Integrand size = 179, antiderivative size = 26 \[ \int \frac {1179648 x^5+131072 x^6+\left (393216 x^7+131072 x^8\right ) \log (3 x)+\left (1572864 x^7+262144 x^8\right ) \log ^2(3 x)+\left (294912 x^9+98304 x^{10}\right ) \log ^3(3 x)+\left (737280 x^9+147456 x^{10}\right ) \log ^4(3 x)+\left (73728 x^{11}+24576 x^{12}\right ) \log ^5(3 x)+\left (147456 x^{11}+32768 x^{12}\right ) \log ^6(3 x)+\left (6144 x^{13}+2048 x^{14}\right ) \log ^7(3 x)+\left (10752 x^{13}+2560 x^{14}\right ) \log ^8(3 x)}{243+405 x+270 x^2+90 x^3+15 x^4+x^5} \, dx=\frac {256 x^6 \left (4+x^2 \log ^2(3 x)\right )^4}{(-3-x)^4} \] Output:
256*x^6/(-3-x)^4*(x^2*ln(3*x)^2+4)^4
Leaf count is larger than twice the leaf count of optimal. \(76\) vs. \(2(26)=52\).
Time = 3.09 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.92 \[ \int \frac {1179648 x^5+131072 x^6+\left (393216 x^7+131072 x^8\right ) \log (3 x)+\left (1572864 x^7+262144 x^8\right ) \log ^2(3 x)+\left (294912 x^9+98304 x^{10}\right ) \log ^3(3 x)+\left (737280 x^9+147456 x^{10}\right ) \log ^4(3 x)+\left (73728 x^{11}+24576 x^{12}\right ) \log ^5(3 x)+\left (147456 x^{11}+32768 x^{12}\right ) \log ^6(3 x)+\left (6144 x^{13}+2048 x^{14}\right ) \log ^7(3 x)+\left (10752 x^{13}+2560 x^{14}\right ) \log ^8(3 x)}{243+405 x+270 x^2+90 x^3+15 x^4+x^5} \, dx=\frac {256 \left (256 \left (-7290-9720 x-4860 x^2-1080 x^3-90 x^4+x^6\right )+256 x^8 \log ^2(3 x)+96 x^{10} \log ^4(3 x)+16 x^{12} \log ^6(3 x)+x^{14} \log ^8(3 x)\right )}{(3+x)^4} \] Input:
Integrate[(1179648*x^5 + 131072*x^6 + (393216*x^7 + 131072*x^8)*Log[3*x] + (1572864*x^7 + 262144*x^8)*Log[3*x]^2 + (294912*x^9 + 98304*x^10)*Log[3*x ]^3 + (737280*x^9 + 147456*x^10)*Log[3*x]^4 + (73728*x^11 + 24576*x^12)*Lo g[3*x]^5 + (147456*x^11 + 32768*x^12)*Log[3*x]^6 + (6144*x^13 + 2048*x^14) *Log[3*x]^7 + (10752*x^13 + 2560*x^14)*Log[3*x]^8)/(243 + 405*x + 270*x^2 + 90*x^3 + 15*x^4 + x^5),x]
Output:
(256*(256*(-7290 - 9720*x - 4860*x^2 - 1080*x^3 - 90*x^4 + x^6) + 256*x^8* Log[3*x]^2 + 96*x^10*Log[3*x]^4 + 16*x^12*Log[3*x]^6 + x^14*Log[3*x]^8))/( 3 + x)^4
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.
| \(\displaystyle \int \frac {131072 x^6+1179648 x^5+\left (2560 x^{14}+10752 x^{13}\right ) \log ^8(3 x)+\left (2048 x^{14}+6144 x^{13}\right ) \log ^7(3 x)+\left (32768 x^{12}+147456 x^{11}\right ) \log ^6(3 x)+\left (24576 x^{12}+73728 x^{11}\right ) \log ^5(3 x)+\left (147456 x^{10}+737280 x^9\right ) \log ^4(3 x)+\left (98304 x^{10}+294912 x^9\right ) \log ^3(3 x)+\left (262144 x^8+1572864 x^7\right ) \log ^2(3 x)+\left (131072 x^8+393216 x^7\right ) \log (3 x)}{x^5+15 x^4+90 x^3+270 x^2+405 x+243} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {131072 x^6+1179648 x^5+\left (2560 x^{14}+10752 x^{13}\right ) \log ^8(3 x)+\left (2048 x^{14}+6144 x^{13}\right ) \log ^7(3 x)+\left (32768 x^{12}+147456 x^{11}\right ) \log ^6(3 x)+\left (24576 x^{12}+73728 x^{11}\right ) \log ^5(3 x)+\left (147456 x^{10}+737280 x^9\right ) \log ^4(3 x)+\left (98304 x^{10}+294912 x^9\right ) \log ^3(3 x)+\left (262144 x^8+1572864 x^7\right ) \log ^2(3 x)+\left (131072 x^8+393216 x^7\right ) \log (3 x)}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {512 x^5 \left (x^2 \log ^2(3 x)+4\right )^3 \left ((5 x+21) x^2 \log ^2(3 x)+4 (x+3) x^2 \log (3 x)+4 (x+9)\right )}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 512 \int \frac {x^5 \left (x^2 \log ^2(3 x)+4\right )^3 \left ((5 x+21) \log ^2(3 x) x^2+4 (x+3) \log (3 x) x^2+4 (x+9)\right )}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 512 \int \left (\frac {(5 x+21) \log ^8(3 x) x^{13}}{(x+3)^5}+\frac {4 \log ^7(3 x) x^{13}}{(x+3)^4}+\frac {32 (2 x+9) \log ^6(3 x) x^{11}}{(x+3)^5}+\frac {48 \log ^5(3 x) x^{11}}{(x+3)^4}+\frac {288 (x+5) \log ^4(3 x) x^9}{(x+3)^5}+\frac {192 \log ^3(3 x) x^9}{(x+3)^4}+\frac {512 (x+6) \log ^2(3 x) x^7}{(x+3)^5}+\frac {256 \log (3 x) x^7}{(x+3)^4}+\frac {256 (x+9) x^5}{(x+3)^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 512 \int \frac {x^5 \left (x^2 \log ^2(3 x)+4\right )^3 \left ((5 x+21) \log ^2(3 x) x^2+4 (x+3) \log (3 x) x^2+4 (x+9)\right )}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 512 \int \left (\frac {(5 x+21) \log ^8(3 x) x^{13}}{(x+3)^5}+\frac {4 \log ^7(3 x) x^{13}}{(x+3)^4}+\frac {32 (2 x+9) \log ^6(3 x) x^{11}}{(x+3)^5}+\frac {48 \log ^5(3 x) x^{11}}{(x+3)^4}+\frac {288 (x+5) \log ^4(3 x) x^9}{(x+3)^5}+\frac {192 \log ^3(3 x) x^9}{(x+3)^4}+\frac {512 (x+6) \log ^2(3 x) x^7}{(x+3)^5}+\frac {256 \log (3 x) x^7}{(x+3)^4}+\frac {256 (x+9) x^5}{(x+3)^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 512 \int \frac {x^5 \left (x^2 \log ^2(3 x)+4\right )^3 \left ((5 x+21) \log ^2(3 x) x^2+4 (x+3) \log (3 x) x^2+4 (x+9)\right )}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 512 \int \left (\frac {(5 x+21) \log ^8(3 x) x^{13}}{(x+3)^5}+\frac {4 \log ^7(3 x) x^{13}}{(x+3)^4}+\frac {32 (2 x+9) \log ^6(3 x) x^{11}}{(x+3)^5}+\frac {48 \log ^5(3 x) x^{11}}{(x+3)^4}+\frac {288 (x+5) \log ^4(3 x) x^9}{(x+3)^5}+\frac {192 \log ^3(3 x) x^9}{(x+3)^4}+\frac {512 (x+6) \log ^2(3 x) x^7}{(x+3)^5}+\frac {256 \log (3 x) x^7}{(x+3)^4}+\frac {256 (x+9) x^5}{(x+3)^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 512 \int \frac {x^5 \left (x^2 \log ^2(3 x)+4\right )^3 \left ((5 x+21) \log ^2(3 x) x^2+4 (x+3) \log (3 x) x^2+4 (x+9)\right )}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 512 \int \left (\frac {(5 x+21) \log ^8(3 x) x^{13}}{(x+3)^5}+\frac {4 \log ^7(3 x) x^{13}}{(x+3)^4}+\frac {32 (2 x+9) \log ^6(3 x) x^{11}}{(x+3)^5}+\frac {48 \log ^5(3 x) x^{11}}{(x+3)^4}+\frac {288 (x+5) \log ^4(3 x) x^9}{(x+3)^5}+\frac {192 \log ^3(3 x) x^9}{(x+3)^4}+\frac {512 (x+6) \log ^2(3 x) x^7}{(x+3)^5}+\frac {256 \log (3 x) x^7}{(x+3)^4}+\frac {256 (x+9) x^5}{(x+3)^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 512 \int \frac {x^5 \left (x^2 \log ^2(3 x)+4\right )^3 \left ((5 x+21) \log ^2(3 x) x^2+4 (x+3) \log (3 x) x^2+4 (x+9)\right )}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 512 \int \left (\frac {(5 x+21) \log ^8(3 x) x^{13}}{(x+3)^5}+\frac {4 \log ^7(3 x) x^{13}}{(x+3)^4}+\frac {32 (2 x+9) \log ^6(3 x) x^{11}}{(x+3)^5}+\frac {48 \log ^5(3 x) x^{11}}{(x+3)^4}+\frac {288 (x+5) \log ^4(3 x) x^9}{(x+3)^5}+\frac {192 \log ^3(3 x) x^9}{(x+3)^4}+\frac {512 (x+6) \log ^2(3 x) x^7}{(x+3)^5}+\frac {256 \log (3 x) x^7}{(x+3)^4}+\frac {256 (x+9) x^5}{(x+3)^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 512 \int \frac {x^5 \left (x^2 \log ^2(3 x)+4\right )^3 \left ((5 x+21) \log ^2(3 x) x^2+4 (x+3) \log (3 x) x^2+4 (x+9)\right )}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 512 \int \left (\frac {(5 x+21) \log ^8(3 x) x^{13}}{(x+3)^5}+\frac {4 \log ^7(3 x) x^{13}}{(x+3)^4}+\frac {32 (2 x+9) \log ^6(3 x) x^{11}}{(x+3)^5}+\frac {48 \log ^5(3 x) x^{11}}{(x+3)^4}+\frac {288 (x+5) \log ^4(3 x) x^9}{(x+3)^5}+\frac {192 \log ^3(3 x) x^9}{(x+3)^4}+\frac {512 (x+6) \log ^2(3 x) x^7}{(x+3)^5}+\frac {256 \log (3 x) x^7}{(x+3)^4}+\frac {256 (x+9) x^5}{(x+3)^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 512 \int \frac {x^5 \left (x^2 \log ^2(3 x)+4\right )^3 \left ((5 x+21) \log ^2(3 x) x^2+4 (x+3) \log (3 x) x^2+4 (x+9)\right )}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 512 \int \left (\frac {(5 x+21) \log ^8(3 x) x^{13}}{(x+3)^5}+\frac {4 \log ^7(3 x) x^{13}}{(x+3)^4}+\frac {32 (2 x+9) \log ^6(3 x) x^{11}}{(x+3)^5}+\frac {48 \log ^5(3 x) x^{11}}{(x+3)^4}+\frac {288 (x+5) \log ^4(3 x) x^9}{(x+3)^5}+\frac {192 \log ^3(3 x) x^9}{(x+3)^4}+\frac {512 (x+6) \log ^2(3 x) x^7}{(x+3)^5}+\frac {256 \log (3 x) x^7}{(x+3)^4}+\frac {256 (x+9) x^5}{(x+3)^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 512 \int \frac {x^5 \left (x^2 \log ^2(3 x)+4\right )^3 \left ((5 x+21) \log ^2(3 x) x^2+4 (x+3) \log (3 x) x^2+4 (x+9)\right )}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 512 \int \left (\frac {(5 x+21) \log ^8(3 x) x^{13}}{(x+3)^5}+\frac {4 \log ^7(3 x) x^{13}}{(x+3)^4}+\frac {32 (2 x+9) \log ^6(3 x) x^{11}}{(x+3)^5}+\frac {48 \log ^5(3 x) x^{11}}{(x+3)^4}+\frac {288 (x+5) \log ^4(3 x) x^9}{(x+3)^5}+\frac {192 \log ^3(3 x) x^9}{(x+3)^4}+\frac {512 (x+6) \log ^2(3 x) x^7}{(x+3)^5}+\frac {256 \log (3 x) x^7}{(x+3)^4}+\frac {256 (x+9) x^5}{(x+3)^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 512 \int \frac {x^5 \left (x^2 \log ^2(3 x)+4\right )^3 \left ((5 x+21) \log ^2(3 x) x^2+4 (x+3) \log (3 x) x^2+4 (x+9)\right )}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 512 \int \left (\frac {(5 x+21) \log ^8(3 x) x^{13}}{(x+3)^5}+\frac {4 \log ^7(3 x) x^{13}}{(x+3)^4}+\frac {32 (2 x+9) \log ^6(3 x) x^{11}}{(x+3)^5}+\frac {48 \log ^5(3 x) x^{11}}{(x+3)^4}+\frac {288 (x+5) \log ^4(3 x) x^9}{(x+3)^5}+\frac {192 \log ^3(3 x) x^9}{(x+3)^4}+\frac {512 (x+6) \log ^2(3 x) x^7}{(x+3)^5}+\frac {256 \log (3 x) x^7}{(x+3)^4}+\frac {256 (x+9) x^5}{(x+3)^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 512 \int \frac {x^5 \left (x^2 \log ^2(3 x)+4\right )^3 \left ((5 x+21) \log ^2(3 x) x^2+4 (x+3) \log (3 x) x^2+4 (x+9)\right )}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 512 \int \left (\frac {(5 x+21) \log ^8(3 x) x^{13}}{(x+3)^5}+\frac {4 \log ^7(3 x) x^{13}}{(x+3)^4}+\frac {32 (2 x+9) \log ^6(3 x) x^{11}}{(x+3)^5}+\frac {48 \log ^5(3 x) x^{11}}{(x+3)^4}+\frac {288 (x+5) \log ^4(3 x) x^9}{(x+3)^5}+\frac {192 \log ^3(3 x) x^9}{(x+3)^4}+\frac {512 (x+6) \log ^2(3 x) x^7}{(x+3)^5}+\frac {256 \log (3 x) x^7}{(x+3)^4}+\frac {256 (x+9) x^5}{(x+3)^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 512 \int \frac {x^5 \left (x^2 \log ^2(3 x)+4\right )^3 \left ((5 x+21) \log ^2(3 x) x^2+4 (x+3) \log (3 x) x^2+4 (x+9)\right )}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 512 \int \left (\frac {(5 x+21) \log ^8(3 x) x^{13}}{(x+3)^5}+\frac {4 \log ^7(3 x) x^{13}}{(x+3)^4}+\frac {32 (2 x+9) \log ^6(3 x) x^{11}}{(x+3)^5}+\frac {48 \log ^5(3 x) x^{11}}{(x+3)^4}+\frac {288 (x+5) \log ^4(3 x) x^9}{(x+3)^5}+\frac {192 \log ^3(3 x) x^9}{(x+3)^4}+\frac {512 (x+6) \log ^2(3 x) x^7}{(x+3)^5}+\frac {256 \log (3 x) x^7}{(x+3)^4}+\frac {256 (x+9) x^5}{(x+3)^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 512 \int \frac {x^5 \left (x^2 \log ^2(3 x)+4\right )^3 \left ((5 x+21) \log ^2(3 x) x^2+4 (x+3) \log (3 x) x^2+4 (x+9)\right )}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 512 \int \left (\frac {(5 x+21) \log ^8(3 x) x^{13}}{(x+3)^5}+\frac {4 \log ^7(3 x) x^{13}}{(x+3)^4}+\frac {32 (2 x+9) \log ^6(3 x) x^{11}}{(x+3)^5}+\frac {48 \log ^5(3 x) x^{11}}{(x+3)^4}+\frac {288 (x+5) \log ^4(3 x) x^9}{(x+3)^5}+\frac {192 \log ^3(3 x) x^9}{(x+3)^4}+\frac {512 (x+6) \log ^2(3 x) x^7}{(x+3)^5}+\frac {256 \log (3 x) x^7}{(x+3)^4}+\frac {256 (x+9) x^5}{(x+3)^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 512 \int \frac {x^5 \left (x^2 \log ^2(3 x)+4\right )^3 \left ((5 x+21) \log ^2(3 x) x^2+4 (x+3) \log (3 x) x^2+4 (x+9)\right )}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 512 \int \left (\frac {(5 x+21) \log ^8(3 x) x^{13}}{(x+3)^5}+\frac {4 \log ^7(3 x) x^{13}}{(x+3)^4}+\frac {32 (2 x+9) \log ^6(3 x) x^{11}}{(x+3)^5}+\frac {48 \log ^5(3 x) x^{11}}{(x+3)^4}+\frac {288 (x+5) \log ^4(3 x) x^9}{(x+3)^5}+\frac {192 \log ^3(3 x) x^9}{(x+3)^4}+\frac {512 (x+6) \log ^2(3 x) x^7}{(x+3)^5}+\frac {256 \log (3 x) x^7}{(x+3)^4}+\frac {256 (x+9) x^5}{(x+3)^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 512 \int \frac {x^5 \left (x^2 \log ^2(3 x)+4\right )^3 \left ((5 x+21) \log ^2(3 x) x^2+4 (x+3) \log (3 x) x^2+4 (x+9)\right )}{(x+3)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 512 \int \left (\frac {(5 x+21) \log ^8(3 x) x^{13}}{(x+3)^5}+\frac {4 \log ^7(3 x) x^{13}}{(x+3)^4}+\frac {32 (2 x+9) \log ^6(3 x) x^{11}}{(x+3)^5}+\frac {48 \log ^5(3 x) x^{11}}{(x+3)^4}+\frac {288 (x+5) \log ^4(3 x) x^9}{(x+3)^5}+\frac {192 \log ^3(3 x) x^9}{(x+3)^4}+\frac {512 (x+6) \log ^2(3 x) x^7}{(x+3)^5}+\frac {256 \log (3 x) x^7}{(x+3)^4}+\frac {256 (x+9) x^5}{(x+3)^5}\right )dx\) |
Input:
Int[(1179648*x^5 + 131072*x^6 + (393216*x^7 + 131072*x^8)*Log[3*x] + (1572 864*x^7 + 262144*x^8)*Log[3*x]^2 + (294912*x^9 + 98304*x^10)*Log[3*x]^3 + (737280*x^9 + 147456*x^10)*Log[3*x]^4 + (73728*x^11 + 24576*x^12)*Log[3*x] ^5 + (147456*x^11 + 32768*x^12)*Log[3*x]^6 + (6144*x^13 + 2048*x^14)*Log[3 *x]^7 + (10752*x^13 + 2560*x^14)*Log[3*x]^8)/(243 + 405*x + 270*x^2 + 90*x ^3 + 15*x^4 + x^5),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(170\) vs. \(2(26)=52\).
Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 6.58
\[\frac {256 x^{14} \ln \left (3 x \right )^{8}}{x^{4}+12 x^{3}+54 x^{2}+108 x +81}+\frac {4096 x^{12} \ln \left (3 x \right )^{6}}{x^{4}+12 x^{3}+54 x^{2}+108 x +81}+\frac {24576 x^{10} \ln \left (3 x \right )^{4}}{x^{4}+12 x^{3}+54 x^{2}+108 x +81}+\frac {65536 x^{8} \ln \left (3 x \right )^{2}}{x^{4}+12 x^{3}+54 x^{2}+108 x +81}+\frac {65536 x^{6}-5898240 x^{4}-70778880 x^{3}-318504960 x^{2}-637009920 x -477757440}{x^{4}+12 x^{3}+54 x^{2}+108 x +81}\]
Input:
int(((2560*x^14+10752*x^13)*ln(3*x)^8+(2048*x^14+6144*x^13)*ln(3*x)^7+(327 68*x^12+147456*x^11)*ln(3*x)^6+(24576*x^12+73728*x^11)*ln(3*x)^5+(147456*x ^10+737280*x^9)*ln(3*x)^4+(98304*x^10+294912*x^9)*ln(3*x)^3+(262144*x^8+15 72864*x^7)*ln(3*x)^2+(131072*x^8+393216*x^7)*ln(3*x)+131072*x^6+1179648*x^ 5)/(x^5+15*x^4+90*x^3+270*x^2+405*x+243),x)
Output:
256*x^14/(x^4+12*x^3+54*x^2+108*x+81)*ln(3*x)^8+4096*x^12/(x^4+12*x^3+54*x ^2+108*x+81)*ln(3*x)^6+24576*x^10/(x^4+12*x^3+54*x^2+108*x+81)*ln(3*x)^4+6 5536*x^8/(x^4+12*x^3+54*x^2+108*x+81)*ln(3*x)^2+65536*(x^6-90*x^4-1080*x^3 -4860*x^2-9720*x-7290)/(x^4+12*x^3+54*x^2+108*x+81)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.46 \[ \int \frac {1179648 x^5+131072 x^6+\left (393216 x^7+131072 x^8\right ) \log (3 x)+\left (1572864 x^7+262144 x^8\right ) \log ^2(3 x)+\left (294912 x^9+98304 x^{10}\right ) \log ^3(3 x)+\left (737280 x^9+147456 x^{10}\right ) \log ^4(3 x)+\left (73728 x^{11}+24576 x^{12}\right ) \log ^5(3 x)+\left (147456 x^{11}+32768 x^{12}\right ) \log ^6(3 x)+\left (6144 x^{13}+2048 x^{14}\right ) \log ^7(3 x)+\left (10752 x^{13}+2560 x^{14}\right ) \log ^8(3 x)}{243+405 x+270 x^2+90 x^3+15 x^4+x^5} \, dx=\frac {256 \, {\left (x^{14} \log \left (3 \, x\right )^{8} + 16 \, x^{12} \log \left (3 \, x\right )^{6} + 96 \, x^{10} \log \left (3 \, x\right )^{4} + 256 \, x^{8} \log \left (3 \, x\right )^{2} + 256 \, x^{6} - 23040 \, x^{4} - 276480 \, x^{3} - 1244160 \, x^{2} - 2488320 \, x - 1866240\right )}}{x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81} \] Input:
integrate(((2560*x^14+10752*x^13)*log(3*x)^8+(2048*x^14+6144*x^13)*log(3*x )^7+(32768*x^12+147456*x^11)*log(3*x)^6+(24576*x^12+73728*x^11)*log(3*x)^5 +(147456*x^10+737280*x^9)*log(3*x)^4+(98304*x^10+294912*x^9)*log(3*x)^3+(2 62144*x^8+1572864*x^7)*log(3*x)^2+(131072*x^8+393216*x^7)*log(3*x)+131072* x^6+1179648*x^5)/(x^5+15*x^4+90*x^3+270*x^2+405*x+243),x, algorithm="frica s")
Output:
256*(x^14*log(3*x)^8 + 16*x^12*log(3*x)^6 + 96*x^10*log(3*x)^4 + 256*x^8*l og(3*x)^2 + 256*x^6 - 23040*x^4 - 276480*x^3 - 1244160*x^2 - 2488320*x - 1 866240)/(x^4 + 12*x^3 + 54*x^2 + 108*x + 81)
Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (24) = 48\).
Time = 0.46 (sec) , antiderivative size = 165, normalized size of antiderivative = 6.35 \[ \int \frac {1179648 x^5+131072 x^6+\left (393216 x^7+131072 x^8\right ) \log (3 x)+\left (1572864 x^7+262144 x^8\right ) \log ^2(3 x)+\left (294912 x^9+98304 x^{10}\right ) \log ^3(3 x)+\left (737280 x^9+147456 x^{10}\right ) \log ^4(3 x)+\left (73728 x^{11}+24576 x^{12}\right ) \log ^5(3 x)+\left (147456 x^{11}+32768 x^{12}\right ) \log ^6(3 x)+\left (6144 x^{13}+2048 x^{14}\right ) \log ^7(3 x)+\left (10752 x^{13}+2560 x^{14}\right ) \log ^8(3 x)}{243+405 x+270 x^2+90 x^3+15 x^4+x^5} \, dx=\frac {256 x^{14} \log {\left (3 x \right )}^{8}}{x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81} + \frac {4096 x^{12} \log {\left (3 x \right )}^{6}}{x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81} + \frac {24576 x^{10} \log {\left (3 x \right )}^{4}}{x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81} + \frac {65536 x^{8} \log {\left (3 x \right )}^{2}}{x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81} + 65536 x^{2} - 786432 x + \frac {- 35389440 x^{3} - 238878720 x^{2} - 573308928 x - 477757440}{x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81} \] Input:
integrate(((2560*x**14+10752*x**13)*ln(3*x)**8+(2048*x**14+6144*x**13)*ln( 3*x)**7+(32768*x**12+147456*x**11)*ln(3*x)**6+(24576*x**12+73728*x**11)*ln (3*x)**5+(147456*x**10+737280*x**9)*ln(3*x)**4+(98304*x**10+294912*x**9)*l n(3*x)**3+(262144*x**8+1572864*x**7)*ln(3*x)**2+(131072*x**8+393216*x**7)* ln(3*x)+131072*x**6+1179648*x**5)/(x**5+15*x**4+90*x**3+270*x**2+405*x+243 ),x)
Output:
256*x**14*log(3*x)**8/(x**4 + 12*x**3 + 54*x**2 + 108*x + 81) + 4096*x**12 *log(3*x)**6/(x**4 + 12*x**3 + 54*x**2 + 108*x + 81) + 24576*x**10*log(3*x )**4/(x**4 + 12*x**3 + 54*x**2 + 108*x + 81) + 65536*x**8*log(3*x)**2/(x** 4 + 12*x**3 + 54*x**2 + 108*x + 81) + 65536*x**2 - 786432*x + (-35389440*x **3 - 238878720*x**2 - 573308928*x - 477757440)/(x**4 + 12*x**3 + 54*x**2 + 108*x + 81)
Leaf count of result is larger than twice the leaf count of optimal. 485 vs. \(2 (24) = 48\).
Time = 0.21 (sec) , antiderivative size = 485, normalized size of antiderivative = 18.65 \[ \int \frac {1179648 x^5+131072 x^6+\left (393216 x^7+131072 x^8\right ) \log (3 x)+\left (1572864 x^7+262144 x^8\right ) \log ^2(3 x)+\left (294912 x^9+98304 x^{10}\right ) \log ^3(3 x)+\left (737280 x^9+147456 x^{10}\right ) \log ^4(3 x)+\left (73728 x^{11}+24576 x^{12}\right ) \log ^5(3 x)+\left (147456 x^{11}+32768 x^{12}\right ) \log ^6(3 x)+\left (6144 x^{13}+2048 x^{14}\right ) \log ^7(3 x)+\left (10752 x^{13}+2560 x^{14}\right ) \log ^8(3 x)}{243+405 x+270 x^2+90 x^3+15 x^4+x^5} \, dx =\text {Too large to display} \] Input:
integrate(((2560*x^14+10752*x^13)*log(3*x)^8+(2048*x^14+6144*x^13)*log(3*x )^7+(32768*x^12+147456*x^11)*log(3*x)^6+(24576*x^12+73728*x^11)*log(3*x)^5 +(147456*x^10+737280*x^9)*log(3*x)^4+(98304*x^10+294912*x^9)*log(3*x)^3+(2 62144*x^8+1572864*x^7)*log(3*x)^2+(131072*x^8+393216*x^7)*log(3*x)+131072* x^6+1179648*x^5)/(x^5+15*x^4+90*x^3+270*x^2+405*x+243),x, algorithm="maxim a")
Output:
65536*x^2 - 786432*x + 256*(x^14*log(3)^8 + 8*x^14*log(3)*log(x)^7 + x^14* log(x)^8 + 16*x^12*log(3)^6 + 96*x^10*log(3)^4 + 256*x^8*log(3)^2 - 136792 9134*log(3)^8 + 4*(7*x^14*log(3)^2 + 4*x^12)*log(x)^6 - 1403004240*log(3)^ 6 + 8*(7*x^14*log(3)^3 + 12*x^12*log(3))*log(x)^5 - 162*(104247*log(3)^8 + 106920*log(3)^6 + 36288*log(3)^4 + 4480*log(3)^2)*x^4 + 2*(35*x^14*log(3) ^4 + 120*x^12*log(3)^2 + 48*x^10)*log(x)^4 - 1944*(104247*log(3)^8 + 10692 0*log(3)^6 + 36288*log(3)^4 + 4480*log(3)^2)*x^3 - 476171136*log(3)^4 + 8* (7*x^14*log(3)^5 + 40*x^12*log(3)^3 + 48*x^10*log(3))*log(x)^3 - 8748*(104 247*log(3)^8 + 106920*log(3)^6 + 36288*log(3)^4 + 4480*log(3)^2)*x^2 + 4*( 7*x^14*log(3)^6 + 60*x^12*log(3)^4 + 144*x^10*log(3)^2 + 64*x^8)*log(x)^2 - 17496*(104247*log(3)^8 + 106920*log(3)^6 + 36288*log(3)^4 + 4480*log(3)^ 2)*x - 58786560*log(3)^2 + 8*(x^14*log(3)^7 + 12*x^12*log(3)^5 + 48*x^10*l og(3)^3 + 64*x^8*log(3))*log(x))/(x^4 + 12*x^3 + 54*x^2 + 108*x + 81) + 88 4736*(80*x^3 + 630*x^2 + 1692*x + 1539)/(x^4 + 12*x^3 + 54*x^2 + 108*x + 8 1) - 2654208*(40*x^3 + 300*x^2 + 780*x + 693)/(x^4 + 12*x^3 + 54*x^2 + 108 *x + 81)
Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (24) = 48\).
Time = 0.17 (sec) , antiderivative size = 358, normalized size of antiderivative = 13.77 \[ \int \frac {1179648 x^5+131072 x^6+\left (393216 x^7+131072 x^8\right ) \log (3 x)+\left (1572864 x^7+262144 x^8\right ) \log ^2(3 x)+\left (294912 x^9+98304 x^{10}\right ) \log ^3(3 x)+\left (737280 x^9+147456 x^{10}\right ) \log ^4(3 x)+\left (73728 x^{11}+24576 x^{12}\right ) \log ^5(3 x)+\left (147456 x^{11}+32768 x^{12}\right ) \log ^6(3 x)+\left (6144 x^{13}+2048 x^{14}\right ) \log ^7(3 x)+\left (10752 x^{13}+2560 x^{14}\right ) \log ^8(3 x)}{243+405 x+270 x^2+90 x^3+15 x^4+x^5} \, dx=256 \, {\left (x^{10} - 12 \, x^{9} + 90 \, x^{8} - 540 \, x^{7} + 2835 \, x^{6} - 13608 \, x^{5} + 61236 \, x^{4} - 262440 \, x^{3} + 1082565 \, x^{2} - 4330260 \, x - \frac {177147 \, {\left (364 \, x^{3} + 3003 \, x^{2} + 8316 \, x + 7722\right )}}{x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81} + 16888014\right )} \log \left (3 \, x\right )^{8} + 4096 \, {\left (x^{8} - 12 \, x^{7} + 90 \, x^{6} - 540 \, x^{5} + 2835 \, x^{4} - 13608 \, x^{3} + 61236 \, x^{2} - 262440 \, x - \frac {19683 \, {\left (220 \, x^{3} + 1782 \, x^{2} + 4860 \, x + 4455\right )}}{x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81} + 1082565\right )} \log \left (3 \, x\right )^{6} + 24576 \, {\left (x^{6} - 12 \, x^{5} + 90 \, x^{4} - 540 \, x^{3} + 2835 \, x^{2} - 13608 \, x - \frac {6561 \, {\left (40 \, x^{3} + 315 \, x^{2} + 840 \, x + 756\right )}}{x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81} + 61236\right )} \log \left (3 \, x\right )^{4} + 65536 \, {\left (x^{4} - 12 \, x^{3} + 90 \, x^{2} - 540 \, x - \frac {243 \, {\left (56 \, x^{3} + 420 \, x^{2} + 1080 \, x + 945\right )}}{x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81} + 2835\right )} \log \left (3 \, x\right )^{2} + 65536 \, x^{2} - 786432 \, x - \frac {1769472 \, {\left (20 \, x^{3} + 135 \, x^{2} + 324 \, x + 270\right )}}{x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81} \] Input:
integrate(((2560*x^14+10752*x^13)*log(3*x)^8+(2048*x^14+6144*x^13)*log(3*x )^7+(32768*x^12+147456*x^11)*log(3*x)^6+(24576*x^12+73728*x^11)*log(3*x)^5 +(147456*x^10+737280*x^9)*log(3*x)^4+(98304*x^10+294912*x^9)*log(3*x)^3+(2 62144*x^8+1572864*x^7)*log(3*x)^2+(131072*x^8+393216*x^7)*log(3*x)+131072* x^6+1179648*x^5)/(x^5+15*x^4+90*x^3+270*x^2+405*x+243),x, algorithm="giac" )
Output:
256*(x^10 - 12*x^9 + 90*x^8 - 540*x^7 + 2835*x^6 - 13608*x^5 + 61236*x^4 - 262440*x^3 + 1082565*x^2 - 4330260*x - 177147*(364*x^3 + 3003*x^2 + 8316* x + 7722)/(x^4 + 12*x^3 + 54*x^2 + 108*x + 81) + 16888014)*log(3*x)^8 + 40 96*(x^8 - 12*x^7 + 90*x^6 - 540*x^5 + 2835*x^4 - 13608*x^3 + 61236*x^2 - 2 62440*x - 19683*(220*x^3 + 1782*x^2 + 4860*x + 4455)/(x^4 + 12*x^3 + 54*x^ 2 + 108*x + 81) + 1082565)*log(3*x)^6 + 24576*(x^6 - 12*x^5 + 90*x^4 - 540 *x^3 + 2835*x^2 - 13608*x - 6561*(40*x^3 + 315*x^2 + 840*x + 756)/(x^4 + 1 2*x^3 + 54*x^2 + 108*x + 81) + 61236)*log(3*x)^4 + 65536*(x^4 - 12*x^3 + 9 0*x^2 - 540*x - 243*(56*x^3 + 420*x^2 + 1080*x + 945)/(x^4 + 12*x^3 + 54*x ^2 + 108*x + 81) + 2835)*log(3*x)^2 + 65536*x^2 - 786432*x - 1769472*(20*x ^3 + 135*x^2 + 324*x + 270)/(x^4 + 12*x^3 + 54*x^2 + 108*x + 81)
Time = 3.47 (sec) , antiderivative size = 170, normalized size of antiderivative = 6.54 \[ \int \frac {1179648 x^5+131072 x^6+\left (393216 x^7+131072 x^8\right ) \log (3 x)+\left (1572864 x^7+262144 x^8\right ) \log ^2(3 x)+\left (294912 x^9+98304 x^{10}\right ) \log ^3(3 x)+\left (737280 x^9+147456 x^{10}\right ) \log ^4(3 x)+\left (73728 x^{11}+24576 x^{12}\right ) \log ^5(3 x)+\left (147456 x^{11}+32768 x^{12}\right ) \log ^6(3 x)+\left (6144 x^{13}+2048 x^{14}\right ) \log ^7(3 x)+\left (10752 x^{13}+2560 x^{14}\right ) \log ^8(3 x)}{243+405 x+270 x^2+90 x^3+15 x^4+x^5} \, dx=65536\,x^2-\frac {35389440\,x^3+238878720\,x^2+573308928\,x+477757440}{x^4+12\,x^3+54\,x^2+108\,x+81}-786432\,x+\frac {65536\,x^8\,{\ln \left (3\,x\right )}^2}{x^4+12\,x^3+54\,x^2+108\,x+81}+\frac {24576\,x^{10}\,{\ln \left (3\,x\right )}^4}{x^4+12\,x^3+54\,x^2+108\,x+81}+\frac {4096\,x^{12}\,{\ln \left (3\,x\right )}^6}{x^4+12\,x^3+54\,x^2+108\,x+81}+\frac {256\,x^{14}\,{\ln \left (3\,x\right )}^8}{x^4+12\,x^3+54\,x^2+108\,x+81} \] Input:
int((log(3*x)*(393216*x^7 + 131072*x^8) + log(3*x)^7*(6144*x^13 + 2048*x^1 4) + log(3*x)^8*(10752*x^13 + 2560*x^14) + log(3*x)^5*(73728*x^11 + 24576* x^12) + log(3*x)^6*(147456*x^11 + 32768*x^12) + log(3*x)^3*(294912*x^9 + 9 8304*x^10) + log(3*x)^4*(737280*x^9 + 147456*x^10) + log(3*x)^2*(1572864*x ^7 + 262144*x^8) + 1179648*x^5 + 131072*x^6)/(405*x + 270*x^2 + 90*x^3 + 1 5*x^4 + x^5 + 243),x)
Output:
65536*x^2 - (573308928*x + 238878720*x^2 + 35389440*x^3 + 477757440)/(108* x + 54*x^2 + 12*x^3 + x^4 + 81) - 786432*x + (65536*x^8*log(3*x)^2)/(108*x + 54*x^2 + 12*x^3 + x^4 + 81) + (24576*x^10*log(3*x)^4)/(108*x + 54*x^2 + 12*x^3 + x^4 + 81) + (4096*x^12*log(3*x)^6)/(108*x + 54*x^2 + 12*x^3 + x^ 4 + 81) + (256*x^14*log(3*x)^8)/(108*x + 54*x^2 + 12*x^3 + x^4 + 81)
Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \frac {1179648 x^5+131072 x^6+\left (393216 x^7+131072 x^8\right ) \log (3 x)+\left (1572864 x^7+262144 x^8\right ) \log ^2(3 x)+\left (294912 x^9+98304 x^{10}\right ) \log ^3(3 x)+\left (737280 x^9+147456 x^{10}\right ) \log ^4(3 x)+\left (73728 x^{11}+24576 x^{12}\right ) \log ^5(3 x)+\left (147456 x^{11}+32768 x^{12}\right ) \log ^6(3 x)+\left (6144 x^{13}+2048 x^{14}\right ) \log ^7(3 x)+\left (10752 x^{13}+2560 x^{14}\right ) \log ^8(3 x)}{243+405 x+270 x^2+90 x^3+15 x^4+x^5} \, dx=\frac {256 x^{6} \left (\mathrm {log}\left (3 x \right )^{8} x^{8}+16 \mathrm {log}\left (3 x \right )^{6} x^{6}+96 \mathrm {log}\left (3 x \right )^{4} x^{4}+256 \mathrm {log}\left (3 x \right )^{2} x^{2}+256\right )}{x^{4}+12 x^{3}+54 x^{2}+108 x +81} \] Input:
int(((2560*x^14+10752*x^13)*log(3*x)^8+(2048*x^14+6144*x^13)*log(3*x)^7+(3 2768*x^12+147456*x^11)*log(3*x)^6+(24576*x^12+73728*x^11)*log(3*x)^5+(1474 56*x^10+737280*x^9)*log(3*x)^4+(98304*x^10+294912*x^9)*log(3*x)^3+(262144* x^8+1572864*x^7)*log(3*x)^2+(131072*x^8+393216*x^7)*log(3*x)+131072*x^6+11 79648*x^5)/(x^5+15*x^4+90*x^3+270*x^2+405*x+243),x)
Output:
(256*x**6*(log(3*x)**8*x**8 + 16*log(3*x)**6*x**6 + 96*log(3*x)**4*x**4 + 256*log(3*x)**2*x**2 + 256))/(x**4 + 12*x**3 + 54*x**2 + 108*x + 81)