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{196} \int -\frac {14 (48147 x+22129)}{(3-2 x)^{11/2} \left (2 x^2+x+1\right )}dx+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}-\frac {1}{14} \int \frac {48147 x+22129}{(3-2 x)^{11/2} \left (2 x^2+x+1\right )}dx\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (-\frac {1}{28} \int \frac {14 (26957 x+5767)}{(3-2 x)^{9/2} \left (2 x^2+x+1\right )}dx-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (-\frac {1}{2} \int \frac {26957 x+5767}{(3-2 x)^{9/2} \left (2 x^2+x+1\right )}dx-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (-\frac {1}{28} \int -\frac {2 (3889-92405 x)}{(3-2 x)^{7/2} \left (2 x^2+x+1\right )}dx-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \int \frac {3889-92405 x}{(3-2 x)^{7/2} \left (2 x^2+x+1\right )}dx-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{28} \int \frac {14 (15423-38491 x)}{(3-2 x)^{5/2} \left (2 x^2+x+1\right )}dx-\frac {38491}{10 (3-2 x)^{5/2}}\right )-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \int \frac {15423-38491 x}{(3-2 x)^{5/2} \left (2 x^2+x+1\right )}dx-\frac {38491}{10 (3-2 x)^{5/2}}\right )-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{28} \int \frac {2 (100183-84627 x)}{(3-2 x)^{3/2} \left (2 x^2+x+1\right )}dx-\frac {28209}{14 (3-2 x)^{3/2}}\right )-\frac {38491}{10 (3-2 x)^{5/2}}\right )-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \int \frac {100183-84627 x}{(3-2 x)^{3/2} \left (2 x^2+x+1\right )}dx-\frac {28209}{14 (3-2 x)^{3/2}}\right )-\frac {38491}{10 (3-2 x)^{5/2}}\right )-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{28} \int \frac {14 (69337-7645 x)}{\sqrt {3-2 x} \left (2 x^2+x+1\right )}dx-\frac {7645}{2 \sqrt {3-2 x}}\right )-\frac {28209}{14 (3-2 x)^{3/2}}\right )-\frac {38491}{10 (3-2 x)^{5/2}}\right )-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \int \frac {69337-7645 x}{\sqrt {3-2 x} \left (2 x^2+x+1\right )}dx-\frac {7645}{2 \sqrt {3-2 x}}\right )-\frac {28209}{14 (3-2 x)^{3/2}}\right )-\frac {38491}{10 (3-2 x)^{5/2}}\right )-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\int -\frac {7645 (3-2 x)+115739}{2 \left ((3-2 x)^2-7 (3-2 x)+14\right )}d\sqrt {3-2 x}-\frac {7645}{2 \sqrt {3-2 x}}\right )-\frac {28209}{14 (3-2 x)^{3/2}}\right )-\frac {38491}{10 (3-2 x)^{5/2}}\right )-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (-\frac {1}{2} \int \frac {7645 (3-2 x)+115739}{(3-2 x)^2-7 (3-2 x)+14}d\sqrt {3-2 x}-\frac {7645}{2 \sqrt {3-2 x}}\right )-\frac {28209}{14 (3-2 x)^{3/2}}\right )-\frac {38491}{10 (3-2 x)^{5/2}}\right )-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (-\frac {\int \frac {115739 \sqrt {7+2 \sqrt {14}}-\left (115739-7645 \sqrt {14}\right ) \sqrt {3-2 x}}{-2 x-\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3}d\sqrt {3-2 x}}{2 \sqrt {14 \left (7+2 \sqrt {14}\right )}}-\frac {\int \frac {\left (115739-7645 \sqrt {14}\right ) \sqrt {3-2 x}+115739 \sqrt {7+2 \sqrt {14}}}{-2 x+\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3}d\sqrt {3-2 x}}{2 \sqrt {14 \left (7+2 \sqrt {14}\right )}}\right )-\frac {7645}{2 \sqrt {3-2 x}}\right )-\frac {28209}{14 (3-2 x)^{3/2}}\right )-\frac {38491}{10 (3-2 x)^{5/2}}\right )-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (-\frac {\frac {1}{2} \sqrt {7+2 \sqrt {14}} \left (115739+7645 \sqrt {14}\right ) \int \frac {1}{-2 x-\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3}d\sqrt {3-2 x}-\frac {1}{2} \left (115739-7645 \sqrt {14}\right ) \int -\frac {\sqrt {7+2 \sqrt {14}}-2 \sqrt {3-2 x}}{-2 x-\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3}d\sqrt {3-2 x}}{2 \sqrt {14 \left (7+2 \sqrt {14}\right )}}-\frac {\frac {1}{2} \sqrt {7+2 \sqrt {14}} \left (115739+7645 \sqrt {14}\right ) \int \frac {1}{-2 x+\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3}d\sqrt {3-2 x}+\frac {1}{2} \left (115739-7645 \sqrt {14}\right ) \int \frac {2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}}{-2 x+\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3}d\sqrt {3-2 x}}{2 \sqrt {14 \left (7+2 \sqrt {14}\right )}}\right )-\frac {7645}{2 \sqrt {3-2 x}}\right )-\frac {28209}{14 (3-2 x)^{3/2}}\right )-\frac {38491}{10 (3-2 x)^{5/2}}\right )-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (-\frac {\frac {1}{2} \sqrt {7+2 \sqrt {14}} \left (115739+7645 \sqrt {14}\right ) \int \frac {1}{-2 x-\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3}d\sqrt {3-2 x}+\frac {1}{2} \left (115739-7645 \sqrt {14}\right ) \int \frac {\sqrt {7+2 \sqrt {14}}-2 \sqrt {3-2 x}}{-2 x-\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3}d\sqrt {3-2 x}}{2 \sqrt {14 \left (7+2 \sqrt {14}\right )}}-\frac {\frac {1}{2} \sqrt {7+2 \sqrt {14}} \left (115739+7645 \sqrt {14}\right ) \int \frac {1}{-2 x+\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3}d\sqrt {3-2 x}+\frac {1}{2} \left (115739-7645 \sqrt {14}\right ) \int \frac {2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}}{-2 x+\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3}d\sqrt {3-2 x}}{2 \sqrt {14 \left (7+2 \sqrt {14}\right )}}\right )-\frac {7645}{2 \sqrt {3-2 x}}\right )-\frac {28209}{14 (3-2 x)^{3/2}}\right )-\frac {38491}{10 (3-2 x)^{5/2}}\right )-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (-\frac {\frac {1}{2} \left (115739-7645 \sqrt {14}\right ) \int \frac {\sqrt {7+2 \sqrt {14}}-2 \sqrt {3-2 x}}{-2 x-\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3}d\sqrt {3-2 x}-\sqrt {7+2 \sqrt {14}} \left (115739+7645 \sqrt {14}\right ) \int \frac {1}{2 x-2 \sqrt {14}+4}d\left (2 \sqrt {3-2 x}-\sqrt {7+2 \sqrt {14}}\right )}{2 \sqrt {14 \left (7+2 \sqrt {14}\right )}}-\frac {\frac {1}{2} \left (115739-7645 \sqrt {14}\right ) \int \frac {2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}}{-2 x+\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3}d\sqrt {3-2 x}-\sqrt {7+2 \sqrt {14}} \left (115739+7645 \sqrt {14}\right ) \int \frac {1}{2 x-2 \sqrt {14}+4}d\left (2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}\right )}{2 \sqrt {14 \left (7+2 \sqrt {14}\right )}}\right )-\frac {7645}{2 \sqrt {3-2 x}}\right )-\frac {28209}{14 (3-2 x)^{3/2}}\right )-\frac {38491}{10 (3-2 x)^{5/2}}\right )-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (-\frac {\frac {1}{2} \left (115739-7645 \sqrt {14}\right ) \int \frac {\sqrt {7+2 \sqrt {14}}-2 \sqrt {3-2 x}}{-2 x-\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3}d\sqrt {3-2 x}+\sqrt {\frac {7+2 \sqrt {14}}{2 \sqrt {14}-7}} \left (115739+7645 \sqrt {14}\right ) \arctan \left (\frac {2 \sqrt {3-2 x}-\sqrt {7+2 \sqrt {14}}}{\sqrt {2 \sqrt {14}-7}}\right )}{2 \sqrt {14 \left (7+2 \sqrt {14}\right )}}-\frac {\frac {1}{2} \left (115739-7645 \sqrt {14}\right ) \int \frac {2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}}{-2 x+\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3}d\sqrt {3-2 x}+\sqrt {\frac {7+2 \sqrt {14}}{2 \sqrt {14}-7}} \left (115739+7645 \sqrt {14}\right ) \arctan \left (\frac {2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}}{\sqrt {2 \sqrt {14}-7}}\right )}{2 \sqrt {14 \left (7+2 \sqrt {14}\right )}}\right )-\frac {7645}{2 \sqrt {3-2 x}}\right )-\frac {28209}{14 (3-2 x)^{3/2}}\right )-\frac {38491}{10 (3-2 x)^{5/2}}\right )-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{42} \left (\frac {15}{28} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{2} \left (-\frac {\sqrt {\frac {7+2 \sqrt {14}}{2 \sqrt {14}-7}} \left (115739+7645 \sqrt {14}\right ) \arctan \left (\frac {2 \sqrt {3-2 x}-\sqrt {7+2 \sqrt {14}}}{\sqrt {2 \sqrt {14}-7}}\right )-\frac {1}{2} \left (115739-7645 \sqrt {14}\right ) \log \left (-2 x-\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3\right )}{2 \sqrt {14 \left (7+2 \sqrt {14}\right )}}-\frac {\sqrt {\frac {7+2 \sqrt {14}}{2 \sqrt {14}-7}} \left (115739+7645 \sqrt {14}\right ) \arctan \left (\frac {2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}}{\sqrt {2 \sqrt {14}-7}}\right )+\frac {1}{2} \left (115739-7645 \sqrt {14}\right ) \log \left (-2 x+\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3\right )}{2 \sqrt {14 \left (7+2 \sqrt {14}\right )}}\right )-\frac {7645}{2 \sqrt {3-2 x}}\right )-\frac {28209}{14 (3-2 x)^{3/2}}\right )-\frac {38491}{10 (3-2 x)^{5/2}}\right )-\frac {92405}{98 (3-2 x)^{7/2}}\right )-\frac {26957}{18 (3-2 x)^{9/2}}\right )+\frac {4377 x+3049}{14 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}\right )+\frac {3049 x+1387}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}\right )+\frac {73 x+23}{42 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}\right )+\frac {x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}\) |
x/(28*(3 - 2*x)^(9/2)*(1 + x + 2*x^2)^4) + ((23 + 73*x)/(42*(3 - 2*x)^(9/2 )*(1 + x + 2*x^2)^3) + ((1387 + 3049*x)/(28*(3 - 2*x)^(9/2)*(1 + x + 2*x^2 )^2) + (15*((3049 + 4377*x)/(14*(3 - 2*x)^(9/2)*(1 + x + 2*x^2)) + (-26957 /(18*(3 - 2*x)^(9/2)) + (-92405/(98*(3 - 2*x)^(7/2)) + (-38491/(10*(3 - 2* x)^(5/2)) + (-28209/(14*(3 - 2*x)^(3/2)) + (-7645/(2*Sqrt[3 - 2*x]) + (-1/ 2*(Sqrt[(7 + 2*Sqrt[14])/(-7 + 2*Sqrt[14])]*(115739 + 7645*Sqrt[14])*ArcTa n[(-Sqrt[7 + 2*Sqrt[14]] + 2*Sqrt[3 - 2*x])/Sqrt[-7 + 2*Sqrt[14]]] - ((115 739 - 7645*Sqrt[14])*Log[3 + Sqrt[14] - Sqrt[7 + 2*Sqrt[14]]*Sqrt[3 - 2*x] - 2*x])/2)/Sqrt[14*(7 + 2*Sqrt[14])] - (Sqrt[(7 + 2*Sqrt[14])/(-7 + 2*Sqr t[14])]*(115739 + 7645*Sqrt[14])*ArcTan[(Sqrt[7 + 2*Sqrt[14]] + 2*Sqrt[3 - 2*x])/Sqrt[-7 + 2*Sqrt[14]]] + ((115739 - 7645*Sqrt[14])*Log[3 + Sqrt[14] + Sqrt[7 + 2*Sqrt[14]]*Sqrt[3 - 2*x] - 2*x])/2)/(2*Sqrt[14*(7 + 2*Sqrt[14 ])]))/2)/14)/2)/14)/2)/14))/28)/42)/28
3.1.47.3.1 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_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c *d^2 - b*d*e + a*e^2))), x] + Simp[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x )^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1 ]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Time = 3.16 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.68
| method | result | size |
| pseudoelliptic | \(-\frac {5 \left (\left (x -\frac {3}{2}\right )^{4} \left (x^{2}+\frac {1}{2} x +\frac {1}{2}\right )^{4} \left (\frac {\sqrt {7+2 \sqrt {14}}\, \left (146319 \sqrt {14}-569986\right ) \left (\ln \left (3-2 x +\sqrt {14}-\sqrt {3-2 x}\, \sqrt {7+2 \sqrt {14}}\right )-\ln \left (3-2 x +\sqrt {14}+\sqrt {3-2 x}\, \sqrt {7+2 \sqrt {14}}\right )\right ) \sqrt {-7+2 \sqrt {14}}}{2}+\left (115739 \sqrt {14}+107030\right ) \left (\arctan \left (\frac {2 \sqrt {3-2 x}-\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right )+\arctan \left (\frac {2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right )\right )\right ) \sqrt {3-2 x}+214060 \sqrt {-7+2 \sqrt {14}}\, \left (\frac {1626349}{76450} x^{10}+x^{12}-\frac {24405799}{2001600} x^{3}+\frac {1627773523}{88070400} x^{4}+\frac {361078207}{11008800} x^{6}+\frac {67601077}{44035200} x^{2}-\frac {53726593}{11008800} x +\frac {36615157}{1376100} x^{8}-\frac {73537943}{4403520} x^{5}+\frac {40289347}{88070400}-\frac {38833907}{1100880} x^{7}-\frac {4242673}{152900} x^{9}-\frac {58789}{7645} x^{11}\right )\right )}{26353376 \left (3-2 x \right )^{\frac {9}{2}} \sqrt {-7+2 \sqrt {14}}\, \left (2 x^{2}+x +1\right )^{4}}\) | \(277\) |
| derivativedivides | \(\frac {\frac {567651623 \sqrt {3-2 x}}{32}-\frac {6194606411 \left (3-2 x \right )^{\frac {3}{2}}}{192}+\frac {9801432515 \left (3-2 x \right )^{\frac {5}{2}}}{384}-\frac {8763772549 \left (3-2 x \right )^{\frac {7}{2}}}{768}+\frac {149630663 \left (3-2 x \right )^{\frac {9}{2}}}{48}-\frac {200063633 \left (3-2 x \right )^{\frac {11}{2}}}{384}+\frac {18969965 \left (3-2 x \right )^{\frac {13}{2}}}{384}-\frac {526135 \left (3-2 x \right )^{\frac {15}{2}}}{256}}{6588344 \left (\left (3-2 x \right )^{2}-7+14 x \right )^{4}}+\frac {5 \left (146319 \sqrt {7+2 \sqrt {14}}\, \sqrt {14}-569986 \sqrt {7+2 \sqrt {14}}\right ) \ln \left (3-2 x +\sqrt {14}+\sqrt {3-2 x}\, \sqrt {7+2 \sqrt {14}}\right )}{13492928512}+\frac {5 \left (-115739 \sqrt {14}-\frac {\left (146319 \sqrt {7+2 \sqrt {14}}\, \sqrt {14}-569986 \sqrt {7+2 \sqrt {14}}\right ) \sqrt {7+2 \sqrt {14}}}{2}\right ) \arctan \left (\frac {2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right )}{3373232128 \sqrt {-7+2 \sqrt {14}}}-\frac {5 \left (146319 \sqrt {7+2 \sqrt {14}}\, \sqrt {14}-569986 \sqrt {7+2 \sqrt {14}}\right ) \ln \left (3-2 x +\sqrt {14}-\sqrt {3-2 x}\, \sqrt {7+2 \sqrt {14}}\right )}{13492928512}-\frac {5 \left (115739 \sqrt {14}+\frac {\left (146319 \sqrt {7+2 \sqrt {14}}\, \sqrt {14}-569986 \sqrt {7+2 \sqrt {14}}\right ) \sqrt {7+2 \sqrt {14}}}{2}\right ) \arctan \left (\frac {2 \sqrt {3-2 x}-\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right )}{3373232128 \sqrt {-7+2 \sqrt {14}}}+\frac {1}{151263 \left (3-2 x \right )^{\frac {9}{2}}}+\frac {5}{235298 \left (3-2 x \right )^{\frac {7}{2}}}+\frac {19}{470596 \left (3-2 x \right )^{\frac {5}{2}}}+\frac {185}{2823576 \left (3-2 x \right )^{\frac {3}{2}}}+\frac {505}{3294172 \sqrt {3-2 x}}\) | \(415\) |
| default | \(\frac {\frac {567651623 \sqrt {3-2 x}}{32}-\frac {6194606411 \left (3-2 x \right )^{\frac {3}{2}}}{192}+\frac {9801432515 \left (3-2 x \right )^{\frac {5}{2}}}{384}-\frac {8763772549 \left (3-2 x \right )^{\frac {7}{2}}}{768}+\frac {149630663 \left (3-2 x \right )^{\frac {9}{2}}}{48}-\frac {200063633 \left (3-2 x \right )^{\frac {11}{2}}}{384}+\frac {18969965 \left (3-2 x \right )^{\frac {13}{2}}}{384}-\frac {526135 \left (3-2 x \right )^{\frac {15}{2}}}{256}}{6588344 \left (\left (3-2 x \right )^{2}-7+14 x \right )^{4}}+\frac {5 \left (146319 \sqrt {7+2 \sqrt {14}}\, \sqrt {14}-569986 \sqrt {7+2 \sqrt {14}}\right ) \ln \left (3-2 x +\sqrt {14}+\sqrt {3-2 x}\, \sqrt {7+2 \sqrt {14}}\right )}{13492928512}+\frac {5 \left (-115739 \sqrt {14}-\frac {\left (146319 \sqrt {7+2 \sqrt {14}}\, \sqrt {14}-569986 \sqrt {7+2 \sqrt {14}}\right ) \sqrt {7+2 \sqrt {14}}}{2}\right ) \arctan \left (\frac {2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right )}{3373232128 \sqrt {-7+2 \sqrt {14}}}-\frac {5 \left (146319 \sqrt {7+2 \sqrt {14}}\, \sqrt {14}-569986 \sqrt {7+2 \sqrt {14}}\right ) \ln \left (3-2 x +\sqrt {14}-\sqrt {3-2 x}\, \sqrt {7+2 \sqrt {14}}\right )}{13492928512}-\frac {5 \left (115739 \sqrt {14}+\frac {\left (146319 \sqrt {7+2 \sqrt {14}}\, \sqrt {14}-569986 \sqrt {7+2 \sqrt {14}}\right ) \sqrt {7+2 \sqrt {14}}}{2}\right ) \arctan \left (\frac {2 \sqrt {3-2 x}-\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right )}{3373232128 \sqrt {-7+2 \sqrt {14}}}+\frac {1}{151263 \left (3-2 x \right )^{\frac {9}{2}}}+\frac {5}{235298 \left (3-2 x \right )^{\frac {7}{2}}}+\frac {19}{470596 \left (3-2 x \right )^{\frac {5}{2}}}+\frac {185}{2823576 \left (3-2 x \right )^{\frac {3}{2}}}+\frac {505}{3294172 \sqrt {3-2 x}}\) | \(415\) |
| trager | \(\text {Expression too large to display}\) | \(508\) |
| risch | \(-\frac {88070400 x^{12}-677249280 x^{11}+1873554048 x^{10}-2443779648 x^{9}+2343370048 x^{8}-3106712560 x^{7}+2888625656 x^{6}-1470758860 x^{5}+1627773523 x^{4}-1073855156 x^{3}+135202154 x^{2}-429812744 x +40289347}{2168506368 \left (2 x -3\right )^{4} \sqrt {3-2 x}\, \left (2 x^{2}+x +1\right )^{4}}+\frac {731595 \ln \left (3-2 x +\sqrt {14}+\sqrt {3-2 x}\, \sqrt {7+2 \sqrt {14}}\right ) \sqrt {7+2 \sqrt {14}}\, \sqrt {14}}{13492928512}-\frac {1424965 \ln \left (3-2 x +\sqrt {14}+\sqrt {3-2 x}\, \sqrt {7+2 \sqrt {14}}\right ) \sqrt {7+2 \sqrt {14}}}{6746464256}-\frac {731595 \arctan \left (\frac {2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right ) \left (7+2 \sqrt {14}\right ) \sqrt {14}}{6746464256 \sqrt {-7+2 \sqrt {14}}}+\frac {1424965 \arctan \left (\frac {2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right ) \left (7+2 \sqrt {14}\right )}{3373232128 \sqrt {-7+2 \sqrt {14}}}-\frac {578695 \arctan \left (\frac {2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right ) \sqrt {14}}{3373232128 \sqrt {-7+2 \sqrt {14}}}-\frac {731595 \ln \left (3-2 x +\sqrt {14}-\sqrt {3-2 x}\, \sqrt {7+2 \sqrt {14}}\right ) \sqrt {7+2 \sqrt {14}}\, \sqrt {14}}{13492928512}+\frac {1424965 \ln \left (3-2 x +\sqrt {14}-\sqrt {3-2 x}\, \sqrt {7+2 \sqrt {14}}\right ) \sqrt {7+2 \sqrt {14}}}{6746464256}-\frac {731595 \arctan \left (\frac {2 \sqrt {3-2 x}-\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right ) \left (7+2 \sqrt {14}\right ) \sqrt {14}}{6746464256 \sqrt {-7+2 \sqrt {14}}}+\frac {1424965 \arctan \left (\frac {2 \sqrt {3-2 x}-\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right ) \left (7+2 \sqrt {14}\right )}{3373232128 \sqrt {-7+2 \sqrt {14}}}-\frac {578695 \arctan \left (\frac {2 \sqrt {3-2 x}-\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right ) \sqrt {14}}{3373232128 \sqrt {-7+2 \sqrt {14}}}\) | \(536\) |
-5/26353376/(3-2*x)^(9/2)*((x-3/2)^4*(x^2+1/2*x+1/2)^4*(1/2*(7+2*14^(1/2)) ^(1/2)*(146319*14^(1/2)-569986)*(ln(3-2*x+14^(1/2)-(3-2*x)^(1/2)*(7+2*14^( 1/2))^(1/2))-ln(3-2*x+14^(1/2)+(3-2*x)^(1/2)*(7+2*14^(1/2))^(1/2)))*(-7+2* 14^(1/2))^(1/2)+(115739*14^(1/2)+107030)*(arctan((2*(3-2*x)^(1/2)-(7+2*14^ (1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))+arctan((2*(3-2*x)^(1/2)+(7+2*14^(1/2) )^(1/2))/(-7+2*14^(1/2))^(1/2))))*(3-2*x)^(1/2)+214060*(-7+2*14^(1/2))^(1/ 2)*(1626349/76450*x^10+x^12-24405799/2001600*x^3+1627773523/88070400*x^4+3 61078207/11008800*x^6+67601077/44035200*x^2-53726593/11008800*x+36615157/1 376100*x^8-73537943/4403520*x^5+40289347/88070400-38833907/1100880*x^7-424 2673/152900*x^9-58789/7645*x^11))/(-7+2*14^(1/2))^(1/2)/(2*x^2+x+1)^4
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.36 \[ \int \frac {1}{(3-2 x)^{11/2} \left (1+x+2 x^2\right )^5} \, dx =\text {Too large to display} \]
1/60718178304*(9*(512*x^13 - 2816*x^12 + 5632*x^11 - 5888*x^10 + 6848*x^9 - 8992*x^8 + 6112*x^7 - 4240*x^6 + 4994*x^5 - 1707*x^4 + 936*x^3 - 1242*x^ 2 - 162*x - 243)*sqrt(314431794275*I*sqrt(7) - 3726162599425)*log(sqrt(314 431794275*I*sqrt(7) - 3726162599425)*(146319*I*sqrt(7) - 115739) + 4081506 61120*sqrt(-2*x + 3)) - 9*(512*x^13 - 2816*x^12 + 5632*x^11 - 5888*x^10 + 6848*x^9 - 8992*x^8 + 6112*x^7 - 4240*x^6 + 4994*x^5 - 1707*x^4 + 936*x^3 - 1242*x^2 - 162*x - 243)*sqrt(314431794275*I*sqrt(7) - 3726162599425)*log (sqrt(314431794275*I*sqrt(7) - 3726162599425)*(-146319*I*sqrt(7) + 115739) + 408150661120*sqrt(-2*x + 3)) - 9*(512*x^13 - 2816*x^12 + 5632*x^11 - 58 88*x^10 + 6848*x^9 - 8992*x^8 + 6112*x^7 - 4240*x^6 + 4994*x^5 - 1707*x^4 + 936*x^3 - 1242*x^2 - 162*x - 243)*sqrt(-314431794275*I*sqrt(7) - 3726162 599425)*log((146319*I*sqrt(7) + 115739)*sqrt(-314431794275*I*sqrt(7) - 372 6162599425) + 408150661120*sqrt(-2*x + 3)) + 9*(512*x^13 - 2816*x^12 + 563 2*x^11 - 5888*x^10 + 6848*x^9 - 8992*x^8 + 6112*x^7 - 4240*x^6 + 4994*x^5 - 1707*x^4 + 936*x^3 - 1242*x^2 - 162*x - 243)*sqrt(-314431794275*I*sqrt(7 ) - 3726162599425)*log((-146319*I*sqrt(7) - 115739)*sqrt(-314431794275*I*s qrt(7) - 3726162599425) + 408150661120*sqrt(-2*x + 3)) + 28*(88070400*x^12 - 677249280*x^11 + 1873554048*x^10 - 2443779648*x^9 + 2343370048*x^8 - 31 06712560*x^7 + 2888625656*x^6 - 1470758860*x^5 + 1627773523*x^4 - 10738551 56*x^3 + 135202154*x^2 - 429812744*x + 40289347)*sqrt(-2*x + 3))/(512*x...
Timed out. \[ \int \frac {1}{(3-2 x)^{11/2} \left (1+x+2 x^2\right )^5} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(3-2 x)^{11/2} \left (1+x+2 x^2\right )^5} \, dx=\int { \frac {1}{{\left (2 \, x^{2} + x + 1\right )}^{5} {\left (-2 \, x + 3\right )}^{\frac {11}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 795 vs. \(2 (298) = 596\).
Time = 0.62 (sec) , antiderivative size = 795, normalized size of antiderivative = 1.95 \[ \int \frac {1}{(3-2 x)^{11/2} \left (1+x+2 x^2\right )^5} \, dx=\text {Too large to display} \]
-5/1511207993344*sqrt(7)*(22935*14^(3/4)*sqrt(7)*(sqrt(14) + 4)*sqrt(-2*sq rt(14) + 8) + 7645*14^(3/4)*sqrt(7)*(sqrt(14) - 4)*sqrt(-2*sqrt(14) + 8) + 53515*14^(3/4)*sqrt(2*sqrt(14) + 8)*(sqrt(14) + 4) + 160545*14^(3/4)*sqrt (2*sqrt(14) + 8)*(sqrt(14) - 4) + 925912*14^(1/4)*sqrt(7)*sqrt(-2*sqrt(14) + 8) + 6481384*14^(1/4)*sqrt(2*sqrt(14) + 8))*arctan(1/28*14^(3/4)*(14^(1 /4)*sqrt(1/2)*sqrt(sqrt(14) + 4) + 2*sqrt(-2*x + 3))/sqrt(-1/8*sqrt(14) + 1/2)) - 5/1511207993344*sqrt(7)*(22935*14^(3/4)*sqrt(7)*(sqrt(14) + 4)*sqr t(-2*sqrt(14) + 8) + 7645*14^(3/4)*sqrt(7)*(sqrt(14) - 4)*sqrt(-2*sqrt(14) + 8) + 53515*14^(3/4)*sqrt(2*sqrt(14) + 8)*(sqrt(14) + 4) + 160545*14^(3/ 4)*sqrt(2*sqrt(14) + 8)*(sqrt(14) - 4) + 925912*14^(1/4)*sqrt(7)*sqrt(-2*s qrt(14) + 8) + 6481384*14^(1/4)*sqrt(2*sqrt(14) + 8))*arctan(-1/28*14^(3/4 )*(14^(1/4)*sqrt(1/2)*sqrt(sqrt(14) + 4) - 2*sqrt(-2*x + 3))/sqrt(-1/8*sqr t(14) + 1/2)) - 5/3022415986688*sqrt(7)*(7645*14^(3/4)*sqrt(7)*sqrt(2*sqrt (14) + 8)*(sqrt(14) + 4) + 22935*14^(3/4)*sqrt(7)*sqrt(2*sqrt(14) + 8)*(sq rt(14) - 4) - 160545*14^(3/4)*(sqrt(14) + 4)*sqrt(-2*sqrt(14) + 8) - 53515 *14^(3/4)*(sqrt(14) - 4)*sqrt(-2*sqrt(14) + 8) + 925912*14^(1/4)*sqrt(7)*s qrt(2*sqrt(14) + 8) - 6481384*14^(1/4)*sqrt(-2*sqrt(14) + 8))*log(14^(1/4) *sqrt(1/2)*sqrt(-2*x + 3)*sqrt(sqrt(14) + 4) - 2*x + sqrt(14) + 3) + 5/302 2415986688*sqrt(7)*(7645*14^(3/4)*sqrt(7)*sqrt(2*sqrt(14) + 8)*(sqrt(14) + 4) + 22935*14^(3/4)*sqrt(7)*sqrt(2*sqrt(14) + 8)*(sqrt(14) - 4) - 1605...
Time = 0.55 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(3-2 x)^{11/2} \left (1+x+2 x^2\right )^5} \, dx =\text {Too large to display} \]
(atan(((3 - 2*x)^(1/2)*(7^(1/2)*12577271771i - 149046503977)^(1/2)*1572158 971375i)/(391663056253676053933850624*((7^(1/2)*181960107187971125i)/19583 1528126838026966925312 - 230036728532618625/27975932589548289566703616)) - (1572158971375*7^(1/2)*(3 - 2*x)^(1/2)*(7^(1/2)*12577271771i - 1490465039 77)^(1/2))/(391663056253676053933850624*((7^(1/2)*181960107187971125i)/195 831528126838026966925312 - 230036728532618625/27975932589548289566703616)) )*(7^(1/2)*12577271771i - 149046503977)^(1/2)*5i)/3373232128 - ((272*x)/44 1 - (164*(2*x - 3)^2)/441 + (1966*(2*x - 3)^3)/3087 - (9091*(2*x - 3)^4)/3 087 - (32070727*(2*x - 3)^5)/5531904 - (41014777*(2*x - 3)^6)/11063808 - ( 141921511*(2*x - 3)^7)/154893312 + (23262655*(2*x - 3)^8)/309786624 + (157 1659*(2*x - 3)^9)/15059072 + (468427*(2*x - 3)^10)/17210368 + (394105*(2*x - 3)^11)/120472576 + (38225*(2*x - 3)^12)/240945152 - 520/441)/(38416*(3 - 2*x)^(9/2) - 76832*(3 - 2*x)^(11/2) + 68600*(3 - 2*x)^(13/2) - 35672*(3 - 2*x)^(15/2) + 11809*(3 - 2*x)^(17/2) - 2548*(3 - 2*x)^(19/2) + 350*(3 - 2*x)^(21/2) - 28*(3 - 2*x)^(23/2) + (3 - 2*x)^(25/2)) - (atan(((3 - 2*x)^( 1/2)*(- 7^(1/2)*12577271771i - 149046503977)^(1/2)*1572158971375i)/(391663 056253676053933850624*((7^(1/2)*181960107187971125i)/195831528126838026966 925312 + 230036728532618625/27975932589548289566703616)) + (1572158971375* 7^(1/2)*(3 - 2*x)^(1/2)*(- 7^(1/2)*12577271771i - 149046503977)^(1/2))/(39 1663056253676053933850624*((7^(1/2)*181960107187971125i)/19583152812683...
Time = 0.02 (sec) , antiderivative size = 4694, normalized size of antiderivative = 11.53 \[ \int \frac {1}{(3-2 x)^{11/2} \left (1+x+2 x^2\right )^5} \, dx =\text {Too large to display} \]
int(( - 1)/(sqrt( - 2*x + 3)*(1024*x**15 - 5120*x**14 + 8960*x**13 - 8960* x**12 + 13440*x**11 - 17024*x**10 + 10080*x**9 - 11360*x**8 + 11860*x**7 - 2660*x**6 + 5159*x**5 - 3255*x**4 - 630*x**3 - 1890*x**2 - 405*x - 243)), x)
( - 3371189760*sqrt(2*sqrt(14) - 7)*sqrt( - 2*x + 3)*sqrt(14)*atan((2*sqrt ( - 2*x + 3) - sqrt(2*sqrt(14) + 7))/sqrt(2*sqrt(14) - 7))*x**12 + 1348475 9040*sqrt(2*sqrt(14) - 7)*sqrt( - 2*x + 3)*sqrt(14)*atan((2*sqrt( - 2*x + 3) - sqrt(2*sqrt(14) + 7))/sqrt(2*sqrt(14) - 7))*x**11 - 16855948800*sqrt( 2*sqrt(14) - 7)*sqrt( - 2*x + 3)*sqrt(14)*atan((2*sqrt( - 2*x + 3) - sqrt( 2*sqrt(14) + 7))/sqrt(2*sqrt(14) - 7))*x**10 + 13484759040*sqrt(2*sqrt(14) - 7)*sqrt( - 2*x + 3)*sqrt(14)*atan((2*sqrt( - 2*x + 3) - sqrt(2*sqrt(14) + 7))/sqrt(2*sqrt(14) - 7))*x**9 - 24862524480*sqrt(2*sqrt(14) - 7)*sqrt( - 2*x + 3)*sqrt(14)*atan((2*sqrt( - 2*x + 3) - sqrt(2*sqrt(14) + 7))/sqrt (2*sqrt(14) - 7))*x**8 + 21912733440*sqrt(2*sqrt(14) - 7)*sqrt( - 2*x + 3) *sqrt(14)*atan((2*sqrt( - 2*x + 3) - sqrt(2*sqrt(14) + 7))/sqrt(2*sqrt(14) - 7))*x**7 - 7374477600*sqrt(2*sqrt(14) - 7)*sqrt( - 2*x + 3)*sqrt(14)*at an((2*sqrt( - 2*x + 3) - sqrt(2*sqrt(14) + 7))/sqrt(2*sqrt(14) - 7))*x**6 + 16855948800*sqrt(2*sqrt(14) - 7)*sqrt( - 2*x + 3)*sqrt(14)*atan((2*sqrt( - 2*x + 3) - sqrt(2*sqrt(14) + 7))/sqrt(2*sqrt(14) - 7))*x**5 - 759834567 0*sqrt(2*sqrt(14) - 7)*sqrt( - 2*x + 3)*sqrt(14)*atan((2*sqrt( - 2*x + 3) - sqrt(2*sqrt(14) + 7))/sqrt(2*sqrt(14) - 7))*x**4 - 158024520*sqrt(2*sqrt (14) - 7)*sqrt( - 2*x + 3)*sqrt(14)*atan((2*sqrt( - 2*x + 3) - sqrt(2*sqrt (14) + 7))/sqrt(2*sqrt(14) - 7))*x**3 - 6399993060*sqrt(2*sqrt(14) - 7)*sq rt( - 2*x + 3)*sqrt(14)*atan((2*sqrt( - 2*x + 3) - sqrt(2*sqrt(14) + 7)...