Integrand size = 462, antiderivative size = 31 \[ \int \frac {-1000-5000 x-6200 x^2-2900 x^3-1000 x^4-1000 x^5-400 x^6+\left (200+1850 x+1600 x^2+350 x^3+300 x^4+300 x^5\right ) \log (5)+\left (-200 x-50 x^2-50 x^4\right ) \log ^2(5)+\left (-200-1850 x-1600 x^2-350 x^3-300 x^4-300 x^5+\left (400 x+100 x^2+100 x^4\right ) \log (5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (-200 x-50 x^2-50 x^4\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )}{256 x^3+448 x^4+288 x^5+144 x^6+104 x^7+48 x^8+8 x^9+\left (-192 x^3-240 x^4-96 x^5-60 x^6-48 x^7-12 x^8\right ) \log (5)+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7\right ) \log ^2(5)+\left (-4 x^3-x^4-x^6\right ) \log ^3(5)+\left (192 x^3+240 x^4+96 x^5+60 x^6+48 x^7+12 x^8+\left (-96 x^3-72 x^4-12 x^5-24 x^6-12 x^7\right ) \log (5)+\left (12 x^3+3 x^4+3 x^6\right ) \log ^2(5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7+\left (-12 x^3-3 x^4-3 x^6\right ) \log (5)\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )+\left (4 x^3+x^4+x^6\right ) \log ^3\left (\frac {x}{4+x+x^3}\right )} \, dx=\left (5-\frac {5}{x \left (-4-2 x+\log (5)-\log \left (\frac {x}{4+x+x^3}\right )\right )}\right )^2 \]
\[ \int \frac {-1000-5000 x-6200 x^2-2900 x^3-1000 x^4-1000 x^5-400 x^6+\left (200+1850 x+1600 x^2+350 x^3+300 x^4+300 x^5\right ) \log (5)+\left (-200 x-50 x^2-50 x^4\right ) \log ^2(5)+\left (-200-1850 x-1600 x^2-350 x^3-300 x^4-300 x^5+\left (400 x+100 x^2+100 x^4\right ) \log (5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (-200 x-50 x^2-50 x^4\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )}{256 x^3+448 x^4+288 x^5+144 x^6+104 x^7+48 x^8+8 x^9+\left (-192 x^3-240 x^4-96 x^5-60 x^6-48 x^7-12 x^8\right ) \log (5)+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7\right ) \log ^2(5)+\left (-4 x^3-x^4-x^6\right ) \log ^3(5)+\left (192 x^3+240 x^4+96 x^5+60 x^6+48 x^7+12 x^8+\left (-96 x^3-72 x^4-12 x^5-24 x^6-12 x^7\right ) \log (5)+\left (12 x^3+3 x^4+3 x^6\right ) \log ^2(5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7+\left (-12 x^3-3 x^4-3 x^6\right ) \log (5)\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )+\left (4 x^3+x^4+x^6\right ) \log ^3\left (\frac {x}{4+x+x^3}\right )} \, dx=\int \frac {-1000-5000 x-6200 x^2-2900 x^3-1000 x^4-1000 x^5-400 x^6+\left (200+1850 x+1600 x^2+350 x^3+300 x^4+300 x^5\right ) \log (5)+\left (-200 x-50 x^2-50 x^4\right ) \log ^2(5)+\left (-200-1850 x-1600 x^2-350 x^3-300 x^4-300 x^5+\left (400 x+100 x^2+100 x^4\right ) \log (5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (-200 x-50 x^2-50 x^4\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )}{256 x^3+448 x^4+288 x^5+144 x^6+104 x^7+48 x^8+8 x^9+\left (-192 x^3-240 x^4-96 x^5-60 x^6-48 x^7-12 x^8\right ) \log (5)+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7\right ) \log ^2(5)+\left (-4 x^3-x^4-x^6\right ) \log ^3(5)+\left (192 x^3+240 x^4+96 x^5+60 x^6+48 x^7+12 x^8+\left (-96 x^3-72 x^4-12 x^5-24 x^6-12 x^7\right ) \log (5)+\left (12 x^3+3 x^4+3 x^6\right ) \log ^2(5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7+\left (-12 x^3-3 x^4-3 x^6\right ) \log (5)\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )+\left (4 x^3+x^4+x^6\right ) \log ^3\left (\frac {x}{4+x+x^3}\right )} \, dx \]
Integrate[(-1000 - 5000*x - 6200*x^2 - 2900*x^3 - 1000*x^4 - 1000*x^5 - 40 0*x^6 + (200 + 1850*x + 1600*x^2 + 350*x^3 + 300*x^4 + 300*x^5)*Log[5] + ( -200*x - 50*x^2 - 50*x^4)*Log[5]^2 + (-200 - 1850*x - 1600*x^2 - 350*x^3 - 300*x^4 - 300*x^5 + (400*x + 100*x^2 + 100*x^4)*Log[5])*Log[x/(4 + x + x^ 3)] + (-200*x - 50*x^2 - 50*x^4)*Log[x/(4 + x + x^3)]^2)/(256*x^3 + 448*x^ 4 + 288*x^5 + 144*x^6 + 104*x^7 + 48*x^8 + 8*x^9 + (-192*x^3 - 240*x^4 - 9 6*x^5 - 60*x^6 - 48*x^7 - 12*x^8)*Log[5] + (48*x^3 + 36*x^4 + 6*x^5 + 12*x ^6 + 6*x^7)*Log[5]^2 + (-4*x^3 - x^4 - x^6)*Log[5]^3 + (192*x^3 + 240*x^4 + 96*x^5 + 60*x^6 + 48*x^7 + 12*x^8 + (-96*x^3 - 72*x^4 - 12*x^5 - 24*x^6 - 12*x^7)*Log[5] + (12*x^3 + 3*x^4 + 3*x^6)*Log[5]^2)*Log[x/(4 + x + x^3)] + (48*x^3 + 36*x^4 + 6*x^5 + 12*x^6 + 6*x^7 + (-12*x^3 - 3*x^4 - 3*x^6)*L og[5])*Log[x/(4 + x + x^3)]^2 + (4*x^3 + x^4 + x^6)*Log[x/(4 + x + x^3)]^3 ),x]
Integrate[(-1000 - 5000*x - 6200*x^2 - 2900*x^3 - 1000*x^4 - 1000*x^5 - 40 0*x^6 + (200 + 1850*x + 1600*x^2 + 350*x^3 + 300*x^4 + 300*x^5)*Log[5] + ( -200*x - 50*x^2 - 50*x^4)*Log[5]^2 + (-200 - 1850*x - 1600*x^2 - 350*x^3 - 300*x^4 - 300*x^5 + (400*x + 100*x^2 + 100*x^4)*Log[5])*Log[x/(4 + x + x^ 3)] + (-200*x - 50*x^2 - 50*x^4)*Log[x/(4 + x + x^3)]^2)/(256*x^3 + 448*x^ 4 + 288*x^5 + 144*x^6 + 104*x^7 + 48*x^8 + 8*x^9 + (-192*x^3 - 240*x^4 - 9 6*x^5 - 60*x^6 - 48*x^7 - 12*x^8)*Log[5] + (48*x^3 + 36*x^4 + 6*x^5 + 12*x ^6 + 6*x^7)*Log[5]^2 + (-4*x^3 - x^4 - x^6)*Log[5]^3 + (192*x^3 + 240*x^4 + 96*x^5 + 60*x^6 + 48*x^7 + 12*x^8 + (-96*x^3 - 72*x^4 - 12*x^5 - 24*x^6 - 12*x^7)*Log[5] + (12*x^3 + 3*x^4 + 3*x^6)*Log[5]^2)*Log[x/(4 + x + x^3)] + (48*x^3 + 36*x^4 + 6*x^5 + 12*x^6 + 6*x^7 + (-12*x^3 - 3*x^4 - 3*x^6)*L og[5])*Log[x/(4 + x + x^3)]^2 + (4*x^3 + x^4 + x^6)*Log[x/(4 + x + x^3)]^3 ), x]
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.
| \(\displaystyle \int \frac {-400 x^6-1000 x^5-1000 x^4-2900 x^3-6200 x^2+\left (-50 x^4-50 x^2-200 x\right ) \log ^2(5)+\left (-50 x^4-50 x^2-200 x\right ) \log ^2\left (\frac {x}{x^3+x+4}\right )+\left (-300 x^5-300 x^4-350 x^3-1600 x^2+\left (100 x^4+100 x^2+400 x\right ) \log (5)-1850 x-200\right ) \log \left (\frac {x}{x^3+x+4}\right )+\left (300 x^5+300 x^4+350 x^3+1600 x^2+1850 x+200\right ) \log (5)-5000 x-1000}{8 x^9+48 x^8+104 x^7+144 x^6+288 x^5+448 x^4+256 x^3+\left (x^6+x^4+4 x^3\right ) \log ^3\left (\frac {x}{x^3+x+4}\right )+\left (-x^6-x^4-4 x^3\right ) \log ^3(5)+\left (6 x^7+12 x^6+6 x^5+36 x^4+48 x^3+\left (-3 x^6-3 x^4-12 x^3\right ) \log (5)\right ) \log ^2\left (\frac {x}{x^3+x+4}\right )+\left (6 x^7+12 x^6+6 x^5+36 x^4+48 x^3\right ) \log ^2(5)+\left (12 x^8+48 x^7+60 x^6+96 x^5+240 x^4+192 x^3+\left (3 x^6+3 x^4+12 x^3\right ) \log ^2(5)+\left (-12 x^7-24 x^6-12 x^5-72 x^4-96 x^3\right ) \log (5)\right ) \log \left (\frac {x}{x^3+x+4}\right )+\left (-12 x^8-48 x^7-60 x^6-96 x^5-240 x^4-192 x^3\right ) \log (5)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {50 \left (-8 x^6-x^5 (20-6 \log (5))-x^4 \left (20+\log ^2(5)-6 \log (5)\right )-\left (x^3+x+4\right ) x \log ^2\left (\frac {x}{x^3+x+4}\right )-x^3 (58-7 \log (5))-x^2 \left (124+\log ^2(5)-32 \log (5)\right )-\left (6 x^5-2 x^4 (\log (5)-3)+7 x^3-2 x^2 (\log (5)-16)+x (37-8 \log (5))+4\right ) \log \left (\frac {x}{x^3+x+4}\right )-x \left (100+4 \log ^2(5)-37 \log (5)\right )+4 (\log (5)-5)\right )}{x^3 \left (x^3+x+4\right ) \left (\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+2 x+4\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 50 \int -\frac {8 x^6+2 (10-\log (125)) x^5+\left (20-6 \log (5)+\log ^2(5)\right ) x^4+(58-7 \log (5)) x^3+\left (124-32 \log (5)+\log ^2(5)\right ) x^2+\left (x^3+x+4\right ) \log ^2\left (\frac {x}{x^3+x+4}\right ) x+\left (100-37 \log (5)+4 \log ^2(5)\right ) x+\left (6 x^5+2 (3-\log (5)) x^4+7 x^3+2 (16-\log (5)) x^2+(37-8 \log (5)) x+4\right ) \log \left (\frac {x}{x^3+x+4}\right )+4 (5-\log (5))}{x^3 \left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -50 \int \frac {8 x^6+2 (10-\log (125)) x^5+\left (20-6 \log (5)+\log ^2(5)\right ) x^4+(58-7 \log (5)) x^3+\left (124-32 \log (5)+\log ^2(5)\right ) x^2+\left (x^3+x+4\right ) \log ^2\left (\frac {x}{x^3+x+4}\right ) x+\left (100-37 \log (5)+4 \log ^2(5)\right ) x+\left (6 x^5+2 (3-\log (5)) x^4+7 x^3+2 (16-\log (5)) x^2+(37-8 \log (5)) x+4\right ) \log \left (\frac {x}{x^3+x+4}\right )+4 (5-\log (5))}{x^3 \left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -50 \int \left (\frac {8 x^3}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}-\frac {2 (-10+\log (125)) x^2}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}+\frac {\left (20-6 \log (5)+\log ^2(5)\right ) x}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}+\frac {58-7 \log (5)}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}+\frac {124-32 \log (5)+\log ^2(5)}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3 x}+\frac {\log ^2\left (\frac {x}{x^3+x+4}\right )}{\left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3 x^2}+\frac {100-37 \log (5)+4 \log ^2(5)}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3 x^2}+\frac {\left (6 x^5+6 \left (1-\frac {\log (5)}{3}\right ) x^4+7 x^3+32 \left (1-\frac {\log (5)}{16}\right ) x^2+37 \left (1-\frac {8 \log (5)}{37}\right ) x+4\right ) \log \left (\frac {x}{x^3+x+4}\right )}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3 x^3}-\frac {4 (-5+\log (5))}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3 x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -50 \left (8 \int \frac {1}{\left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+(5-\log (5)) \int \frac {1}{x^3 \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+\frac {1}{4} \left (100-37 \log (5)+4 \log ^2(5)\right ) \int \frac {1}{x^2 \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx-\frac {1}{4} (5-\log (5)) \int \frac {1}{x^2 \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx-\frac {1}{16} \left (100-37 \log (5)+4 \log ^2(5)\right ) \int \frac {1}{x \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+\frac {1}{4} \left (124-32 \log (5)+\log ^2(5)\right ) \int \frac {1}{x \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+\frac {1}{16} (5-\log (5)) \int \frac {1}{x \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+\frac {1}{16} \left (100-37 \log (5)+4 \log ^2(5)\right ) \int \frac {1}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx-\frac {1}{4} \left (124-32 \log (5)+\log ^2(5)\right ) \int \frac {1}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx-\frac {17}{16} (5-\log (5)) \int \frac {1}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+(58-7 \log (5)) \int \frac {1}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx-32 \int \frac {1}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx-\frac {1}{4} \left (100-37 \log (5)+4 \log ^2(5)\right ) \int \frac {x}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+\left (20-6 \log (5)+\log ^2(5)\right ) \int \frac {x}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+\frac {1}{4} (5-\log (5)) \int \frac {x}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx-8 \int \frac {x}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+2 (10-\log (125)) \int \frac {x^2}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+\frac {1}{16} \left (100-37 \log (5)+4 \log ^2(5)\right ) \int \frac {x^2}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx-\frac {1}{4} \left (124-32 \log (5)+\log ^2(5)\right ) \int \frac {x^2}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx-\frac {1}{16} (5-\log (5)) \int \frac {x^2}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+\int \frac {\log \left (\frac {x}{x^3+x+4}\right )}{x^3 \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+(9-\log (25)) \int \frac {\log \left (\frac {x}{x^3+x+4}\right )}{x^2 \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+\frac {23}{4} \int \frac {\log \left (\frac {x}{x^3+x+4}\right )}{x \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+\frac {1}{4} \int \frac {\log \left (\frac {x}{x^3+x+4}\right )}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx-3 \int \frac {x \log \left (\frac {x}{x^3+x+4}\right )}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+\frac {1}{4} \int \frac {x^2 \log \left (\frac {x}{x^3+x+4}\right )}{\left (x^3+x+4\right ) \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx+\int \frac {\log ^2\left (\frac {x}{x^3+x+4}\right )}{x^2 \left (2 x+\log \left (\frac {x}{5 \left (x^3+x+4\right )}\right )+4\right )^3}dx\right )\) |
Int[(-1000 - 5000*x - 6200*x^2 - 2900*x^3 - 1000*x^4 - 1000*x^5 - 400*x^6 + (200 + 1850*x + 1600*x^2 + 350*x^3 + 300*x^4 + 300*x^5)*Log[5] + (-200*x - 50*x^2 - 50*x^4)*Log[5]^2 + (-200 - 1850*x - 1600*x^2 - 350*x^3 - 300*x ^4 - 300*x^5 + (400*x + 100*x^2 + 100*x^4)*Log[5])*Log[x/(4 + x + x^3)] + (-200*x - 50*x^2 - 50*x^4)*Log[x/(4 + x + x^3)]^2)/(256*x^3 + 448*x^4 + 28 8*x^5 + 144*x^6 + 104*x^7 + 48*x^8 + 8*x^9 + (-192*x^3 - 240*x^4 - 96*x^5 - 60*x^6 - 48*x^7 - 12*x^8)*Log[5] + (48*x^3 + 36*x^4 + 6*x^5 + 12*x^6 + 6 *x^7)*Log[5]^2 + (-4*x^3 - x^4 - x^6)*Log[5]^3 + (192*x^3 + 240*x^4 + 96*x ^5 + 60*x^6 + 48*x^7 + 12*x^8 + (-96*x^3 - 72*x^4 - 12*x^5 - 24*x^6 - 12*x ^7)*Log[5] + (12*x^3 + 3*x^4 + 3*x^6)*Log[5]^2)*Log[x/(4 + x + x^3)] + (48 *x^3 + 36*x^4 + 6*x^5 + 12*x^6 + 6*x^7 + (-12*x^3 - 3*x^4 - 3*x^6)*Log[5]) *Log[x/(4 + x + x^3)]^2 + (4*x^3 + x^4 + x^6)*Log[x/(4 + x + x^3)]^3),x]
3.2.24.3.1 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[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.76 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.84
| method | result | size |
| risch | \(-\frac {25 \left (2 x \ln \left (5\right )-4 x^{2}-2 x \ln \left (\frac {x}{x^{3}+x +4}\right )-8 x -1\right )}{x^{2} {\left (\ln \left (5\right )-\ln \left (\frac {x}{x^{3}+x +4}\right )-4-2 x \right )}^{2}}\) | \(57\) |
| parallelrisch | \(\frac {25+200 x -50 x \ln \left (5\right )+100 x^{2}+50 x \ln \left (\frac {x}{x^{3}+x +4}\right )}{x^{2} \left (\ln \left (5\right )^{2}-4 x \ln \left (5\right )-2 \ln \left (5\right ) \ln \left (\frac {x}{x^{3}+x +4}\right )+4 x^{2}+4 x \ln \left (\frac {x}{x^{3}+x +4}\right )+\ln \left (\frac {x}{x^{3}+x +4}\right )^{2}-8 \ln \left (5\right )+16 x +8 \ln \left (\frac {x}{x^{3}+x +4}\right )+16\right )}\) | \(114\) |
int(((-50*x^4-50*x^2-200*x)*ln(x/(x^3+x+4))^2+((100*x^4+100*x^2+400*x)*ln( 5)-300*x^5-300*x^4-350*x^3-1600*x^2-1850*x-200)*ln(x/(x^3+x+4))+(-50*x^4-5 0*x^2-200*x)*ln(5)^2+(300*x^5+300*x^4+350*x^3+1600*x^2+1850*x+200)*ln(5)-4 00*x^6-1000*x^5-1000*x^4-2900*x^3-6200*x^2-5000*x-1000)/((x^6+x^4+4*x^3)*l n(x/(x^3+x+4))^3+((-3*x^6-3*x^4-12*x^3)*ln(5)+6*x^7+12*x^6+6*x^5+36*x^4+48 *x^3)*ln(x/(x^3+x+4))^2+((3*x^6+3*x^4+12*x^3)*ln(5)^2+(-12*x^7-24*x^6-12*x ^5-72*x^4-96*x^3)*ln(5)+12*x^8+48*x^7+60*x^6+96*x^5+240*x^4+192*x^3)*ln(x/ (x^3+x+4))+(-x^6-x^4-4*x^3)*ln(5)^3+(6*x^7+12*x^6+6*x^5+36*x^4+48*x^3)*ln( 5)^2+(-12*x^8-48*x^7-60*x^6-96*x^5-240*x^4-192*x^3)*ln(5)+8*x^9+48*x^8+104 *x^7+144*x^6+288*x^5+448*x^4+256*x^3),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (32) = 64\).
Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.81 \[ \int \frac {-1000-5000 x-6200 x^2-2900 x^3-1000 x^4-1000 x^5-400 x^6+\left (200+1850 x+1600 x^2+350 x^3+300 x^4+300 x^5\right ) \log (5)+\left (-200 x-50 x^2-50 x^4\right ) \log ^2(5)+\left (-200-1850 x-1600 x^2-350 x^3-300 x^4-300 x^5+\left (400 x+100 x^2+100 x^4\right ) \log (5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (-200 x-50 x^2-50 x^4\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )}{256 x^3+448 x^4+288 x^5+144 x^6+104 x^7+48 x^8+8 x^9+\left (-192 x^3-240 x^4-96 x^5-60 x^6-48 x^7-12 x^8\right ) \log (5)+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7\right ) \log ^2(5)+\left (-4 x^3-x^4-x^6\right ) \log ^3(5)+\left (192 x^3+240 x^4+96 x^5+60 x^6+48 x^7+12 x^8+\left (-96 x^3-72 x^4-12 x^5-24 x^6-12 x^7\right ) \log (5)+\left (12 x^3+3 x^4+3 x^6\right ) \log ^2(5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7+\left (-12 x^3-3 x^4-3 x^6\right ) \log (5)\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )+\left (4 x^3+x^4+x^6\right ) \log ^3\left (\frac {x}{4+x+x^3}\right )} \, dx=\frac {25 \, {\left (4 \, x^{2} - 2 \, x \log \left (5\right ) + 2 \, x \log \left (\frac {x}{x^{3} + x + 4}\right ) + 8 \, x + 1\right )}}{4 \, x^{4} + x^{2} \log \left (5\right )^{2} + x^{2} \log \left (\frac {x}{x^{3} + x + 4}\right )^{2} + 16 \, x^{3} + 16 \, x^{2} - 4 \, {\left (x^{3} + 2 \, x^{2}\right )} \log \left (5\right ) + 2 \, {\left (2 \, x^{3} - x^{2} \log \left (5\right ) + 4 \, x^{2}\right )} \log \left (\frac {x}{x^{3} + x + 4}\right )} \]
integrate(((-50*x^4-50*x^2-200*x)*log(x/(x^3+x+4))^2+((100*x^4+100*x^2+400 *x)*log(5)-300*x^5-300*x^4-350*x^3-1600*x^2-1850*x-200)*log(x/(x^3+x+4))+( -50*x^4-50*x^2-200*x)*log(5)^2+(300*x^5+300*x^4+350*x^3+1600*x^2+1850*x+20 0)*log(5)-400*x^6-1000*x^5-1000*x^4-2900*x^3-6200*x^2-5000*x-1000)/((x^6+x ^4+4*x^3)*log(x/(x^3+x+4))^3+((-3*x^6-3*x^4-12*x^3)*log(5)+6*x^7+12*x^6+6* x^5+36*x^4+48*x^3)*log(x/(x^3+x+4))^2+((3*x^6+3*x^4+12*x^3)*log(5)^2+(-12* x^7-24*x^6-12*x^5-72*x^4-96*x^3)*log(5)+12*x^8+48*x^7+60*x^6+96*x^5+240*x^ 4+192*x^3)*log(x/(x^3+x+4))+(-x^6-x^4-4*x^3)*log(5)^3+(6*x^7+12*x^6+6*x^5+ 36*x^4+48*x^3)*log(5)^2+(-12*x^8-48*x^7-60*x^6-96*x^5-240*x^4-192*x^3)*log (5)+8*x^9+48*x^8+104*x^7+144*x^6+288*x^5+448*x^4+256*x^3),x, algorithm=\
25*(4*x^2 - 2*x*log(5) + 2*x*log(x/(x^3 + x + 4)) + 8*x + 1)/(4*x^4 + x^2* log(5)^2 + x^2*log(x/(x^3 + x + 4))^2 + 16*x^3 + 16*x^2 - 4*(x^3 + 2*x^2)* log(5) + 2*(2*x^3 - x^2*log(5) + 4*x^2)*log(x/(x^3 + x + 4)))
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (24) = 48\).
Time = 0.24 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.68 \[ \int \frac {-1000-5000 x-6200 x^2-2900 x^3-1000 x^4-1000 x^5-400 x^6+\left (200+1850 x+1600 x^2+350 x^3+300 x^4+300 x^5\right ) \log (5)+\left (-200 x-50 x^2-50 x^4\right ) \log ^2(5)+\left (-200-1850 x-1600 x^2-350 x^3-300 x^4-300 x^5+\left (400 x+100 x^2+100 x^4\right ) \log (5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (-200 x-50 x^2-50 x^4\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )}{256 x^3+448 x^4+288 x^5+144 x^6+104 x^7+48 x^8+8 x^9+\left (-192 x^3-240 x^4-96 x^5-60 x^6-48 x^7-12 x^8\right ) \log (5)+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7\right ) \log ^2(5)+\left (-4 x^3-x^4-x^6\right ) \log ^3(5)+\left (192 x^3+240 x^4+96 x^5+60 x^6+48 x^7+12 x^8+\left (-96 x^3-72 x^4-12 x^5-24 x^6-12 x^7\right ) \log (5)+\left (12 x^3+3 x^4+3 x^6\right ) \log ^2(5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7+\left (-12 x^3-3 x^4-3 x^6\right ) \log (5)\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )+\left (4 x^3+x^4+x^6\right ) \log ^3\left (\frac {x}{4+x+x^3}\right )} \, dx=\frac {100 x^{2} + 50 x \log {\left (\frac {x}{x^{3} + x + 4} \right )} - 50 x \log {\left (5 \right )} + 200 x + 25}{4 x^{4} - 4 x^{3} \log {\left (5 \right )} + 16 x^{3} + x^{2} \log {\left (\frac {x}{x^{3} + x + 4} \right )}^{2} - 8 x^{2} \log {\left (5 \right )} + x^{2} \log {\left (5 \right )}^{2} + 16 x^{2} + \left (4 x^{3} - 2 x^{2} \log {\left (5 \right )} + 8 x^{2}\right ) \log {\left (\frac {x}{x^{3} + x + 4} \right )}} \]
integrate(((-50*x**4-50*x**2-200*x)*ln(x/(x**3+x+4))**2+((100*x**4+100*x** 2+400*x)*ln(5)-300*x**5-300*x**4-350*x**3-1600*x**2-1850*x-200)*ln(x/(x**3 +x+4))+(-50*x**4-50*x**2-200*x)*ln(5)**2+(300*x**5+300*x**4+350*x**3+1600* x**2+1850*x+200)*ln(5)-400*x**6-1000*x**5-1000*x**4-2900*x**3-6200*x**2-50 00*x-1000)/((x**6+x**4+4*x**3)*ln(x/(x**3+x+4))**3+((-3*x**6-3*x**4-12*x** 3)*ln(5)+6*x**7+12*x**6+6*x**5+36*x**4+48*x**3)*ln(x/(x**3+x+4))**2+((3*x* *6+3*x**4+12*x**3)*ln(5)**2+(-12*x**7-24*x**6-12*x**5-72*x**4-96*x**3)*ln( 5)+12*x**8+48*x**7+60*x**6+96*x**5+240*x**4+192*x**3)*ln(x/(x**3+x+4))+(-x **6-x**4-4*x**3)*ln(5)**3+(6*x**7+12*x**6+6*x**5+36*x**4+48*x**3)*ln(5)**2 +(-12*x**8-48*x**7-60*x**6-96*x**5-240*x**4-192*x**3)*ln(5)+8*x**9+48*x**8 +104*x**7+144*x**6+288*x**5+448*x**4+256*x**3),x)
(100*x**2 + 50*x*log(x/(x**3 + x + 4)) - 50*x*log(5) + 200*x + 25)/(4*x**4 - 4*x**3*log(5) + 16*x**3 + x**2*log(x/(x**3 + x + 4))**2 - 8*x**2*log(5) + x**2*log(5)**2 + 16*x**2 + (4*x**3 - 2*x**2*log(5) + 8*x**2)*log(x/(x** 3 + x + 4)))
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (32) = 64\).
Time = 0.41 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.26 \[ \int \frac {-1000-5000 x-6200 x^2-2900 x^3-1000 x^4-1000 x^5-400 x^6+\left (200+1850 x+1600 x^2+350 x^3+300 x^4+300 x^5\right ) \log (5)+\left (-200 x-50 x^2-50 x^4\right ) \log ^2(5)+\left (-200-1850 x-1600 x^2-350 x^3-300 x^4-300 x^5+\left (400 x+100 x^2+100 x^4\right ) \log (5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (-200 x-50 x^2-50 x^4\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )}{256 x^3+448 x^4+288 x^5+144 x^6+104 x^7+48 x^8+8 x^9+\left (-192 x^3-240 x^4-96 x^5-60 x^6-48 x^7-12 x^8\right ) \log (5)+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7\right ) \log ^2(5)+\left (-4 x^3-x^4-x^6\right ) \log ^3(5)+\left (192 x^3+240 x^4+96 x^5+60 x^6+48 x^7+12 x^8+\left (-96 x^3-72 x^4-12 x^5-24 x^6-12 x^7\right ) \log (5)+\left (12 x^3+3 x^4+3 x^6\right ) \log ^2(5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7+\left (-12 x^3-3 x^4-3 x^6\right ) \log (5)\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )+\left (4 x^3+x^4+x^6\right ) \log ^3\left (\frac {x}{4+x+x^3}\right )} \, dx=\frac {25 \, {\left (4 \, x^{2} - 2 \, x {\left (\log \left (5\right ) - 4\right )} - 2 \, x \log \left (x^{3} + x + 4\right ) + 2 \, x \log \left (x\right ) + 1\right )}}{4 \, x^{4} - 4 \, x^{3} {\left (\log \left (5\right ) - 4\right )} + x^{2} \log \left (x^{3} + x + 4\right )^{2} + x^{2} \log \left (x\right )^{2} + {\left (\log \left (5\right )^{2} - 8 \, \log \left (5\right ) + 16\right )} x^{2} - 2 \, {\left (2 \, x^{3} - x^{2} {\left (\log \left (5\right ) - 4\right )} + x^{2} \log \left (x\right )\right )} \log \left (x^{3} + x + 4\right ) + 2 \, {\left (2 \, x^{3} - x^{2} {\left (\log \left (5\right ) - 4\right )}\right )} \log \left (x\right )} \]
integrate(((-50*x^4-50*x^2-200*x)*log(x/(x^3+x+4))^2+((100*x^4+100*x^2+400 *x)*log(5)-300*x^5-300*x^4-350*x^3-1600*x^2-1850*x-200)*log(x/(x^3+x+4))+( -50*x^4-50*x^2-200*x)*log(5)^2+(300*x^5+300*x^4+350*x^3+1600*x^2+1850*x+20 0)*log(5)-400*x^6-1000*x^5-1000*x^4-2900*x^3-6200*x^2-5000*x-1000)/((x^6+x ^4+4*x^3)*log(x/(x^3+x+4))^3+((-3*x^6-3*x^4-12*x^3)*log(5)+6*x^7+12*x^6+6* x^5+36*x^4+48*x^3)*log(x/(x^3+x+4))^2+((3*x^6+3*x^4+12*x^3)*log(5)^2+(-12* x^7-24*x^6-12*x^5-72*x^4-96*x^3)*log(5)+12*x^8+48*x^7+60*x^6+96*x^5+240*x^ 4+192*x^3)*log(x/(x^3+x+4))+(-x^6-x^4-4*x^3)*log(5)^3+(6*x^7+12*x^6+6*x^5+ 36*x^4+48*x^3)*log(5)^2+(-12*x^8-48*x^7-60*x^6-96*x^5-240*x^4-192*x^3)*log (5)+8*x^9+48*x^8+104*x^7+144*x^6+288*x^5+448*x^4+256*x^3),x, algorithm=\
25*(4*x^2 - 2*x*(log(5) - 4) - 2*x*log(x^3 + x + 4) + 2*x*log(x) + 1)/(4*x ^4 - 4*x^3*(log(5) - 4) + x^2*log(x^3 + x + 4)^2 + x^2*log(x)^2 + (log(5)^ 2 - 8*log(5) + 16)*x^2 - 2*(2*x^3 - x^2*(log(5) - 4) + x^2*log(x))*log(x^3 + x + 4) + 2*(2*x^3 - x^2*(log(5) - 4))*log(x))
Exception generated. \[ \int \frac {-1000-5000 x-6200 x^2-2900 x^3-1000 x^4-1000 x^5-400 x^6+\left (200+1850 x+1600 x^2+350 x^3+300 x^4+300 x^5\right ) \log (5)+\left (-200 x-50 x^2-50 x^4\right ) \log ^2(5)+\left (-200-1850 x-1600 x^2-350 x^3-300 x^4-300 x^5+\left (400 x+100 x^2+100 x^4\right ) \log (5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (-200 x-50 x^2-50 x^4\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )}{256 x^3+448 x^4+288 x^5+144 x^6+104 x^7+48 x^8+8 x^9+\left (-192 x^3-240 x^4-96 x^5-60 x^6-48 x^7-12 x^8\right ) \log (5)+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7\right ) \log ^2(5)+\left (-4 x^3-x^4-x^6\right ) \log ^3(5)+\left (192 x^3+240 x^4+96 x^5+60 x^6+48 x^7+12 x^8+\left (-96 x^3-72 x^4-12 x^5-24 x^6-12 x^7\right ) \log (5)+\left (12 x^3+3 x^4+3 x^6\right ) \log ^2(5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7+\left (-12 x^3-3 x^4-3 x^6\right ) \log (5)\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )+\left (4 x^3+x^4+x^6\right ) \log ^3\left (\frac {x}{4+x+x^3}\right )} \, dx=\text {Exception raised: TypeError} \]
integrate(((-50*x^4-50*x^2-200*x)*log(x/(x^3+x+4))^2+((100*x^4+100*x^2+400 *x)*log(5)-300*x^5-300*x^4-350*x^3-1600*x^2-1850*x-200)*log(x/(x^3+x+4))+( -50*x^4-50*x^2-200*x)*log(5)^2+(300*x^5+300*x^4+350*x^3+1600*x^2+1850*x+20 0)*log(5)-400*x^6-1000*x^5-1000*x^4-2900*x^3-6200*x^2-5000*x-1000)/((x^6+x ^4+4*x^3)*log(x/(x^3+x+4))^3+((-3*x^6-3*x^4-12*x^3)*log(5)+6*x^7+12*x^6+6* x^5+36*x^4+48*x^3)*log(x/(x^3+x+4))^2+((3*x^6+3*x^4+12*x^3)*log(5)^2+(-12* x^7-24*x^6-12*x^5-72*x^4-96*x^3)*log(5)+12*x^8+48*x^7+60*x^6+96*x^5+240*x^ 4+192*x^3)*log(x/(x^3+x+4))+(-x^6-x^4-4*x^3)*log(5)^3+(6*x^7+12*x^6+6*x^5+ 36*x^4+48*x^3)*log(5)^2+(-12*x^8-48*x^7-60*x^6-96*x^5-240*x^4-192*x^3)*log (5)+8*x^9+48*x^8+104*x^7+144*x^6+288*x^5+448*x^4+256*x^3),x, algorithm=\
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {-1000-5000 x-6200 x^2-2900 x^3-1000 x^4-1000 x^5-400 x^6+\left (200+1850 x+1600 x^2+350 x^3+300 x^4+300 x^5\right ) \log (5)+\left (-200 x-50 x^2-50 x^4\right ) \log ^2(5)+\left (-200-1850 x-1600 x^2-350 x^3-300 x^4-300 x^5+\left (400 x+100 x^2+100 x^4\right ) \log (5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (-200 x-50 x^2-50 x^4\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )}{256 x^3+448 x^4+288 x^5+144 x^6+104 x^7+48 x^8+8 x^9+\left (-192 x^3-240 x^4-96 x^5-60 x^6-48 x^7-12 x^8\right ) \log (5)+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7\right ) \log ^2(5)+\left (-4 x^3-x^4-x^6\right ) \log ^3(5)+\left (192 x^3+240 x^4+96 x^5+60 x^6+48 x^7+12 x^8+\left (-96 x^3-72 x^4-12 x^5-24 x^6-12 x^7\right ) \log (5)+\left (12 x^3+3 x^4+3 x^6\right ) \log ^2(5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7+\left (-12 x^3-3 x^4-3 x^6\right ) \log (5)\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )+\left (4 x^3+x^4+x^6\right ) \log ^3\left (\frac {x}{4+x+x^3}\right )} \, dx=\int -\frac {5000\,x+\ln \left (\frac {x}{x^3+x+4}\right )\,\left (1850\,x-\ln \left (5\right )\,\left (100\,x^4+100\,x^2+400\,x\right )+1600\,x^2+350\,x^3+300\,x^4+300\,x^5+200\right )+{\ln \left (\frac {x}{x^3+x+4}\right )}^2\,\left (50\,x^4+50\,x^2+200\,x\right )+{\ln \left (5\right )}^2\,\left (50\,x^4+50\,x^2+200\,x\right )+6200\,x^2+2900\,x^3+1000\,x^4+1000\,x^5+400\,x^6-\ln \left (5\right )\,\left (300\,x^5+300\,x^4+350\,x^3+1600\,x^2+1850\,x+200\right )+1000}{{\ln \left (\frac {x}{x^3+x+4}\right )}^2\,\left (48\,x^3-\ln \left (5\right )\,\left (3\,x^6+3\,x^4+12\,x^3\right )+36\,x^4+6\,x^5+12\,x^6+6\,x^7\right )+{\ln \left (\frac {x}{x^3+x+4}\right )}^3\,\left (x^6+x^4+4\,x^3\right )-{\ln \left (5\right )}^3\,\left (x^6+x^4+4\,x^3\right )-\ln \left (5\right )\,\left (12\,x^8+48\,x^7+60\,x^6+96\,x^5+240\,x^4+192\,x^3\right )+{\ln \left (5\right )}^2\,\left (6\,x^7+12\,x^6+6\,x^5+36\,x^4+48\,x^3\right )+\ln \left (\frac {x}{x^3+x+4}\right )\,\left ({\ln \left (5\right )}^2\,\left (3\,x^6+3\,x^4+12\,x^3\right )-\ln \left (5\right )\,\left (12\,x^7+24\,x^6+12\,x^5+72\,x^4+96\,x^3\right )+192\,x^3+240\,x^4+96\,x^5+60\,x^6+48\,x^7+12\,x^8\right )+256\,x^3+448\,x^4+288\,x^5+144\,x^6+104\,x^7+48\,x^8+8\,x^9} \,d x \]
int(-(5000*x + log(x/(x + x^3 + 4))*(1850*x - log(5)*(400*x + 100*x^2 + 10 0*x^4) + 1600*x^2 + 350*x^3 + 300*x^4 + 300*x^5 + 200) + log(x/(x + x^3 + 4))^2*(200*x + 50*x^2 + 50*x^4) + log(5)^2*(200*x + 50*x^2 + 50*x^4) + 620 0*x^2 + 2900*x^3 + 1000*x^4 + 1000*x^5 + 400*x^6 - log(5)*(1850*x + 1600*x ^2 + 350*x^3 + 300*x^4 + 300*x^5 + 200) + 1000)/(log(x/(x + x^3 + 4))^2*(4 8*x^3 - log(5)*(12*x^3 + 3*x^4 + 3*x^6) + 36*x^4 + 6*x^5 + 12*x^6 + 6*x^7) + log(x/(x + x^3 + 4))^3*(4*x^3 + x^4 + x^6) - log(5)^3*(4*x^3 + x^4 + x^ 6) - log(5)*(192*x^3 + 240*x^4 + 96*x^5 + 60*x^6 + 48*x^7 + 12*x^8) + log( 5)^2*(48*x^3 + 36*x^4 + 6*x^5 + 12*x^6 + 6*x^7) + log(x/(x + x^3 + 4))*(lo g(5)^2*(12*x^3 + 3*x^4 + 3*x^6) - log(5)*(96*x^3 + 72*x^4 + 12*x^5 + 24*x^ 6 + 12*x^7) + 192*x^3 + 240*x^4 + 96*x^5 + 60*x^6 + 48*x^7 + 12*x^8) + 256 *x^3 + 448*x^4 + 288*x^5 + 144*x^6 + 104*x^7 + 48*x^8 + 8*x^9),x)
int(-(5000*x + log(x/(x + x^3 + 4))*(1850*x - log(5)*(400*x + 100*x^2 + 10 0*x^4) + 1600*x^2 + 350*x^3 + 300*x^4 + 300*x^5 + 200) + log(x/(x + x^3 + 4))^2*(200*x + 50*x^2 + 50*x^4) + log(5)^2*(200*x + 50*x^2 + 50*x^4) + 620 0*x^2 + 2900*x^3 + 1000*x^4 + 1000*x^5 + 400*x^6 - log(5)*(1850*x + 1600*x ^2 + 350*x^3 + 300*x^4 + 300*x^5 + 200) + 1000)/(log(x/(x + x^3 + 4))^2*(4 8*x^3 - log(5)*(12*x^3 + 3*x^4 + 3*x^6) + 36*x^4 + 6*x^5 + 12*x^6 + 6*x^7) + log(x/(x + x^3 + 4))^3*(4*x^3 + x^4 + x^6) - log(5)^3*(4*x^3 + x^4 + x^ 6) - log(5)*(192*x^3 + 240*x^4 + 96*x^5 + 60*x^6 + 48*x^7 + 12*x^8) + log( 5)^2*(48*x^3 + 36*x^4 + 6*x^5 + 12*x^6 + 6*x^7) + log(x/(x + x^3 + 4))*(lo g(5)^2*(12*x^3 + 3*x^4 + 3*x^6) - log(5)*(96*x^3 + 72*x^4 + 12*x^5 + 24*x^ 6 + 12*x^7) + 192*x^3 + 240*x^4 + 96*x^5 + 60*x^6 + 48*x^7 + 12*x^8) + 256 *x^3 + 448*x^4 + 288*x^5 + 144*x^6 + 104*x^7 + 48*x^8 + 8*x^9), x)