Integrand size = 142, antiderivative size = 21 \[ \int \frac {1100 x+1080 x^2+332 x^3+32 x^4+\left (1000+1040 x+328 x^2+32 x^3\right ) \log (5)+\left (6000 x+10000 x^2+5760 x^3+1408 x^4+128 x^5+\left (5000+9000 x+5440 x^2+1376 x^3+128 x^4\right ) \log (5)\right ) \log (x)+\left (2500 x+8000 x^2+7200 x^3+2560 x^4+320 x^5+\left (4000 x+4800 x^2+1920 x^3+256 x^4\right ) \log (5)\right ) \log ^2(x)}{x} \, dx=4 (x+\log (5)) \left (5+x+(-5-2 x)^2 \log (x)\right )^2 \] Output:
4*(ln(5)+x)*(5+x+(-2*x-5)^2*ln(x))^2
Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(21)=42\).
Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 6.14 \[ \int \frac {1100 x+1080 x^2+332 x^3+32 x^4+\left (1000+1040 x+328 x^2+32 x^3\right ) \log (5)+\left (6000 x+10000 x^2+5760 x^3+1408 x^4+128 x^5+\left (5000+9000 x+5440 x^2+1376 x^3+128 x^4\right ) \log (5)\right ) \log (x)+\left (2500 x+8000 x^2+7200 x^3+2560 x^4+320 x^5+\left (4000 x+4800 x^2+1920 x^3+256 x^4\right ) \log (5)\right ) \log ^2(x)}{x} \, dx=4 \left (x^3+25 \log (5)+5 x (5+452 \log (5)-150 \log (125))+x^2 (10-299 \log (5)+100 \log (125))+\frac {2}{3} \left (12 x^4+375 \log (5)+375 x (1+\log (5))+4 x^3 (30+\log (125))+15 x^2 (25+\log (390625))\right ) \log (x)+\left (16 x^5+625 \log (5)+16 x^4 (10+\log (5))+125 x (5+8 \log (5))+200 x^2 (5+\log (125))+40 x^3 (15+\log (625))\right ) \log ^2(x)\right ) \] Input:
Integrate[(1100*x + 1080*x^2 + 332*x^3 + 32*x^4 + (1000 + 1040*x + 328*x^2 + 32*x^3)*Log[5] + (6000*x + 10000*x^2 + 5760*x^3 + 1408*x^4 + 128*x^5 + (5000 + 9000*x + 5440*x^2 + 1376*x^3 + 128*x^4)*Log[5])*Log[x] + (2500*x + 8000*x^2 + 7200*x^3 + 2560*x^4 + 320*x^5 + (4000*x + 4800*x^2 + 1920*x^3 + 256*x^4)*Log[5])*Log[x]^2)/x,x]
Output:
4*(x^3 + 25*Log[5] + 5*x*(5 + 452*Log[5] - 150*Log[125]) + x^2*(10 - 299*L og[5] + 100*Log[125]) + (2*(12*x^4 + 375*Log[5] + 375*x*(1 + Log[5]) + 4*x ^3*(30 + Log[125]) + 15*x^2*(25 + Log[390625]))*Log[x])/3 + (16*x^5 + 625* Log[5] + 16*x^4*(10 + Log[5]) + 125*x*(5 + 8*Log[5]) + 200*x^2*(5 + Log[12 5]) + 40*x^3*(15 + Log[625]))*Log[x]^2)
Leaf count is larger than twice the leaf count of optimal. \(294\) vs. \(2(21)=42\).
Time = 0.72 (sec) , antiderivative size = 294, normalized size of antiderivative = 14.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2010, 2009}
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.
| \(\displaystyle \int \frac {32 x^4+332 x^3+1080 x^2+\left (32 x^3+328 x^2+1040 x+1000\right ) \log (5)+\left (320 x^5+2560 x^4+7200 x^3+8000 x^2+\left (256 x^4+1920 x^3+4800 x^2+4000 x\right ) \log (5)+2500 x\right ) \log ^2(x)+\left (128 x^5+1408 x^4+5760 x^3+10000 x^2+\left (128 x^4+1376 x^3+5440 x^2+9000 x+5000\right ) \log (5)+6000 x\right ) \log (x)+1100 x}{x} \, dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (\frac {4 (x+5) \left (8 x^3+x^2 (43+8 \log (5))+x (55+42 \log (5))+50 \log (5)\right )}{x}+\frac {8 (2 x+5) \left (8 x^4+4 x^3 (17+\log (25))+2 x^2 (95+33 \log (5))+25 x (6+7 \log (5))+125 \log (5)\right ) \log (x)}{x}+4 (2 x+5)^3 (10 x+5+8 \log (5)) \log ^2(x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 64 x^5 \log ^2(x)+8 x^4+64 x^4 (10+\log (5)) \log ^2(x)+16 x^4 (22+\log (25)) \log (x)-32 x^4 (10+\log (5)) \log (x)-4 x^4 (22+\log (25))+8 x^4 (10+\log (5))+160 x^3 (15+\log (625)) \log ^2(x)-\frac {320}{3} x^3 (15+\log (625)) \log (x)+\frac {32}{3} x^3 (180+33 \log (5)+5 \log (25)) \log (x)+\frac {320}{9} x^3 (15+\log (625))-\frac {32}{9} x^3 (180+33 \log (5)+5 \log (25))+\frac {4}{3} x^3 (83+8 \log (5))+800 x^2 (5+\log (125)) \log ^2(x)-800 x^2 (5+\log (125)) \log (x)+40 x^2 (125+68 \log (5)) \log (x)+400 x^2 (5+\log (125))-20 x^2 (125+68 \log (5))+4 x^2 (135+41 \log (5))+500 x (5+8 \log (5)) \log ^2(x)+2500 \log (5) \log ^2(x)+3000 x (2+\log (125)) \log (x)-1000 x (5+8 \log (5)) \log (x)-3000 x (2+\log (125))+20 x (55+52 \log (5))+1000 x (5+8 \log (5))+1000 \log (5) \log (x)\) |
Input:
Int[(1100*x + 1080*x^2 + 332*x^3 + 32*x^4 + (1000 + 1040*x + 328*x^2 + 32* x^3)*Log[5] + (6000*x + 10000*x^2 + 5760*x^3 + 1408*x^4 + 128*x^5 + (5000 + 9000*x + 5440*x^2 + 1376*x^3 + 128*x^4)*Log[5])*Log[x] + (2500*x + 8000* x^2 + 7200*x^3 + 2560*x^4 + 320*x^5 + (4000*x + 4800*x^2 + 1920*x^3 + 256* x^4)*Log[5])*Log[x]^2)/x,x]
Output:
8*x^4 + 8*x^4*(10 + Log[5]) + 1000*x*(5 + 8*Log[5]) + (4*x^3*(83 + 8*Log[5 ]))/3 + 4*x^2*(135 + 41*Log[5]) + 20*x*(55 + 52*Log[5]) - 20*x^2*(125 + 68 *Log[5]) - 4*x^4*(22 + Log[25]) - (32*x^3*(180 + 33*Log[5] + 5*Log[25]))/9 - 3000*x*(2 + Log[125]) + 400*x^2*(5 + Log[125]) + (320*x^3*(15 + Log[625 ]))/9 + 1000*Log[5]*Log[x] - 32*x^4*(10 + Log[5])*Log[x] - 1000*x*(5 + 8*L og[5])*Log[x] + 40*x^2*(125 + 68*Log[5])*Log[x] + 16*x^4*(22 + Log[25])*Lo g[x] + (32*x^3*(180 + 33*Log[5] + 5*Log[25])*Log[x])/3 + 3000*x*(2 + Log[1 25])*Log[x] - 800*x^2*(5 + Log[125])*Log[x] - (320*x^3*(15 + Log[625])*Log [x])/3 + 64*x^5*Log[x]^2 + 2500*Log[5]*Log[x]^2 + 64*x^4*(10 + Log[5])*Log [x]^2 + 500*x*(5 + 8*Log[5])*Log[x]^2 + 800*x^2*(5 + Log[125])*Log[x]^2 + 160*x^3*(15 + Log[625])*Log[x]^2
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(132\) vs. \(2(21)=42\).
Time = 48.71 (sec) , antiderivative size = 133, normalized size of antiderivative = 6.33
| method | result | size |
| risch | \(\left (64 x^{4} \ln \left (5\right )+64 x^{5}+640 x^{3} \ln \left (5\right )+640 x^{4}+2400 x^{2} \ln \left (5\right )+2400 x^{3}+4000 x \ln \left (5\right )+4000 x^{2}+2500 \ln \left (5\right )+2500 x \right ) \ln \left (x \right )^{2}+\left (32 x^{3} \ln \left (5\right )+32 x^{4}+320 x^{2} \ln \left (5\right )+320 x^{3}+1000 x \ln \left (5\right )+1000 x^{2}+1000 x \right ) \ln \left (x \right )+4 x^{2} \ln \left (5\right )+4 x^{3}+1000 \ln \left (5\right ) \ln \left (x \right )+40 x \ln \left (5\right )+40 x^{2}+100 x\) | \(133\) |
| norman | \(\left (40+4 \ln \left (5\right )\right ) x^{2}+\left (100+40 \ln \left (5\right )\right ) x +1000 \ln \left (5\right ) \ln \left (x \right )+\left (320+32 \ln \left (5\right )\right ) x^{3} \ln \left (x \right )+\left (640+64 \ln \left (5\right )\right ) x^{4} \ln \left (x \right )^{2}+\left (1000+320 \ln \left (5\right )\right ) x^{2} \ln \left (x \right )+\left (1000+1000 \ln \left (5\right )\right ) x \ln \left (x \right )+\left (2400+640 \ln \left (5\right )\right ) x^{3} \ln \left (x \right )^{2}+\left (2500+4000 \ln \left (5\right )\right ) x \ln \left (x \right )^{2}+\left (4000+2400 \ln \left (5\right )\right ) x^{2} \ln \left (x \right )^{2}+4 x^{3}+32 x^{4} \ln \left (x \right )+64 x^{5} \ln \left (x \right )^{2}+2500 \ln \left (x \right )^{2} \ln \left (5\right )\) | \(143\) |
| parallelrisch | \(100 x +32 x^{4} \ln \left (x \right )+640 x^{4} \ln \left (x \right )^{2}+1000 x \ln \left (5\right ) \ln \left (x \right )+2500 \ln \left (x \right )^{2} \ln \left (5\right )+1000 \ln \left (5\right ) \ln \left (x \right )+64 x^{5} \ln \left (x \right )^{2}+2400 x^{3} \ln \left (x \right )^{2}+320 x^{3} \ln \left (x \right )+4000 x^{2} \ln \left (x \right )^{2}+320 x^{2} \ln \left (5\right ) \ln \left (x \right )+4 x^{2} \ln \left (5\right )+40 x \ln \left (5\right )+1000 x^{2} \ln \left (x \right )+1000 x \ln \left (x \right )+2500 x \ln \left (x \right )^{2}+40 x^{2}+4 x^{3}+2400 \ln \left (x \right )^{2} \ln \left (5\right ) x^{2}+640 \ln \left (x \right )^{2} \ln \left (5\right ) x^{3}+64 \ln \left (x \right )^{2} \ln \left (5\right ) x^{4}+32 \ln \left (x \right ) \ln \left (5\right ) x^{3}+4000 \ln \left (x \right )^{2} \ln \left (5\right ) x\) | \(177\) |
| default | \(100 x +32 x^{4} \ln \left (x \right )+640 x^{4} \ln \left (x \right )^{2}+2500 \ln \left (x \right )^{2} \ln \left (5\right )+1000 \ln \left (5\right ) \ln \left (x \right )+64 x^{5} \ln \left (x \right )^{2}+2400 x^{3} \ln \left (x \right )^{2}+320 x^{3} \ln \left (x \right )+4000 x^{2} \ln \left (x \right )^{2}+4800 \ln \left (5\right ) \left (\frac {x^{2} \ln \left (x \right )^{2}}{2}-\frac {x^{2} \ln \left (x \right )}{2}+\frac {x^{2}}{4}\right )+256 \ln \left (5\right ) \left (\frac {x^{4} \ln \left (x \right )^{2}}{4}-\frac {x^{4} \ln \left (x \right )}{8}+\frac {x^{4}}{32}\right )+128 \ln \left (5\right ) \left (\frac {x^{4} \ln \left (x \right )}{4}-\frac {x^{4}}{16}\right )+1920 \ln \left (5\right ) \left (\frac {x^{3} \ln \left (x \right )^{2}}{3}-\frac {2 x^{3} \ln \left (x \right )}{9}+\frac {2 x^{3}}{27}\right )+1376 \ln \left (5\right ) \left (\frac {x^{3} \ln \left (x \right )}{3}-\frac {x^{3}}{9}\right )+9000 \ln \left (5\right ) \left (x \ln \left (x \right )-x \right )+4000 \ln \left (5\right ) \left (x \ln \left (x \right )^{2}-2 x \ln \left (x \right )+2 x \right )+5440 \ln \left (5\right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+\frac {32 x^{3} \ln \left (5\right )}{3}+164 x^{2} \ln \left (5\right )+1040 x \ln \left (5\right )+1000 x^{2} \ln \left (x \right )+1000 x \ln \left (x \right )+2500 x \ln \left (x \right )^{2}+40 x^{2}+4 x^{3}\) | \(277\) |
| parts | \(100 x +32 x^{4} \ln \left (x \right )+640 x^{4} \ln \left (x \right )^{2}+2500 \ln \left (x \right )^{2} \ln \left (5\right )+1000 \ln \left (5\right ) \ln \left (x \right )+64 x^{5} \ln \left (x \right )^{2}+2400 x^{3} \ln \left (x \right )^{2}+320 x^{3} \ln \left (x \right )+4000 x^{2} \ln \left (x \right )^{2}+4800 \ln \left (5\right ) \left (\frac {x^{2} \ln \left (x \right )^{2}}{2}-\frac {x^{2} \ln \left (x \right )}{2}+\frac {x^{2}}{4}\right )+256 \ln \left (5\right ) \left (\frac {x^{4} \ln \left (x \right )^{2}}{4}-\frac {x^{4} \ln \left (x \right )}{8}+\frac {x^{4}}{32}\right )+128 \ln \left (5\right ) \left (\frac {x^{4} \ln \left (x \right )}{4}-\frac {x^{4}}{16}\right )+1920 \ln \left (5\right ) \left (\frac {x^{3} \ln \left (x \right )^{2}}{3}-\frac {2 x^{3} \ln \left (x \right )}{9}+\frac {2 x^{3}}{27}\right )+1376 \ln \left (5\right ) \left (\frac {x^{3} \ln \left (x \right )}{3}-\frac {x^{3}}{9}\right )+9000 \ln \left (5\right ) \left (x \ln \left (x \right )-x \right )+4000 \ln \left (5\right ) \left (x \ln \left (x \right )^{2}-2 x \ln \left (x \right )+2 x \right )+5440 \ln \left (5\right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+\frac {32 x^{3} \ln \left (5\right )}{3}+164 x^{2} \ln \left (5\right )+1040 x \ln \left (5\right )+1000 x^{2} \ln \left (x \right )+1000 x \ln \left (x \right )+2500 x \ln \left (x \right )^{2}+40 x^{2}+4 x^{3}\) | \(277\) |
Input:
int((((256*x^4+1920*x^3+4800*x^2+4000*x)*ln(5)+320*x^5+2560*x^4+7200*x^3+8 000*x^2+2500*x)*ln(x)^2+((128*x^4+1376*x^3+5440*x^2+9000*x+5000)*ln(5)+128 *x^5+1408*x^4+5760*x^3+10000*x^2+6000*x)*ln(x)+(32*x^3+328*x^2+1040*x+1000 )*ln(5)+32*x^4+332*x^3+1080*x^2+1100*x)/x,x,method=_RETURNVERBOSE)
Output:
(64*x^4*ln(5)+64*x^5+640*x^3*ln(5)+640*x^4+2400*x^2*ln(5)+2400*x^3+4000*x* ln(5)+4000*x^2+2500*ln(5)+2500*x)*ln(x)^2+(32*x^3*ln(5)+32*x^4+320*x^2*ln( 5)+320*x^3+1000*x*ln(5)+1000*x^2+1000*x)*ln(x)+4*x^2*ln(5)+4*x^3+1000*ln(5 )*ln(x)+40*x*ln(5)+40*x^2+100*x
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (21) = 42\).
Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 5.67 \[ \int \frac {1100 x+1080 x^2+332 x^3+32 x^4+\left (1000+1040 x+328 x^2+32 x^3\right ) \log (5)+\left (6000 x+10000 x^2+5760 x^3+1408 x^4+128 x^5+\left (5000+9000 x+5440 x^2+1376 x^3+128 x^4\right ) \log (5)\right ) \log (x)+\left (2500 x+8000 x^2+7200 x^3+2560 x^4+320 x^5+\left (4000 x+4800 x^2+1920 x^3+256 x^4\right ) \log (5)\right ) \log ^2(x)}{x} \, dx=4 \, x^{3} + 4 \, {\left (16 \, x^{5} + 160 \, x^{4} + 600 \, x^{3} + 1000 \, x^{2} + {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )} \log \left (5\right ) + 625 \, x\right )} \log \left (x\right )^{2} + 40 \, x^{2} + 4 \, {\left (x^{2} + 10 \, x\right )} \log \left (5\right ) + 8 \, {\left (4 \, x^{4} + 40 \, x^{3} + 125 \, x^{2} + {\left (4 \, x^{3} + 40 \, x^{2} + 125 \, x + 125\right )} \log \left (5\right ) + 125 \, x\right )} \log \left (x\right ) + 100 \, x \] Input:
integrate((((256*x^4+1920*x^3+4800*x^2+4000*x)*log(5)+320*x^5+2560*x^4+720 0*x^3+8000*x^2+2500*x)*log(x)^2+((128*x^4+1376*x^3+5440*x^2+9000*x+5000)*l og(5)+128*x^5+1408*x^4+5760*x^3+10000*x^2+6000*x)*log(x)+(32*x^3+328*x^2+1 040*x+1000)*log(5)+32*x^4+332*x^3+1080*x^2+1100*x)/x,x, algorithm="fricas" )
Output:
4*x^3 + 4*(16*x^5 + 160*x^4 + 600*x^3 + 1000*x^2 + (16*x^4 + 160*x^3 + 600 *x^2 + 1000*x + 625)*log(5) + 625*x)*log(x)^2 + 40*x^2 + 4*(x^2 + 10*x)*lo g(5) + 8*(4*x^4 + 40*x^3 + 125*x^2 + (4*x^3 + 40*x^2 + 125*x + 125)*log(5) + 125*x)*log(x) + 100*x
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (24) = 48\).
Time = 0.20 (sec) , antiderivative size = 143, normalized size of antiderivative = 6.81 \[ \int \frac {1100 x+1080 x^2+332 x^3+32 x^4+\left (1000+1040 x+328 x^2+32 x^3\right ) \log (5)+\left (6000 x+10000 x^2+5760 x^3+1408 x^4+128 x^5+\left (5000+9000 x+5440 x^2+1376 x^3+128 x^4\right ) \log (5)\right ) \log (x)+\left (2500 x+8000 x^2+7200 x^3+2560 x^4+320 x^5+\left (4000 x+4800 x^2+1920 x^3+256 x^4\right ) \log (5)\right ) \log ^2(x)}{x} \, dx=4 x^{3} + x^{2} \cdot \left (4 \log {\left (5 \right )} + 40\right ) + x \left (40 \log {\left (5 \right )} + 100\right ) + \left (32 x^{4} + 32 x^{3} \log {\left (5 \right )} + 320 x^{3} + 320 x^{2} \log {\left (5 \right )} + 1000 x^{2} + 1000 x + 1000 x \log {\left (5 \right )}\right ) \log {\left (x \right )} + \left (64 x^{5} + 64 x^{4} \log {\left (5 \right )} + 640 x^{4} + 640 x^{3} \log {\left (5 \right )} + 2400 x^{3} + 2400 x^{2} \log {\left (5 \right )} + 4000 x^{2} + 2500 x + 4000 x \log {\left (5 \right )} + 2500 \log {\left (5 \right )}\right ) \log {\left (x \right )}^{2} + 1000 \log {\left (5 \right )} \log {\left (x \right )} \] Input:
integrate((((256*x**4+1920*x**3+4800*x**2+4000*x)*ln(5)+320*x**5+2560*x**4 +7200*x**3+8000*x**2+2500*x)*ln(x)**2+((128*x**4+1376*x**3+5440*x**2+9000* x+5000)*ln(5)+128*x**5+1408*x**4+5760*x**3+10000*x**2+6000*x)*ln(x)+(32*x* *3+328*x**2+1040*x+1000)*ln(5)+32*x**4+332*x**3+1080*x**2+1100*x)/x,x)
Output:
4*x**3 + x**2*(4*log(5) + 40) + x*(40*log(5) + 100) + (32*x**4 + 32*x**3*l og(5) + 320*x**3 + 320*x**2*log(5) + 1000*x**2 + 1000*x + 1000*x*log(5))*l og(x) + (64*x**5 + 64*x**4*log(5) + 640*x**4 + 640*x**3*log(5) + 2400*x**3 + 2400*x**2*log(5) + 4000*x**2 + 2500*x + 4000*x*log(5) + 2500*log(5))*lo g(x)**2 + 1000*log(5)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (21) = 42\).
Time = 0.05 (sec) , antiderivative size = 306, normalized size of antiderivative = 14.57 \[ \int \frac {1100 x+1080 x^2+332 x^3+32 x^4+\left (1000+1040 x+328 x^2+32 x^3\right ) \log (5)+\left (6000 x+10000 x^2+5760 x^3+1408 x^4+128 x^5+\left (5000+9000 x+5440 x^2+1376 x^3+128 x^4\right ) \log (5)\right ) \log (x)+\left (2500 x+8000 x^2+7200 x^3+2560 x^4+320 x^5+\left (4000 x+4800 x^2+1920 x^3+256 x^4\right ) \log (5)\right ) \log ^2(x)}{x} \, dx=\frac {64}{25} \, {\left (25 \, \log \left (x\right )^{2} - 10 \, \log \left (x\right ) + 2\right )} x^{5} + 8 \, {\left (8 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1\right )} x^{4} \log \left (5\right ) + \frac {128}{5} \, x^{5} \log \left (x\right ) + 80 \, {\left (8 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1\right )} x^{4} - \frac {128}{25} \, x^{5} + \frac {640}{9} \, {\left (9 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 2\right )} x^{3} \log \left (5\right ) + 352 \, x^{4} \log \left (x\right ) + \frac {800}{3} \, {\left (9 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 2\right )} x^{3} - 80 \, x^{4} + 1200 \, {\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} \log \left (5\right ) + \frac {32}{3} \, x^{3} \log \left (5\right ) + 1920 \, x^{3} \log \left (x\right ) + 2000 \, {\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} - \frac {1588}{3} \, x^{3} + 4000 \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x \log \left (5\right ) + 164 \, x^{2} \log \left (5\right ) + 5000 \, x^{2} \log \left (x\right ) + 2500 \, \log \left (5\right ) \log \left (x\right )^{2} + 2500 \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x - 1960 \, x^{2} + 8 \, {\left (4 \, x^{4} \log \left (x\right ) - x^{4}\right )} \log \left (5\right ) + \frac {1376}{9} \, {\left (3 \, x^{3} \log \left (x\right ) - x^{3}\right )} \log \left (5\right ) + 1360 \, {\left (2 \, x^{2} \log \left (x\right ) - x^{2}\right )} \log \left (5\right ) + 9000 \, {\left (x \log \left (x\right ) - x\right )} \log \left (5\right ) + 1040 \, x \log \left (5\right ) + 6000 \, x \log \left (x\right ) + 1000 \, \log \left (5\right ) \log \left (x\right ) - 4900 \, x \] Input:
integrate((((256*x^4+1920*x^3+4800*x^2+4000*x)*log(5)+320*x^5+2560*x^4+720 0*x^3+8000*x^2+2500*x)*log(x)^2+((128*x^4+1376*x^3+5440*x^2+9000*x+5000)*l og(5)+128*x^5+1408*x^4+5760*x^3+10000*x^2+6000*x)*log(x)+(32*x^3+328*x^2+1 040*x+1000)*log(5)+32*x^4+332*x^3+1080*x^2+1100*x)/x,x, algorithm="maxima" )
Output:
64/25*(25*log(x)^2 - 10*log(x) + 2)*x^5 + 8*(8*log(x)^2 - 4*log(x) + 1)*x^ 4*log(5) + 128/5*x^5*log(x) + 80*(8*log(x)^2 - 4*log(x) + 1)*x^4 - 128/25* x^5 + 640/9*(9*log(x)^2 - 6*log(x) + 2)*x^3*log(5) + 352*x^4*log(x) + 800/ 3*(9*log(x)^2 - 6*log(x) + 2)*x^3 - 80*x^4 + 1200*(2*log(x)^2 - 2*log(x) + 1)*x^2*log(5) + 32/3*x^3*log(5) + 1920*x^3*log(x) + 2000*(2*log(x)^2 - 2* log(x) + 1)*x^2 - 1588/3*x^3 + 4000*(log(x)^2 - 2*log(x) + 2)*x*log(5) + 1 64*x^2*log(5) + 5000*x^2*log(x) + 2500*log(5)*log(x)^2 + 2500*(log(x)^2 - 2*log(x) + 2)*x - 1960*x^2 + 8*(4*x^4*log(x) - x^4)*log(5) + 1376/9*(3*x^3 *log(x) - x^3)*log(5) + 1360*(2*x^2*log(x) - x^2)*log(5) + 9000*(x*log(x) - x)*log(5) + 1040*x*log(5) + 6000*x*log(x) + 1000*log(5)*log(x) - 4900*x
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (21) = 42\).
Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 5.86 \[ \int \frac {1100 x+1080 x^2+332 x^3+32 x^4+\left (1000+1040 x+328 x^2+32 x^3\right ) \log (5)+\left (6000 x+10000 x^2+5760 x^3+1408 x^4+128 x^5+\left (5000+9000 x+5440 x^2+1376 x^3+128 x^4\right ) \log (5)\right ) \log (x)+\left (2500 x+8000 x^2+7200 x^3+2560 x^4+320 x^5+\left (4000 x+4800 x^2+1920 x^3+256 x^4\right ) \log (5)\right ) \log ^2(x)}{x} \, dx=4 \, x^{3} + 4 \, x^{2} {\left (\log \left (5\right ) + 10\right )} + 4 \, {\left (16 \, x^{5} + 16 \, x^{4} {\left (\log \left (5\right ) + 10\right )} + 40 \, x^{3} {\left (4 \, \log \left (5\right ) + 15\right )} + 200 \, x^{2} {\left (3 \, \log \left (5\right ) + 5\right )} + 125 \, x {\left (8 \, \log \left (5\right ) + 5\right )} + 625 \, \log \left (5\right )\right )} \log \left (x\right )^{2} + 20 \, x {\left (2 \, \log \left (5\right ) + 5\right )} + 8 \, {\left (4 \, x^{4} + 4 \, x^{3} {\left (\log \left (5\right ) + 10\right )} + 5 \, x^{2} {\left (8 \, \log \left (5\right ) + 25\right )} + 125 \, x {\left (\log \left (5\right ) + 1\right )}\right )} \log \left (x\right ) + 1000 \, \log \left (5\right ) \log \left (x\right ) \] Input:
integrate((((256*x^4+1920*x^3+4800*x^2+4000*x)*log(5)+320*x^5+2560*x^4+720 0*x^3+8000*x^2+2500*x)*log(x)^2+((128*x^4+1376*x^3+5440*x^2+9000*x+5000)*l og(5)+128*x^5+1408*x^4+5760*x^3+10000*x^2+6000*x)*log(x)+(32*x^3+328*x^2+1 040*x+1000)*log(5)+32*x^4+332*x^3+1080*x^2+1100*x)/x,x, algorithm="giac")
Output:
4*x^3 + 4*x^2*(log(5) + 10) + 4*(16*x^5 + 16*x^4*(log(5) + 10) + 40*x^3*(4 *log(5) + 15) + 200*x^2*(3*log(5) + 5) + 125*x*(8*log(5) + 5) + 625*log(5) )*log(x)^2 + 20*x*(2*log(5) + 5) + 8*(4*x^4 + 4*x^3*(log(5) + 10) + 5*x^2* (8*log(5) + 25) + 125*x*(log(5) + 1))*log(x) + 1000*log(5)*log(x)
Time = 3.49 (sec) , antiderivative size = 126, normalized size of antiderivative = 6.00 \[ \int \frac {1100 x+1080 x^2+332 x^3+32 x^4+\left (1000+1040 x+328 x^2+32 x^3\right ) \log (5)+\left (6000 x+10000 x^2+5760 x^3+1408 x^4+128 x^5+\left (5000+9000 x+5440 x^2+1376 x^3+128 x^4\right ) \log (5)\right ) \log (x)+\left (2500 x+8000 x^2+7200 x^3+2560 x^4+320 x^5+\left (4000 x+4800 x^2+1920 x^3+256 x^4\right ) \log (5)\right ) \log ^2(x)}{x} \, dx=2500\,\ln \left (5\right )\,{\ln \left (x\right )}^2+x\,\left (\left (4000\,\ln \left (5\right )+2500\right )\,{\ln \left (x\right )}^2+\left (1000\,\ln \left (5\right )+1000\right )\,\ln \left (x\right )+40\,\ln \left (5\right )+100\right )+x^2\,\left (\left (2400\,\ln \left (5\right )+4000\right )\,{\ln \left (x\right )}^2+\left (320\,\ln \left (5\right )+1000\right )\,\ln \left (x\right )+\ln \left (625\right )+40\right )+64\,x^5\,{\ln \left (x\right )}^2+x^4\,\left (\left (64\,\ln \left (5\right )+640\right )\,{\ln \left (x\right )}^2+32\,\ln \left (x\right )\right )+1000\,\ln \left (5\right )\,\ln \left (x\right )+x^3\,\left (\left (640\,\ln \left (5\right )+2400\right )\,{\ln \left (x\right )}^2+\left (32\,\ln \left (5\right )+320\right )\,\ln \left (x\right )+4\right ) \] Input:
int((1100*x + log(x)^2*(2500*x + log(5)*(4000*x + 4800*x^2 + 1920*x^3 + 25 6*x^4) + 8000*x^2 + 7200*x^3 + 2560*x^4 + 320*x^5) + log(5)*(1040*x + 328* x^2 + 32*x^3 + 1000) + 1080*x^2 + 332*x^3 + 32*x^4 + log(x)*(6000*x + log( 5)*(9000*x + 5440*x^2 + 1376*x^3 + 128*x^4 + 5000) + 10000*x^2 + 5760*x^3 + 1408*x^4 + 128*x^5))/x,x)
Output:
2500*log(5)*log(x)^2 + x*(40*log(5) + log(x)*(1000*log(5) + 1000) + log(x) ^2*(4000*log(5) + 2500) + 100) + x^2*(log(625) + log(x)*(320*log(5) + 1000 ) + log(x)^2*(2400*log(5) + 4000) + 40) + 64*x^5*log(x)^2 + x^4*(32*log(x) + log(x)^2*(64*log(5) + 640)) + 1000*log(5)*log(x) + x^3*(log(x)*(32*log( 5) + 320) + log(x)^2*(640*log(5) + 2400) + 4)
Time = 0.17 (sec) , antiderivative size = 176, normalized size of antiderivative = 8.38 \[ \int \frac {1100 x+1080 x^2+332 x^3+32 x^4+\left (1000+1040 x+328 x^2+32 x^3\right ) \log (5)+\left (6000 x+10000 x^2+5760 x^3+1408 x^4+128 x^5+\left (5000+9000 x+5440 x^2+1376 x^3+128 x^4\right ) \log (5)\right ) \log (x)+\left (2500 x+8000 x^2+7200 x^3+2560 x^4+320 x^5+\left (4000 x+4800 x^2+1920 x^3+256 x^4\right ) \log (5)\right ) \log ^2(x)}{x} \, dx=64 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (5\right ) x^{4}+640 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (5\right ) x^{3}+2400 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (5\right ) x^{2}+4000 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (5\right ) x +2500 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (5\right )+64 \mathrm {log}\left (x \right )^{2} x^{5}+640 \mathrm {log}\left (x \right )^{2} x^{4}+2400 \mathrm {log}\left (x \right )^{2} x^{3}+4000 \mathrm {log}\left (x \right )^{2} x^{2}+2500 \mathrm {log}\left (x \right )^{2} x +32 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x^{3}+320 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x^{2}+1000 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x +1000 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right )+32 \,\mathrm {log}\left (x \right ) x^{4}+320 \,\mathrm {log}\left (x \right ) x^{3}+1000 \,\mathrm {log}\left (x \right ) x^{2}+1000 \,\mathrm {log}\left (x \right ) x +4 \,\mathrm {log}\left (5\right ) x^{2}+40 \,\mathrm {log}\left (5\right ) x +4 x^{3}+40 x^{2}+100 x \] Input:
int((((256*x^4+1920*x^3+4800*x^2+4000*x)*log(5)+320*x^5+2560*x^4+7200*x^3+ 8000*x^2+2500*x)*log(x)^2+((128*x^4+1376*x^3+5440*x^2+9000*x+5000)*log(5)+ 128*x^5+1408*x^4+5760*x^3+10000*x^2+6000*x)*log(x)+(32*x^3+328*x^2+1040*x+ 1000)*log(5)+32*x^4+332*x^3+1080*x^2+1100*x)/x,x)
Output:
4*(16*log(x)**2*log(5)*x**4 + 160*log(x)**2*log(5)*x**3 + 600*log(x)**2*lo g(5)*x**2 + 1000*log(x)**2*log(5)*x + 625*log(x)**2*log(5) + 16*log(x)**2* x**5 + 160*log(x)**2*x**4 + 600*log(x)**2*x**3 + 1000*log(x)**2*x**2 + 625 *log(x)**2*x + 8*log(x)*log(5)*x**3 + 80*log(x)*log(5)*x**2 + 250*log(x)*l og(5)*x + 250*log(x)*log(5) + 8*log(x)*x**4 + 80*log(x)*x**3 + 250*log(x)* x**2 + 250*log(x)*x + log(5)*x**2 + 10*log(5)*x + x**3 + 10*x**2 + 25*x)