\[ \int \frac {-1000-5000 x-6200 x^2-2900 x^3-1000 x^4-1000 x^5-400 x^6+\left (200+1850 x+1600 x^2+350 x^3+300 x^4+300 x^5\right ) \log (5)+\left (-200 x-50 x^2-50 x^4\right ) \log ^2(5)+\left (-200-1850 x-1600 x^2-350 x^3-300 x^4-300 x^5+\left (400 x+100 x^2+100 x^4\right ) \log (5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (-200 x-50 x^2-50 x^4\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )}{256 x^3+448 x^4+288 x^5+144 x^6+104 x^7+48 x^8+8 x^9+\left (-192 x^3-240 x^4-96 x^5-60 x^6-48 x^7-12 x^8\right ) \log (5)+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7\right ) \log ^2(5)+\left (-4 x^3-x^4-x^6\right ) \log ^3(5)+\left (192 x^3+240 x^4+96 x^5+60 x^6+48 x^7+12 x^8+\left (-96 x^3-72 x^4-12 x^5-24 x^6-12 x^7\right ) \log (5)+\left (12 x^3+3 x^4+3 x^6\right ) \log ^2(5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7+\left (-12 x^3-3 x^4-3 x^6\right ) \log (5)\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )+\left (4 x^3+x^4+x^6\right ) \log ^3\left (\frac {x}{4+x+x^3}\right )} \, dx \]
Optimal antiderivative \[ \left (5-\frac {5}{\left (\ln \! \left (5\right )-\ln \! \left (\frac {x}{x^{3}+x +4}\right )-4-2 x \right ) x}\right )^{2} \]
command
Integrate[(-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 + 288*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]
Mathematica 13.1 output
\[ \int \frac {-1000-5000 x-6200 x^2-2900 x^3-1000 x^4-1000 x^5-400 x^6+\left (200+1850 x+1600 x^2+350 x^3+300 x^4+300 x^5\right ) \log (5)+\left (-200 x-50 x^2-50 x^4\right ) \log ^2(5)+\left (-200-1850 x-1600 x^2-350 x^3-300 x^4-300 x^5+\left (400 x+100 x^2+100 x^4\right ) \log (5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (-200 x-50 x^2-50 x^4\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )}{256 x^3+448 x^4+288 x^5+144 x^6+104 x^7+48 x^8+8 x^9+\left (-192 x^3-240 x^4-96 x^5-60 x^6-48 x^7-12 x^8\right ) \log (5)+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7\right ) \log ^2(5)+\left (-4 x^3-x^4-x^6\right ) \log ^3(5)+\left (192 x^3+240 x^4+96 x^5+60 x^6+48 x^7+12 x^8+\left (-96 x^3-72 x^4-12 x^5-24 x^6-12 x^7\right ) \log (5)+\left (12 x^3+3 x^4+3 x^6\right ) \log ^2(5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7+\left (-12 x^3-3 x^4-3 x^6\right ) \log (5)\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )+\left (4 x^3+x^4+x^6\right ) \log ^3\left (\frac {x}{4+x+x^3}\right )} \, dx \]
Mathematica 12.3 output
\[ -\frac {25 \left (x \left (32+136 x+96 x^2-48 x^3+66 x^4+45 x^5-18 x^6+20 x^7+6 x^8-x^9+2 x^{10}\right ) \log ^3\left (\frac {x}{5 \left (4+x+x^3\right )}\right )+\left (4+x+x^3\right ) \log ^2\left (\frac {x}{5 \left (4+x+x^3\right )}\right ) \left (-16+8 x^9+x^7 (13-2 \log (5))+2 x (-1+8 \log (5))+8 x^2 (23+8 \log (5))+x^6 (68+8 \log (5))-4 x^4 (11+9 \log (5))+x^5 (-2+26 \log (5))+x^3 (227+32 \log (5))+x^8 (-2+\log (625))-2 x \left (8+32 x+16 x^2-18 x^3+13 x^4+4 x^5-x^6+2 x^7\right ) \log \left (\frac {x}{4+x+x^3}\right )\right )+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right ) \left (8 x^{13}+2 x^{12} (6+5 \log (5))+x^4 \left (1356-31 \log (5)-48 \log ^2(5)\right )+x^7 \left (754+50 \log (5)-18 \log ^2(5)\right )+x^{10} \left (160+21 \log (5)-\log ^2(5)\right )+2 x^9 \left (71+73 \log (5)+3 \log ^2(5)\right )+8 x \left (-63+9 \log (5)+4 \log ^2(5)\right )+x^8 \left (-92+63 \log (5)+20 \log ^2(5)\right )+4 x^3 \left (655+385 \log (5)+24 \log ^2(5)\right )+2 x^5 \left (94+142 \log (5)+33 \log ^2(5)\right )+x^6 \left (1072+727 \log (5)+45 \log ^2(5)\right )+2 x^2 \left (404+603 \log (5)+68 \log ^2(5)\right )-32 (7+\log (25))+2 x^{11} \left (-14+\log ^2(5)+\log (625)\right )-\left (4+x+x^3\right ) \left (-16+10 x^9+x^7 (11-2 \log (5))+4 x^8 (2+\log (5))+2 x (11+8 \log (5))+8 x^2 (37+8 \log (5))+x^6 (98+8 \log (5))-4 x^4 (23+9 \log (5))+x^5 (20+26 \log (5))+x^3 (315+32 \log (5))\right ) \log \left (\frac {x}{4+x+x^3}\right )+x \left (32+136 x+96 x^2-48 x^3+66 x^4+45 x^5-18 x^6+20 x^7+6 x^8-x^9+2 x^{10}\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )\right )-2 \left (8 x^{14}-8 x^{13} (1+\log (5))+16 (1+7 \log (5))-4 x^{10} (1+25 \log (5))+x^6 \left (230-536 \log (5)-115 \log ^2(5)\right )+x \left (224+220 \log (5)-48 \log ^2(5)\right )+x^8 \left (656-38 \log (5)-30 \log ^2(5)\right )+x^{12} \left (2+6 \log (5)-\log ^2(5)\right )+x^7 \left (730-549 \log (5)+13 \log ^2(5)\right )+2 x^4 \left (1250-539 \log (5)+20 \log ^2(5)\right )-x^9 \left (14+23 \log (5)+24 \log ^2(5)\right )-2 x^5 \left (-544+83 \log (5)+38 \log ^2(5)\right )-4 x^2 \left (-262+149 \log (5)+59 \log ^2(5)\right )-2 x^3 \left (-1156+871 \log (5)+116 \log ^2(5)\right )-5 x^{11} \left (-26+\log ^2(5)+\log (25)\right )+\left (-112+100 x^{10}+8 x^{13}+x^4 (1078-80 \log (5))+x^7 (549-26 \log (5))+10 x^{11} (1+\log (5))+4 x (-55+24 \log (5))+x^9 (23+48 \log (5))+x^8 (38+60 \log (5))+2 x^5 (83+76 \log (5))+x^6 (536+230 \log (5))+2 x^3 (871+232 \log (5))+x^2 (596+472 \log (5))+x^{12} (-6+\log (25))\right ) \log \left (\frac {x}{4+x+x^3}\right )-x \left (48+236 x+232 x^2-40 x^3+76 x^4+115 x^5-13 x^6+30 x^7+24 x^8+5 x^{10}+x^{11}\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )\right )\right )}{4 x^2 \left (2+4 x+x^2-x^3+x^4\right )^3 \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^2} \]