22.21 Problem number 3124

$$\int \frac{-1000-5000x-6200x^2-2900x^3-1000x^4-1000x^5-400x^6+(200+1850x+1600x^2+350x^3+300x^4+300x^5)\log (5)+(-200x-50x^2-50x^4){\log }^2(5)+(-200-1850x-1600x^2-350x^3-300x^4-300x^5+(400x+100x^2+100x^4)\log (5))\log \left(\frac{x}{4+x+x^3}\right)+(-200x-50x^2-50x^4){\log }^2\left(\frac{x}{4+x+x^3}\right)}{256x^3+448x^4+288x^5+144x^6+104x^7+48x^8+8x^9+(-192x^3-240x^4-96x^5-60x^6-48x^7-12x^8)\log (5)+(48x^3+36x^4+6x^5+12x^6+6x^7){\log }^2(5)+(-4x^3-x^4-x^6){\log }^3(5)+(192x^3+240x^4+96x^5+60x^6+48x^7+12x^8+(-96x^3-72x^4-12x^5-24x^6-12x^7)\log (5)+(12x^3+3x^4+3x^6){\log }^2(5))\log \left(\frac{x}{4+x+x^3}\right)+(48x^3+36x^4+6x^5+12x^6+6x^7+(-12x^3-3x^4-3x^6)\log (5)){\log }^2\left(\frac{x}{4+x+x^3}\right)+(4x^3+x^4+x^6){\log }^3\left(\frac{x}{4+x+x^3}\right)}dx$$

Optimal antiderivative $${(5-\frac{5}{(\ln \left(5\right)-\ln \left(\frac{x}{x^3+x+4}\right)-4-2x)x})}^2$$

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

$$\int \frac{-1000-5000x-6200x^2-2900x^3-1000x^4-1000x^5-400x^6+(200+1850x+1600x^2+350x^3+300x^4+300x^5)\log (5)+(-200x-50x^2-50x^4){\log }^2(5)+(-200-1850x-1600x^2-350x^3-300x^4-300x^5+(400x+100x^2+100x^4)\log (5))\log \left(\frac{x}{4+x+x^3}\right)+(-200x-50x^2-50x^4){\log }^2\left(\frac{x}{4+x+x^3}\right)}{256x^3+448x^4+288x^5+144x^6+104x^7+48x^8+8x^9+(-192x^3-240x^4-96x^5-60x^6-48x^7-12x^8)\log (5)+(48x^3+36x^4+6x^5+12x^6+6x^7){\log }^2(5)+(-4x^3-x^4-x^6){\log }^3(5)+(192x^3+240x^4+96x^5+60x^6+48x^7+12x^8+(-96x^3-72x^4-12x^5-24x^6-12x^7)\log (5)+(12x^3+3x^4+3x^6){\log }^2(5))\log \left(\frac{x}{4+x+x^3}\right)+(48x^3+36x^4+6x^5+12x^6+6x^7+(-12x^3-3x^4-3x^6)\log (5)){\log }^2\left(\frac{x}{4+x+x^3}\right)+(4x^3+x^4+x^6){\log }^3\left(\frac{x}{4+x+x^3}\right)}dx$$

Mathematica 12.3 output

$$-\frac{25(x(32+136x+96x^2-48x^3+66x^4+45x^5-18x^6+20x^7+6x^8-x^9+2x^{10}){\log }^3\left(\frac{x}{5(4+x+x^3)}\right)+(4+x+x^3){\log }^2\left(\frac{x}{5(4+x+x^3)}\right)(-16+8x^9+x^7(13-2\log (5))+2x(-1+8\log (5))+8x^2(23+8\log (5))+x^6(68+8\log (5))-4x^4(11+9\log (5))+x^5(-2+26\log (5))+x^3(227+32\log (5))+x^8(-2+\log (625))-2x(8+32x+16x^2-18x^3+13x^4+4x^5-x^6+2x^7)\log \left(\frac{x}{4+x+x^3}\right))+\log \left(\frac{x}{5(4+x+x^3)}\right)(8x^{13}+2x^{12}(6+5\log (5))+x^4(1356-31\log (5)-48{\log }^2(5))+x^7(754+50\log (5)-18{\log }^2(5))+x^{10}(160+21\log (5)-{\log }^2(5))+2x^9(71+73\log (5)+3{\log }^2(5))+8x(-63+9\log (5)+4{\log }^2(5))+x^8(-92+63\log (5)+20{\log }^2(5))+4x^3(655+385\log (5)+24{\log }^2(5))+2x^5(94+142\log (5)+33{\log }^2(5))+x^6(1072+727\log (5)+45{\log }^2(5))+2x^2(404+603\log (5)+68{\log }^2(5))-32(7+\log (25))+2x^{11}(-14+{\log }^2(5)+\log (625))-(4+x+x^3)(-16+10x^9+x^7(11-2\log (5))+4x^8(2+\log (5))+2x(11+8\log (5))+8x^2(37+8\log (5))+x^6(98+8\log (5))-4x^4(23+9\log (5))+x^5(20+26\log (5))+x^3(315+32\log (5)))\log \left(\frac{x}{4+x+x^3}\right)+x(32+136x+96x^2-48x^3+66x^4+45x^5-18x^6+20x^7+6x^8-x^9+2x^{10}){\log }^2\left(\frac{x}{4+x+x^3}\right))-2(8x^{14}-8x^{13}(1+\log (5))+16(1+7\log (5))-4x^{10}(1+25\log (5))+x^6(230-536\log (5)-115{\log }^2(5))+x(224+220\log (5)-48{\log }^2(5))+x^8(656-38\log (5)-30{\log }^2(5))+x^{12}(2+6\log (5)-{\log }^2(5))+x^7(730-549\log (5)+13{\log }^2(5))+2x^4(1250-539\log (5)+20{\log }^2(5))-x^9(14+23\log (5)+24{\log }^2(5))-2x^5(-544+83\log (5)+38{\log }^2(5))-4x^2(-262+149\log (5)+59{\log }^2(5))-2x^3(-1156+871\log (5)+116{\log }^2(5))-5x^{11}(-26+{\log }^2(5)+\log (25))+(-112+100x^{10}+8x^{13}+x^4(1078-80\log (5))+x^7(549-26\log (5))+10x^{11}(1+\log (5))+4x(-55+24\log (5))+x^9(23+48\log (5))+x^8(38+60\log (5))+2x^5(83+76\log (5))+x^6(536+230\log (5))+2x^3(871+232\log (5))+x^2(596+472\log (5))+x^{12}(-6+\log (25)))\log \left(\frac{x}{4+x+x^3}\right)-x(48+236x+232x^2-40x^3+76x^4+115x^5-13x^6+30x^7+24x^8+5x^{10}+x^{11}){\log }^2\left(\frac{x}{4+x+x^3}\right)))}{4x^2{(2+4x+x^2-x^3+x^4)}^3{(4+2x+\log \left(\frac{x}{5(4+x+x^3)}\right))}^2}$$