\(\int \frac {-40000-65000 x-5000 x^2-5000 x^3-16250 x^4-12500 x^5+3750 x^7+3750 x^8+1250 x^9+e^2 (-1250 x^2-2500 x^3)+e (-15000 x-26250 x^2-1250 x^3-1250 x^5-1250 x^6)+(-3200-5200 x-400 x^2-400 x^3-1300 x^4-1000 x^5+300 x^7+300 x^8+100 x^9+e^2 (-100 x^2-200 x^3)+e (-1200 x-2100 x^2-100 x^3-100 x^5-100 x^6)) \log (4)+(-64-104 x-8 x^2-8 x^3-26 x^4-20 x^5+6 x^7+6 x^8+2 x^9+e^2 (-2 x^2-4 x^3)+e (-24 x-42 x^2-2 x^3-2 x^5-2 x^6)) \log ^2(4)+(15000 x+26250 x^2+1250 x^3+1250 x^5+1250 x^6+e (2500 x^2+5000 x^3)+(1200 x+2100 x^2+100 x^3+100 x^5+100 x^6+e (200 x^2+400 x^3)) \log (4)+(24 x+42 x^2+2 x^3+2 x^5+2 x^6+e (4 x^2+8 x^3)) \log ^2(4)) \log (x)+(-1250 x^2-2500 x^3+(-100 x^2-200 x^3) \log (4)+(-2 x^2-4 x^3) \log ^2(4)) \log ^2(x)}{x^5+3 x^6+3 x^7+x^8} \, dx\) [729]

Optimal result

Integrand size = 439, antiderivative size = 31 \[ \int \frac {-40000-65000 x-5000 x^2-5000 x^3-16250 x^4-12500 x^5+3750 x^7+3750 x^8+1250 x^9+e^2 \left (-1250 x^2-2500 x^3\right )+e \left (-15000 x-26250 x^2-1250 x^3-1250 x^5-1250 x^6\right )+\left (-3200-5200 x-400 x^2-400 x^3-1300 x^4-1000 x^5+300 x^7+300 x^8+100 x^9+e^2 \left (-100 x^2-200 x^3\right )+e \left (-1200 x-2100 x^2-100 x^3-100 x^5-100 x^6\right )\right ) \log (4)+\left (-64-104 x-8 x^2-8 x^3-26 x^4-20 x^5+6 x^7+6 x^8+2 x^9+e^2 \left (-2 x^2-4 x^3\right )+e \left (-24 x-42 x^2-2 x^3-2 x^5-2 x^6\right )\right ) \log ^2(4)+\left (15000 x+26250 x^2+1250 x^3+1250 x^5+1250 x^6+e \left (2500 x^2+5000 x^3\right )+\left (1200 x+2100 x^2+100 x^3+100 x^5+100 x^6+e \left (200 x^2+400 x^3\right )\right ) \log (4)+\left (24 x+42 x^2+2 x^3+2 x^5+2 x^6+e \left (4 x^2+8 x^3\right )\right ) \log ^2(4)\right ) \log (x)+\left (-1250 x^2-2500 x^3+\left (-100 x^2-200 x^3\right ) \log (4)+\left (-2 x^2-4 x^3\right ) \log ^2(4)\right ) \log ^2(x)}{x^5+3 x^6+3 x^7+x^8} \, dx=(25+\log (4))^2 \left (x-\frac {-e-\frac {4}{x}+\log (x)}{x+x^2}\right )^2 \] Output:

(25+2*ln(2))^2*(x-(ln(x)-exp(1)-4/x)/(x^2+x))^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {-40000-65000 x-5000 x^2-5000 x^3-16250 x^4-12500 x^5+3750 x^7+3750 x^8+1250 x^9+e^2 \left (-1250 x^2-2500 x^3\right )+e \left (-15000 x-26250 x^2-1250 x^3-1250 x^5-1250 x^6\right )+\left (-3200-5200 x-400 x^2-400 x^3-1300 x^4-1000 x^5+300 x^7+300 x^8+100 x^9+e^2 \left (-100 x^2-200 x^3\right )+e \left (-1200 x-2100 x^2-100 x^3-100 x^5-100 x^6\right )\right ) \log (4)+\left (-64-104 x-8 x^2-8 x^3-26 x^4-20 x^5+6 x^7+6 x^8+2 x^9+e^2 \left (-2 x^2-4 x^3\right )+e \left (-24 x-42 x^2-2 x^3-2 x^5-2 x^6\right )\right ) \log ^2(4)+\left (15000 x+26250 x^2+1250 x^3+1250 x^5+1250 x^6+e \left (2500 x^2+5000 x^3\right )+\left (1200 x+2100 x^2+100 x^3+100 x^5+100 x^6+e \left (200 x^2+400 x^3\right )\right ) \log (4)+\left (24 x+42 x^2+2 x^3+2 x^5+2 x^6+e \left (4 x^2+8 x^3\right )\right ) \log ^2(4)\right ) \log (x)+\left (-1250 x^2-2500 x^3+\left (-100 x^2-200 x^3\right ) \log (4)+\left (-2 x^2-4 x^3\right ) \log ^2(4)\right ) \log ^2(x)}{x^5+3 x^6+3 x^7+x^8} \, dx=\frac {(25+\log (4))^2 \left (4+e x+x^3+x^4-x \log (x)\right )^2}{x^4 (1+x)^2} \] Input:

Integrate[(-40000 - 65000*x - 5000*x^2 - 5000*x^3 - 16250*x^4 - 12500*x^5 
+ 3750*x^7 + 3750*x^8 + 1250*x^9 + E^2*(-1250*x^2 - 2500*x^3) + E*(-15000* 
x - 26250*x^2 - 1250*x^3 - 1250*x^5 - 1250*x^6) + (-3200 - 5200*x - 400*x^ 
2 - 400*x^3 - 1300*x^4 - 1000*x^5 + 300*x^7 + 300*x^8 + 100*x^9 + E^2*(-10 
0*x^2 - 200*x^3) + E*(-1200*x - 2100*x^2 - 100*x^3 - 100*x^5 - 100*x^6))*L 
og[4] + (-64 - 104*x - 8*x^2 - 8*x^3 - 26*x^4 - 20*x^5 + 6*x^7 + 6*x^8 + 2 
*x^9 + E^2*(-2*x^2 - 4*x^3) + E*(-24*x - 42*x^2 - 2*x^3 - 2*x^5 - 2*x^6))* 
Log[4]^2 + (15000*x + 26250*x^2 + 1250*x^3 + 1250*x^5 + 1250*x^6 + E*(2500 
*x^2 + 5000*x^3) + (1200*x + 2100*x^2 + 100*x^3 + 100*x^5 + 100*x^6 + E*(2 
00*x^2 + 400*x^3))*Log[4] + (24*x + 42*x^2 + 2*x^3 + 2*x^5 + 2*x^6 + E*(4* 
x^2 + 8*x^3))*Log[4]^2)*Log[x] + (-1250*x^2 - 2500*x^3 + (-100*x^2 - 200*x 
^3)*Log[4] + (-2*x^2 - 4*x^3)*Log[4]^2)*Log[x]^2)/(x^5 + 3*x^6 + 3*x^7 + x 
^8),x]
 

Output:

((25 + Log[4])^2*(4 + E*x + x^3 + x^4 - x*Log[x])^2)/(x^4*(1 + x)^2)
 

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 4.56 (sec) , antiderivative size = 553, normalized size of antiderivative = 17.84, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {2026, 2007, 7239, 27, 25, 7293, 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.

\(\displaystyle \int \frac {1250 x^9+3750 x^8+3750 x^7-12500 x^5-16250 x^4-5000 x^3-5000 x^2+e^2 \left (-2500 x^3-1250 x^2\right )+\left (-2500 x^3-1250 x^2+\left (-4 x^3-2 x^2\right ) \log ^2(4)+\left (-200 x^3-100 x^2\right ) \log (4)\right ) \log ^2(x)+e \left (-1250 x^6-1250 x^5-1250 x^3-26250 x^2-15000 x\right )+\left (1250 x^6+1250 x^5+1250 x^3+26250 x^2+e \left (5000 x^3+2500 x^2\right )+\left (2 x^6+2 x^5+2 x^3+42 x^2+e \left (8 x^3+4 x^2\right )+24 x\right ) \log ^2(4)+\left (100 x^6+100 x^5+100 x^3+2100 x^2+e \left (400 x^3+200 x^2\right )+1200 x\right ) \log (4)+15000 x\right ) \log (x)+\left (2 x^9+6 x^8+6 x^7-20 x^5-26 x^4-8 x^3-8 x^2+e^2 \left (-4 x^3-2 x^2\right )+e \left (-2 x^6-2 x^5-2 x^3-42 x^2-24 x\right )-104 x-64\right ) \log ^2(4)+\left (100 x^9+300 x^8+300 x^7-1000 x^5-1300 x^4-400 x^3-400 x^2+e^2 \left (-200 x^3-100 x^2\right )+e \left (-100 x^6-100 x^5-100 x^3-2100 x^2-1200 x\right )-5200 x-3200\right ) \log (4)-65000 x-40000}{x^8+3 x^7+3 x^6+x^5} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {1250 x^9+3750 x^8+3750 x^7-12500 x^5-16250 x^4-5000 x^3-5000 x^2+e^2 \left (-2500 x^3-1250 x^2\right )+\left (-2500 x^3-1250 x^2+\left (-4 x^3-2 x^2\right ) \log ^2(4)+\left (-200 x^3-100 x^2\right ) \log (4)\right ) \log ^2(x)+e \left (-1250 x^6-1250 x^5-1250 x^3-26250 x^2-15000 x\right )+\left (1250 x^6+1250 x^5+1250 x^3+26250 x^2+e \left (5000 x^3+2500 x^2\right )+\left (2 x^6+2 x^5+2 x^3+42 x^2+e \left (8 x^3+4 x^2\right )+24 x\right ) \log ^2(4)+\left (100 x^6+100 x^5+100 x^3+2100 x^2+e \left (400 x^3+200 x^2\right )+1200 x\right ) \log (4)+15000 x\right ) \log (x)+\left (2 x^9+6 x^8+6 x^7-20 x^5-26 x^4-8 x^3-8 x^2+e^2 \left (-4 x^3-2 x^2\right )+e \left (-2 x^6-2 x^5-2 x^3-42 x^2-24 x\right )-104 x-64\right ) \log ^2(4)+\left (100 x^9+300 x^8+300 x^7-1000 x^5-1300 x^4-400 x^3-400 x^2+e^2 \left (-200 x^3-100 x^2\right )+e \left (-100 x^6-100 x^5-100 x^3-2100 x^2-1200 x\right )-5200 x-3200\right ) \log (4)-65000 x-40000}{x^5 \left (x^3+3 x^2+3 x+1\right )}dx\)

\(\Big \downarrow \) 2007

\(\displaystyle \int \frac {1250 x^9+3750 x^8+3750 x^7-12500 x^5-16250 x^4-5000 x^3-5000 x^2+e^2 \left (-2500 x^3-1250 x^2\right )+\left (-2500 x^3-1250 x^2+\left (-4 x^3-2 x^2\right ) \log ^2(4)+\left (-200 x^3-100 x^2\right ) \log (4)\right ) \log ^2(x)+e \left (-1250 x^6-1250 x^5-1250 x^3-26250 x^2-15000 x\right )+\left (1250 x^6+1250 x^5+1250 x^3+26250 x^2+e \left (5000 x^3+2500 x^2\right )+\left (2 x^6+2 x^5+2 x^3+42 x^2+e \left (8 x^3+4 x^2\right )+24 x\right ) \log ^2(4)+\left (100 x^6+100 x^5+100 x^3+2100 x^2+e \left (400 x^3+200 x^2\right )+1200 x\right ) \log (4)+15000 x\right ) \log (x)+\left (2 x^9+6 x^8+6 x^7-20 x^5-26 x^4-8 x^3-8 x^2+e^2 \left (-4 x^3-2 x^2\right )+e \left (-2 x^6-2 x^5-2 x^3-42 x^2-24 x\right )-104 x-64\right ) \log ^2(4)+\left (100 x^9+300 x^8+300 x^7-1000 x^5-1300 x^4-400 x^3-400 x^2+e^2 \left (-200 x^3-100 x^2\right )+e \left (-100 x^6-100 x^5-100 x^3-2100 x^2-1200 x\right )-5200 x-3200\right ) \log (4)-65000 x-40000}{x^5 (x+1)^3}dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (25+\log (4))^2 \left (x^9+3 x^8+3 x^7-e x^6-(10+e) x^5-13 x^4-\left (4+e+2 e^2\right ) x^3-\left (4+21 e+e^2\right ) x^2-(2 x+1) x^2 \log ^2(x)+\left (x^5+x^4+(1+4 e) x^2+(21+2 e) x+12\right ) x \log (x)-4 (13+3 e) x-32\right )}{x^5 (x+1)^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 (25+\log (4))^2 \int -\frac {-x^9-3 x^8-3 x^7+e x^6+(10+e) x^5+13 x^4+\left (4+e+2 e^2\right ) x^3+(2 x+1) \log ^2(x) x^2+\left (4+21 e+e^2\right ) x^2-\left (x^5+x^4+(1+4 e) x^2+(21+2 e) x+12\right ) \log (x) x+4 (13+3 e) x+32}{x^5 (x+1)^3}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -2 (25+\log (4))^2 \int \frac {-x^9-3 x^8-3 x^7+e x^6+(10+e) x^5+13 x^4+\left (4+e+2 e^2\right ) x^3+(2 x+1) \log ^2(x) x^2+\left (4+21 e+e^2\right ) x^2-\left (x^5+x^4+(1+4 e) x^2+(21+2 e) x+12\right ) \log (x) x+4 (13+3 e) x+32}{x^5 (x+1)^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 (25+\log (4))^2 \int \left (-\frac {x^4}{(x+1)^3}-\frac {3 x^3}{(x+1)^3}-\frac {3 x^2}{(x+1)^3}+\frac {e x}{(x+1)^3}+\frac {10+e}{(x+1)^3}+\frac {13}{(x+1)^3 x}+\frac {4+e+2 e^2}{(x+1)^3 x^2}+\frac {(2 x+1) \log ^2(x)}{(x+1)^3 x^3}+\frac {4+21 e+e^2}{(x+1)^3 x^3}+\frac {\left (-x^5-x^4-(1+4 e) x^2-21 \left (1+\frac {2 e}{21}\right ) x-12\right ) \log (x)}{(x+1)^3 x^4}+\frac {4 (13+3 e)}{(x+1)^3 x^4}+\frac {32}{(x+1)^3 x^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 (25+\log (4))^2 \left (\operatorname {PolyLog}\left (2,-\frac {1}{x}\right )+\operatorname {PolyLog}(2,-x)-\frac {8}{x^4}-\frac {4 (13+3 e)}{3 x^3}+\frac {100}{3 x^3}+\frac {4 \log (x)}{x^3}+\frac {e x^2}{2 (x+1)^2}-\frac {x^2}{2}-\frac {4+21 e+e^2}{2 x^2}+\frac {6 (13+3 e)}{x^2}-\frac {15-2 e}{4 x^2}-\frac {385}{4 x^2}-\frac {\log ^2(x)}{2 x^2}-\frac {(15-2 e) \log (x)}{2 x^2}-\frac {\log (x)}{2 x^2}-\frac {2 \left (4+e+2 e^2\right )}{x+1}+\frac {3 \left (4+21 e+e^2\right )}{x+1}-\frac {16 (13+3 e)}{x+1}+\frac {4-e}{x+1}+\frac {172}{x+1}-\frac {4+e+2 e^2}{2 (x+1)^2}+\frac {4+21 e+e^2}{2 (x+1)^2}-\frac {2 (13+3 e)}{(x+1)^2}-\frac {10+e}{2 (x+1)^2}+\frac {23}{(x+1)^2}-\frac {4+e+2 e^2}{x}+\frac {3 \left (4+21 e+e^2\right )}{x}-\frac {24 (13+3 e)}{x}+\frac {2 (5-e)}{x}+\frac {322}{x}+\frac {x \log ^2(x)}{x+1}-\frac {\log ^2(x)}{2 (x+1)^2}-\frac {3 \log ^2(x)}{2}+\frac {\log ^2(x)}{x}+\frac {2 (6-e) x \log (x)}{x+1}-\frac {x \log (x)}{x+1}-\log \left (\frac {1}{x}+1\right ) \log (x)-\frac {(4-e) \log (x)}{(x+1)^2}-3 \left (4+e+2 e^2\right ) \log (x)+6 \left (4+21 e+e^2\right ) \log (x)-40 (13+3 e) \log (x)+(4-e) \log (x)+493 \log (x)+\log (x) \log (x+1)+3 \left (4+e+2 e^2\right ) \log (x+1)-6 \left (4+21 e+e^2\right ) \log (x+1)+40 (13+3 e) \log (x+1)-2 (6-e) \log (x+1)-(4-e) \log (x+1)-492 \log (x+1)+\frac {2 (5-e) \log (x)}{x}+\frac {2 \log (x)}{x}\right )\)

Input:

Int[(-40000 - 65000*x - 5000*x^2 - 5000*x^3 - 16250*x^4 - 12500*x^5 + 3750 
*x^7 + 3750*x^8 + 1250*x^9 + E^2*(-1250*x^2 - 2500*x^3) + E*(-15000*x - 26 
250*x^2 - 1250*x^3 - 1250*x^5 - 1250*x^6) + (-3200 - 5200*x - 400*x^2 - 40 
0*x^3 - 1300*x^4 - 1000*x^5 + 300*x^7 + 300*x^8 + 100*x^9 + E^2*(-100*x^2 
- 200*x^3) + E*(-1200*x - 2100*x^2 - 100*x^3 - 100*x^5 - 100*x^6))*Log[4] 
+ (-64 - 104*x - 8*x^2 - 8*x^3 - 26*x^4 - 20*x^5 + 6*x^7 + 6*x^8 + 2*x^9 + 
 E^2*(-2*x^2 - 4*x^3) + E*(-24*x - 42*x^2 - 2*x^3 - 2*x^5 - 2*x^6))*Log[4] 
^2 + (15000*x + 26250*x^2 + 1250*x^3 + 1250*x^5 + 1250*x^6 + E*(2500*x^2 + 
 5000*x^3) + (1200*x + 2100*x^2 + 100*x^3 + 100*x^5 + 100*x^6 + E*(200*x^2 
 + 400*x^3))*Log[4] + (24*x + 42*x^2 + 2*x^3 + 2*x^5 + 2*x^6 + E*(4*x^2 + 
8*x^3))*Log[4]^2)*Log[x] + (-1250*x^2 - 2500*x^3 + (-100*x^2 - 200*x^3)*Lo 
g[4] + (-2*x^2 - 4*x^3)*Log[4]^2)*Log[x]^2)/(x^5 + 3*x^6 + 3*x^7 + x^8),x]
 

Output:

-2*(25 + Log[4])^2*(-8/x^4 + 100/(3*x^3) - (4*(13 + 3*E))/(3*x^3) - 385/(4 
*x^2) - (15 - 2*E)/(4*x^2) + (6*(13 + 3*E))/x^2 - (4 + 21*E + E^2)/(2*x^2) 
 + 322/x + (2*(5 - E))/x - (24*(13 + 3*E))/x + (3*(4 + 21*E + E^2))/x - (4 
 + E + 2*E^2)/x - x^2/2 + 23/(1 + x)^2 - (10 + E)/(2*(1 + x)^2) - (2*(13 + 
 3*E))/(1 + x)^2 + (4 + 21*E + E^2)/(2*(1 + x)^2) - (4 + E + 2*E^2)/(2*(1 
+ x)^2) + (E*x^2)/(2*(1 + x)^2) + 172/(1 + x) + (4 - E)/(1 + x) - (16*(13 
+ 3*E))/(1 + x) + (3*(4 + 21*E + E^2))/(1 + x) - (2*(4 + E + 2*E^2))/(1 + 
x) + 493*Log[x] + (4 - E)*Log[x] - 40*(13 + 3*E)*Log[x] + 6*(4 + 21*E + E^ 
2)*Log[x] - 3*(4 + E + 2*E^2)*Log[x] + (4*Log[x])/x^3 - Log[x]/(2*x^2) - ( 
(15 - 2*E)*Log[x])/(2*x^2) + (2*Log[x])/x + (2*(5 - E)*Log[x])/x - ((4 - E 
)*Log[x])/(1 + x)^2 - (x*Log[x])/(1 + x) + (2*(6 - E)*x*Log[x])/(1 + x) - 
Log[1 + x^(-1)]*Log[x] - (3*Log[x]^2)/2 - Log[x]^2/(2*x^2) + Log[x]^2/x - 
Log[x]^2/(2*(1 + x)^2) + (x*Log[x]^2)/(1 + x) - 492*Log[1 + x] - (4 - E)*L 
og[1 + x] - 2*(6 - E)*Log[1 + x] + 40*(13 + 3*E)*Log[1 + x] - 6*(4 + 21*E 
+ E^2)*Log[1 + x] + 3*(4 + E + 2*E^2)*Log[1 + x] + Log[x]*Log[1 + x] + Pol 
yLog[2, -x^(-1)] + PolyLog[2, -x])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2007

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, 
x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex 
pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol 
yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(356\) vs. \(2(34)=68\).

Time = 28.83 (sec) , antiderivative size = 357, normalized size of antiderivative = 11.52

method	result	size
risch	\(\frac {\left (4 \ln \left (2\right )^{2}+100 \ln \left (2\right )+625\right ) \ln \left (x \right )^{2}}{x^{2} \left (x^{2}+2 x +1\right )}-\frac {2 \left (4 x^{4} \ln \left (2\right )^{2}+4 x^{3} \ln \left (2\right )^{2}+100 x^{4} \ln \left (2\right )+4 \,{\mathrm e} \ln \left (2\right )^{2} x +100 x^{3} \ln \left (2\right )+625 x^{4}+100 x \,{\mathrm e} \ln \left (2\right )+625 x^{3}+625 x \,{\mathrm e}+16 \ln \left (2\right )^{2}+400 \ln \left (2\right )+2500\right ) \ln \left (x \right )}{x^{3} \left (x^{2}+2 x +1\right )}+\frac {10000+1250 x^{5} {\mathrm e}+8 x^{7} \ln \left (2\right )^{2}+200 x^{7} \ln \left (2\right )+32 x^{4} \ln \left (2\right )^{2}+32 x^{3} \ln \left (2\right )^{2}+800 x \,{\mathrm e} \ln \left (2\right )+800 x^{4} \ln \left (2\right )+100 x^{6} \ln \left (2\right )+800 x^{3} \ln \left (2\right )+1250 x^{4} {\mathrm e}+625 x^{2} {\mathrm e}^{2}+5000 x \,{\mathrm e}+1600 \ln \left (2\right )+64 \ln \left (2\right )^{2}+1250 x^{7}+625 x^{8}+5000 x^{4}+5000 x^{3}+625 x^{6}+4 \ln \left (2\right )^{2} x^{8}+100 \ln \left (2\right ) x^{8}+4 \ln \left (2\right )^{2} x^{6}+32 \,{\mathrm e} \ln \left (2\right )^{2} x +200 \,{\mathrm e} \ln \left (2\right ) x^{5}+8 \,{\mathrm e} \ln \left (2\right )^{2} x^{5}+4 \,{\mathrm e}^{2} \ln \left (2\right )^{2} x^{2}+100 \,{\mathrm e}^{2} \ln \left (2\right ) x^{2}+8 \,{\mathrm e} \ln \left (2\right )^{2} x^{4}+200 \,{\mathrm e} \ln \left (2\right ) x^{4}}{x^{4} \left (x^{2}+2 x +1\right )}\)	\(357\)
parallelrisch	\(\frac {10000-8 x^{5} \ln \left (2\right )^{2}+1250 x^{5} {\mathrm e}+8 x^{7} \ln \left (2\right )^{2}+200 x^{7} \ln \left (2\right )+28 x^{4} \ln \left (2\right )^{2}+32 x^{3} \ln \left (2\right )^{2}-1250 x^{5} \ln \left (x \right )+800 x \,{\mathrm e} \ln \left (2\right )+700 x^{4} \ln \left (2\right )-200 x^{5} \ln \left (2\right )+800 x^{3} \ln \left (2\right )+1250 x^{4} {\mathrm e}+625 x^{2} {\mathrm e}^{2}-1250 x^{4} \ln \left (x \right )-5000 x \ln \left (x \right )+5000 x \,{\mathrm e}+100 x^{2} \ln \left (2\right ) \ln \left (x \right )^{2}-8 \ln \left (x \right ) {\mathrm e} \ln \left (2\right )^{2} x^{2}-200 \ln \left (x \right ) {\mathrm e} \ln \left (2\right ) x^{2}+625 x^{2} \ln \left (x \right )^{2}+1600 \ln \left (2\right )+64 \ln \left (2\right )^{2}+1250 x^{7}+625 x^{8}+4375 x^{4}+5000 x^{3}-1250 x^{5}-800 x \ln \left (2\right ) \ln \left (x \right )-1250 \ln \left (x \right ) {\mathrm e} x^{2}-8 \ln \left (x \right ) \ln \left (2\right )^{2} x^{4}+4 \ln \left (2\right )^{2} x^{8}+100 \ln \left (2\right ) x^{8}-32 \ln \left (x \right ) \ln \left (2\right )^{2} x +32 \,{\mathrm e} \ln \left (2\right )^{2} x -200 \ln \left (x \right ) \ln \left (2\right ) x^{5}+200 \,{\mathrm e} \ln \left (2\right ) x^{5}-8 \ln \left (x \right ) \ln \left (2\right )^{2} x^{5}+8 \,{\mathrm e} \ln \left (2\right )^{2} x^{5}-200 \ln \left (x \right ) \ln \left (2\right ) x^{4}+4 \,{\mathrm e}^{2} \ln \left (2\right )^{2} x^{2}+100 \,{\mathrm e}^{2} \ln \left (2\right ) x^{2}+4 \ln \left (x \right )^{2} \ln \left (2\right )^{2} x^{2}+8 \,{\mathrm e} \ln \left (2\right )^{2} x^{4}+200 \,{\mathrm e} \ln \left (2\right ) x^{4}}{x^{4} \left (x^{2}+2 x +1\right )}\)	\(381\)
orering	\(\text {Expression too large to display}\)	\(4186\)

Input:

int(((4*(-4*x^3-2*x^2)*ln(2)^2+2*(-200*x^3-100*x^2)*ln(2)-2500*x^3-1250*x^ 
2)*ln(x)^2+(4*((8*x^3+4*x^2)*exp(1)+2*x^6+2*x^5+2*x^3+42*x^2+24*x)*ln(2)^2 
+2*((400*x^3+200*x^2)*exp(1)+100*x^6+100*x^5+100*x^3+2100*x^2+1200*x)*ln(2 
)+(5000*x^3+2500*x^2)*exp(1)+1250*x^6+1250*x^5+1250*x^3+26250*x^2+15000*x) 
*ln(x)+4*((-4*x^3-2*x^2)*exp(1)^2+(-2*x^6-2*x^5-2*x^3-42*x^2-24*x)*exp(1)+ 
2*x^9+6*x^8+6*x^7-20*x^5-26*x^4-8*x^3-8*x^2-104*x-64)*ln(2)^2+2*((-200*x^3 
-100*x^2)*exp(1)^2+(-100*x^6-100*x^5-100*x^3-2100*x^2-1200*x)*exp(1)+100*x 
^9+300*x^8+300*x^7-1000*x^5-1300*x^4-400*x^3-400*x^2-5200*x-3200)*ln(2)+(- 
2500*x^3-1250*x^2)*exp(1)^2+(-1250*x^6-1250*x^5-1250*x^3-26250*x^2-15000*x 
)*exp(1)+1250*x^9+3750*x^8+3750*x^7-12500*x^5-16250*x^4-5000*x^3-5000*x^2- 
65000*x-40000)/(x^8+3*x^7+3*x^6+x^5),x,method=_RETURNVERBOSE)
 

Output:

(4*ln(2)^2+100*ln(2)+625)/x^2/(x^2+2*x+1)*ln(x)^2-2*(4*x^4*ln(2)^2+4*x^3*l 
n(2)^2+100*x^4*ln(2)+4*exp(1)*ln(2)^2*x+100*x^3*ln(2)+625*x^4+100*x*exp(1) 
*ln(2)+625*x^3+625*x*exp(1)+16*ln(2)^2+400*ln(2)+2500)/x^3/(x^2+2*x+1)*ln( 
x)+(10000+1250*x^5*exp(1)+8*x^7*ln(2)^2+200*x^7*ln(2)+32*x^4*ln(2)^2+32*x^ 
3*ln(2)^2+800*x*exp(1)*ln(2)+800*x^4*ln(2)+100*x^6*ln(2)+800*x^3*ln(2)+125 
0*x^4*exp(1)+625*x^2*exp(2)+5000*x*exp(1)+1600*ln(2)+64*ln(2)^2+1250*x^7+6 
25*x^8+5000*x^4+5000*x^3+625*x^6+4*ln(2)^2*x^8+100*ln(2)*x^8+4*ln(2)^2*x^6 
+32*exp(1)*ln(2)^2*x+200*exp(1)*ln(2)*x^5+8*exp(1)*ln(2)^2*x^5+4*exp(2)*ln 
(2)^2*x^2+100*exp(2)*ln(2)*x^2+8*exp(1)*ln(2)^2*x^4+200*exp(1)*ln(2)*x^4)/ 
x^4/(x^2+2*x+1)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (33) = 66\).

Time = 0.08 (sec) , antiderivative size = 253, normalized size of antiderivative = 8.16 \[ \int \frac {-40000-65000 x-5000 x^2-5000 x^3-16250 x^4-12500 x^5+3750 x^7+3750 x^8+1250 x^9+e^2 \left (-1250 x^2-2500 x^3\right )+e \left (-15000 x-26250 x^2-1250 x^3-1250 x^5-1250 x^6\right )+\left (-3200-5200 x-400 x^2-400 x^3-1300 x^4-1000 x^5+300 x^7+300 x^8+100 x^9+e^2 \left (-100 x^2-200 x^3\right )+e \left (-1200 x-2100 x^2-100 x^3-100 x^5-100 x^6\right )\right ) \log (4)+\left (-64-104 x-8 x^2-8 x^3-26 x^4-20 x^5+6 x^7+6 x^8+2 x^9+e^2 \left (-2 x^2-4 x^3\right )+e \left (-24 x-42 x^2-2 x^3-2 x^5-2 x^6\right )\right ) \log ^2(4)+\left (15000 x+26250 x^2+1250 x^3+1250 x^5+1250 x^6+e \left (2500 x^2+5000 x^3\right )+\left (1200 x+2100 x^2+100 x^3+100 x^5+100 x^6+e \left (200 x^2+400 x^3\right )\right ) \log (4)+\left (24 x+42 x^2+2 x^3+2 x^5+2 x^6+e \left (4 x^2+8 x^3\right )\right ) \log ^2(4)\right ) \log (x)+\left (-1250 x^2-2500 x^3+\left (-100 x^2-200 x^3\right ) \log (4)+\left (-2 x^2-4 x^3\right ) \log ^2(4)\right ) \log ^2(x)}{x^5+3 x^6+3 x^7+x^8} \, dx=\frac {625 \, x^{8} + 1250 \, x^{7} + 625 \, x^{6} + 5000 \, x^{4} + 5000 \, x^{3} + 625 \, x^{2} e^{2} + 4 \, {\left (x^{8} + 2 \, x^{7} + x^{6} + 8 \, x^{4} + 8 \, x^{3} + x^{2} e^{2} + 2 \, {\left (x^{5} + x^{4} + 4 \, x\right )} e + 16\right )} \log \left (2\right )^{2} + {\left (4 \, x^{2} \log \left (2\right )^{2} + 100 \, x^{2} \log \left (2\right ) + 625 \, x^{2}\right )} \log \left (x\right )^{2} + 1250 \, {\left (x^{5} + x^{4} + 4 \, x\right )} e + 100 \, {\left (x^{8} + 2 \, x^{7} + x^{6} + 8 \, x^{4} + 8 \, x^{3} + x^{2} e^{2} + 2 \, {\left (x^{5} + x^{4} + 4 \, x\right )} e + 16\right )} \log \left (2\right ) - 2 \, {\left (625 \, x^{5} + 625 \, x^{4} + 625 \, x^{2} e + 4 \, {\left (x^{5} + x^{4} + x^{2} e + 4 \, x\right )} \log \left (2\right )^{2} + 100 \, {\left (x^{5} + x^{4} + x^{2} e + 4 \, x\right )} \log \left (2\right ) + 2500 \, x\right )} \log \left (x\right ) + 10000}{x^{6} + 2 \, x^{5} + x^{4}} \] Input:

integrate(((4*(-4*x^3-2*x^2)*log(2)^2+2*(-200*x^3-100*x^2)*log(2)-2500*x^3 
-1250*x^2)*log(x)^2+(4*((8*x^3+4*x^2)*exp(1)+2*x^6+2*x^5+2*x^3+42*x^2+24*x 
)*log(2)^2+2*((400*x^3+200*x^2)*exp(1)+100*x^6+100*x^5+100*x^3+2100*x^2+12 
00*x)*log(2)+(5000*x^3+2500*x^2)*exp(1)+1250*x^6+1250*x^5+1250*x^3+26250*x 
^2+15000*x)*log(x)+4*((-4*x^3-2*x^2)*exp(1)^2+(-2*x^6-2*x^5-2*x^3-42*x^2-2 
4*x)*exp(1)+2*x^9+6*x^8+6*x^7-20*x^5-26*x^4-8*x^3-8*x^2-104*x-64)*log(2)^2 
+2*((-200*x^3-100*x^2)*exp(1)^2+(-100*x^6-100*x^5-100*x^3-2100*x^2-1200*x) 
*exp(1)+100*x^9+300*x^8+300*x^7-1000*x^5-1300*x^4-400*x^3-400*x^2-5200*x-3 
200)*log(2)+(-2500*x^3-1250*x^2)*exp(1)^2+(-1250*x^6-1250*x^5-1250*x^3-262 
50*x^2-15000*x)*exp(1)+1250*x^9+3750*x^8+3750*x^7-12500*x^5-16250*x^4-5000 
*x^3-5000*x^2-65000*x-40000)/(x^8+3*x^7+3*x^6+x^5),x, algorithm="fricas")
 

Output:

(625*x^8 + 1250*x^7 + 625*x^6 + 5000*x^4 + 5000*x^3 + 625*x^2*e^2 + 4*(x^8 
 + 2*x^7 + x^6 + 8*x^4 + 8*x^3 + x^2*e^2 + 2*(x^5 + x^4 + 4*x)*e + 16)*log 
(2)^2 + (4*x^2*log(2)^2 + 100*x^2*log(2) + 625*x^2)*log(x)^2 + 1250*(x^5 + 
 x^4 + 4*x)*e + 100*(x^8 + 2*x^7 + x^6 + 8*x^4 + 8*x^3 + x^2*e^2 + 2*(x^5 
+ x^4 + 4*x)*e + 16)*log(2) - 2*(625*x^5 + 625*x^4 + 625*x^2*e + 4*(x^5 + 
x^4 + x^2*e + 4*x)*log(2)^2 + 100*(x^5 + x^4 + x^2*e + 4*x)*log(2) + 2500* 
x)*log(x) + 10000)/(x^6 + 2*x^5 + x^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {-40000-65000 x-5000 x^2-5000 x^3-16250 x^4-12500 x^5+3750 x^7+3750 x^8+1250 x^9+e^2 \left (-1250 x^2-2500 x^3\right )+e \left (-15000 x-26250 x^2-1250 x^3-1250 x^5-1250 x^6\right )+\left (-3200-5200 x-400 x^2-400 x^3-1300 x^4-1000 x^5+300 x^7+300 x^8+100 x^9+e^2 \left (-100 x^2-200 x^3\right )+e \left (-1200 x-2100 x^2-100 x^3-100 x^5-100 x^6\right )\right ) \log (4)+\left (-64-104 x-8 x^2-8 x^3-26 x^4-20 x^5+6 x^7+6 x^8+2 x^9+e^2 \left (-2 x^2-4 x^3\right )+e \left (-24 x-42 x^2-2 x^3-2 x^5-2 x^6\right )\right ) \log ^2(4)+\left (15000 x+26250 x^2+1250 x^3+1250 x^5+1250 x^6+e \left (2500 x^2+5000 x^3\right )+\left (1200 x+2100 x^2+100 x^3+100 x^5+100 x^6+e \left (200 x^2+400 x^3\right )\right ) \log (4)+\left (24 x+42 x^2+2 x^3+2 x^5+2 x^6+e \left (4 x^2+8 x^3\right )\right ) \log ^2(4)\right ) \log (x)+\left (-1250 x^2-2500 x^3+\left (-100 x^2-200 x^3\right ) \log (4)+\left (-2 x^2-4 x^3\right ) \log ^2(4)\right ) \log ^2(x)}{x^5+3 x^6+3 x^7+x^8} \, dx=\text {Timed out} \] Input:

integrate(((4*(-4*x**3-2*x**2)*ln(2)**2+2*(-200*x**3-100*x**2)*ln(2)-2500* 
x**3-1250*x**2)*ln(x)**2+(4*((8*x**3+4*x**2)*exp(1)+2*x**6+2*x**5+2*x**3+4 
2*x**2+24*x)*ln(2)**2+2*((400*x**3+200*x**2)*exp(1)+100*x**6+100*x**5+100* 
x**3+2100*x**2+1200*x)*ln(2)+(5000*x**3+2500*x**2)*exp(1)+1250*x**6+1250*x 
**5+1250*x**3+26250*x**2+15000*x)*ln(x)+4*((-4*x**3-2*x**2)*exp(1)**2+(-2* 
x**6-2*x**5-2*x**3-42*x**2-24*x)*exp(1)+2*x**9+6*x**8+6*x**7-20*x**5-26*x* 
*4-8*x**3-8*x**2-104*x-64)*ln(2)**2+2*((-200*x**3-100*x**2)*exp(1)**2+(-10 
0*x**6-100*x**5-100*x**3-2100*x**2-1200*x)*exp(1)+100*x**9+300*x**8+300*x* 
*7-1000*x**5-1300*x**4-400*x**3-400*x**2-5200*x-3200)*ln(2)+(-2500*x**3-12 
50*x**2)*exp(1)**2+(-1250*x**6-1250*x**5-1250*x**3-26250*x**2-15000*x)*exp 
(1)+1250*x**9+3750*x**8+3750*x**7-12500*x**5-16250*x**4-5000*x**3-5000*x** 
2-65000*x-40000)/(x**8+3*x**7+3*x**6+x**5),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1988 vs. \(2 (33) = 66\).

Time = 0.25 (sec) , antiderivative size = 1988, normalized size of antiderivative = 64.13 \[ \int \frac {-40000-65000 x-5000 x^2-5000 x^3-16250 x^4-12500 x^5+3750 x^7+3750 x^8+1250 x^9+e^2 \left (-1250 x^2-2500 x^3\right )+e \left (-15000 x-26250 x^2-1250 x^3-1250 x^5-1250 x^6\right )+\left (-3200-5200 x-400 x^2-400 x^3-1300 x^4-1000 x^5+300 x^7+300 x^8+100 x^9+e^2 \left (-100 x^2-200 x^3\right )+e \left (-1200 x-2100 x^2-100 x^3-100 x^5-100 x^6\right )\right ) \log (4)+\left (-64-104 x-8 x^2-8 x^3-26 x^4-20 x^5+6 x^7+6 x^8+2 x^9+e^2 \left (-2 x^2-4 x^3\right )+e \left (-24 x-42 x^2-2 x^3-2 x^5-2 x^6\right )\right ) \log ^2(4)+\left (15000 x+26250 x^2+1250 x^3+1250 x^5+1250 x^6+e \left (2500 x^2+5000 x^3\right )+\left (1200 x+2100 x^2+100 x^3+100 x^5+100 x^6+e \left (200 x^2+400 x^3\right )\right ) \log (4)+\left (24 x+42 x^2+2 x^3+2 x^5+2 x^6+e \left (4 x^2+8 x^3\right )\right ) \log ^2(4)\right ) \log (x)+\left (-1250 x^2-2500 x^3+\left (-100 x^2-200 x^3\right ) \log (4)+\left (-2 x^2-4 x^3\right ) \log ^2(4)\right ) \log ^2(x)}{x^5+3 x^6+3 x^7+x^8} \, dx=\text {Too large to display} \] Input:

integrate(((4*(-4*x^3-2*x^2)*log(2)^2+2*(-200*x^3-100*x^2)*log(2)-2500*x^3 
-1250*x^2)*log(x)^2+(4*((8*x^3+4*x^2)*exp(1)+2*x^6+2*x^5+2*x^3+42*x^2+24*x 
)*log(2)^2+2*((400*x^3+200*x^2)*exp(1)+100*x^6+100*x^5+100*x^3+2100*x^2+12 
00*x)*log(2)+(5000*x^3+2500*x^2)*exp(1)+1250*x^6+1250*x^5+1250*x^3+26250*x 
^2+15000*x)*log(x)+4*((-4*x^3-2*x^2)*exp(1)^2+(-2*x^6-2*x^5-2*x^3-42*x^2-2 
4*x)*exp(1)+2*x^9+6*x^8+6*x^7-20*x^5-26*x^4-8*x^3-8*x^2-104*x-64)*log(2)^2 
+2*((-200*x^3-100*x^2)*exp(1)^2+(-100*x^6-100*x^5-100*x^3-2100*x^2-1200*x) 
*exp(1)+100*x^9+300*x^8+300*x^7-1000*x^5-1300*x^4-400*x^3-400*x^2-5200*x-3 
200)*log(2)+(-2500*x^3-1250*x^2)*exp(1)^2+(-1250*x^6-1250*x^5-1250*x^3-262 
50*x^2-15000*x)*exp(1)+1250*x^9+3750*x^8+3750*x^7-12500*x^5-16250*x^4-5000 
*x^3-5000*x^2-65000*x-40000)/(x^8+3*x^7+3*x^6+x^5),x, algorithm="maxima")
 

Output:

-4*((12*x^3 + 18*x^2 + 4*x - 1)/(x^4 + 2*x^3 + x^2) - 12*log(x + 1) + 12*l 
og(x))*e^2*log(2)^2 + 8*((6*x^2 + 9*x + 2)/(x^3 + 2*x^2 + x) - 6*log(x + 1 
) + 6*log(x))*e^2*log(2)^2 + 16*((60*x^4 + 90*x^3 + 20*x^2 - 5*x + 2)/(x^5 
 + 2*x^4 + x^3) - 60*log(x + 1) + 60*log(x))*e*log(2)^2 - 84*((12*x^3 + 18 
*x^2 + 4*x - 1)/(x^4 + 2*x^3 + x^2) - 12*log(x + 1) + 12*log(x))*e*log(2)^ 
2 + 4*((6*x^2 + 9*x + 2)/(x^3 + 2*x^2 + x) - 6*log(x + 1) + 6*log(x))*e*lo 
g(2)^2 - 100*((12*x^3 + 18*x^2 + 4*x - 1)/(x^4 + 2*x^3 + x^2) - 12*log(x + 
 1) + 12*log(x))*e^2*log(2) + 200*((6*x^2 + 9*x + 2)/(x^3 + 2*x^2 + x) - 6 
*log(x + 1) + 6*log(x))*e^2*log(2) + 400*((60*x^4 + 90*x^3 + 20*x^2 - 5*x 
+ 2)/(x^5 + 2*x^4 + x^3) - 60*log(x + 1) + 60*log(x))*e*log(2) - 2100*((12 
*x^3 + 18*x^2 + 4*x - 1)/(x^4 + 2*x^3 + x^2) - 12*log(x + 1) + 12*log(x))* 
e*log(2) + 100*((6*x^2 + 9*x + 2)/(x^3 + 2*x^2 + x) - 6*log(x + 1) + 6*log 
(x))*e*log(2) + 4*(x^2 - 6*x + (8*x + 7)/(x^2 + 2*x + 1) + 12*log(x + 1))* 
log(2)^2 + 12*(2*x - (6*x + 5)/(x^2 + 2*x + 1) - 6*log(x + 1))*log(2)^2 - 
64*((60*x^5 + 90*x^4 + 20*x^3 - 5*x^2 + 2*x - 1)/(x^6 + 2*x^5 + x^4) - 60* 
log(x + 1) + 60*log(x))*log(2)^2 + 208/3*((60*x^4 + 90*x^3 + 20*x^2 - 5*x 
+ 2)/(x^5 + 2*x^4 + x^3) - 60*log(x + 1) + 60*log(x))*log(2)^2 - 16*((12*x 
^3 + 18*x^2 + 4*x - 1)/(x^4 + 2*x^3 + x^2) - 12*log(x + 1) + 12*log(x))*lo 
g(2)^2 + 16*((6*x^2 + 9*x + 2)/(x^3 + 2*x^2 + x) - 6*log(x + 1) + 6*log(x) 
)*log(2)^2 + 12*((4*x + 3)/(x^2 + 2*x + 1) + 2*log(x + 1))*log(2)^2 - 5...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (33) = 66\).

Time = 0.14 (sec) , antiderivative size = 375, normalized size of antiderivative = 12.10 \[ \int \frac {-40000-65000 x-5000 x^2-5000 x^3-16250 x^4-12500 x^5+3750 x^7+3750 x^8+1250 x^9+e^2 \left (-1250 x^2-2500 x^3\right )+e \left (-15000 x-26250 x^2-1250 x^3-1250 x^5-1250 x^6\right )+\left (-3200-5200 x-400 x^2-400 x^3-1300 x^4-1000 x^5+300 x^7+300 x^8+100 x^9+e^2 \left (-100 x^2-200 x^3\right )+e \left (-1200 x-2100 x^2-100 x^3-100 x^5-100 x^6\right )\right ) \log (4)+\left (-64-104 x-8 x^2-8 x^3-26 x^4-20 x^5+6 x^7+6 x^8+2 x^9+e^2 \left (-2 x^2-4 x^3\right )+e \left (-24 x-42 x^2-2 x^3-2 x^5-2 x^6\right )\right ) \log ^2(4)+\left (15000 x+26250 x^2+1250 x^3+1250 x^5+1250 x^6+e \left (2500 x^2+5000 x^3\right )+\left (1200 x+2100 x^2+100 x^3+100 x^5+100 x^6+e \left (200 x^2+400 x^3\right )\right ) \log (4)+\left (24 x+42 x^2+2 x^3+2 x^5+2 x^6+e \left (4 x^2+8 x^3\right )\right ) \log ^2(4)\right ) \log (x)+\left (-1250 x^2-2500 x^3+\left (-100 x^2-200 x^3\right ) \log (4)+\left (-2 x^2-4 x^3\right ) \log ^2(4)\right ) \log ^2(x)}{x^5+3 x^6+3 x^7+x^8} \, dx=\frac {4 \, x^{8} \log \left (2\right )^{2} + 100 \, x^{8} \log \left (2\right ) + 8 \, x^{7} \log \left (2\right )^{2} + 625 \, x^{8} + 200 \, x^{7} \log \left (2\right ) + 4 \, x^{6} \log \left (2\right )^{2} + 8 \, x^{5} e \log \left (2\right )^{2} - 8 \, x^{5} \log \left (2\right )^{2} \log \left (x\right ) + 1250 \, x^{7} + 100 \, x^{6} \log \left (2\right ) + 200 \, x^{5} e \log \left (2\right ) + 8 \, x^{4} e \log \left (2\right )^{2} - 200 \, x^{5} \log \left (2\right ) \log \left (x\right ) - 8 \, x^{4} \log \left (2\right )^{2} \log \left (x\right ) + 625 \, x^{6} + 1250 \, x^{5} e + 200 \, x^{4} e \log \left (2\right ) + 32 \, x^{4} \log \left (2\right )^{2} - 1250 \, x^{5} \log \left (x\right ) - 200 \, x^{4} \log \left (2\right ) \log \left (x\right ) - 8 \, x^{2} e \log \left (2\right )^{2} \log \left (x\right ) + 4 \, x^{2} \log \left (2\right )^{2} \log \left (x\right )^{2} + 1250 \, x^{4} e + 800 \, x^{4} \log \left (2\right ) + 32 \, x^{3} \log \left (2\right )^{2} + 4 \, x^{2} e^{2} \log \left (2\right )^{2} - 1250 \, x^{4} \log \left (x\right ) - 200 \, x^{2} e \log \left (2\right ) \log \left (x\right ) + 100 \, x^{2} \log \left (2\right ) \log \left (x\right )^{2} + 5000 \, x^{4} + 800 \, x^{3} \log \left (2\right ) + 100 \, x^{2} e^{2} \log \left (2\right ) + 32 \, x e \log \left (2\right )^{2} - 1250 \, x^{2} e \log \left (x\right ) - 32 \, x \log \left (2\right )^{2} \log \left (x\right ) + 625 \, x^{2} \log \left (x\right )^{2} + 5000 \, x^{3} + 625 \, x^{2} e^{2} + 800 \, x e \log \left (2\right ) - 800 \, x \log \left (2\right ) \log \left (x\right ) + 5000 \, x e + 64 \, \log \left (2\right )^{2} - 5000 \, x \log \left (x\right ) + 1600 \, \log \left (2\right ) + 10000}{x^{6} + 2 \, x^{5} + x^{4}} \] Input:

integrate(((4*(-4*x^3-2*x^2)*log(2)^2+2*(-200*x^3-100*x^2)*log(2)-2500*x^3 
-1250*x^2)*log(x)^2+(4*((8*x^3+4*x^2)*exp(1)+2*x^6+2*x^5+2*x^3+42*x^2+24*x 
)*log(2)^2+2*((400*x^3+200*x^2)*exp(1)+100*x^6+100*x^5+100*x^3+2100*x^2+12 
00*x)*log(2)+(5000*x^3+2500*x^2)*exp(1)+1250*x^6+1250*x^5+1250*x^3+26250*x 
^2+15000*x)*log(x)+4*((-4*x^3-2*x^2)*exp(1)^2+(-2*x^6-2*x^5-2*x^3-42*x^2-2 
4*x)*exp(1)+2*x^9+6*x^8+6*x^7-20*x^5-26*x^4-8*x^3-8*x^2-104*x-64)*log(2)^2 
+2*((-200*x^3-100*x^2)*exp(1)^2+(-100*x^6-100*x^5-100*x^3-2100*x^2-1200*x) 
*exp(1)+100*x^9+300*x^8+300*x^7-1000*x^5-1300*x^4-400*x^3-400*x^2-5200*x-3 
200)*log(2)+(-2500*x^3-1250*x^2)*exp(1)^2+(-1250*x^6-1250*x^5-1250*x^3-262 
50*x^2-15000*x)*exp(1)+1250*x^9+3750*x^8+3750*x^7-12500*x^5-16250*x^4-5000 
*x^3-5000*x^2-65000*x-40000)/(x^8+3*x^7+3*x^6+x^5),x, algorithm="giac")
 

Output:

(4*x^8*log(2)^2 + 100*x^8*log(2) + 8*x^7*log(2)^2 + 625*x^8 + 200*x^7*log( 
2) + 4*x^6*log(2)^2 + 8*x^5*e*log(2)^2 - 8*x^5*log(2)^2*log(x) + 1250*x^7 
+ 100*x^6*log(2) + 200*x^5*e*log(2) + 8*x^4*e*log(2)^2 - 200*x^5*log(2)*lo 
g(x) - 8*x^4*log(2)^2*log(x) + 625*x^6 + 1250*x^5*e + 200*x^4*e*log(2) + 3 
2*x^4*log(2)^2 - 1250*x^5*log(x) - 200*x^4*log(2)*log(x) - 8*x^2*e*log(2)^ 
2*log(x) + 4*x^2*log(2)^2*log(x)^2 + 1250*x^4*e + 800*x^4*log(2) + 32*x^3* 
log(2)^2 + 4*x^2*e^2*log(2)^2 - 1250*x^4*log(x) - 200*x^2*e*log(2)*log(x) 
+ 100*x^2*log(2)*log(x)^2 + 5000*x^4 + 800*x^3*log(2) + 100*x^2*e^2*log(2) 
 + 32*x*e*log(2)^2 - 1250*x^2*e*log(x) - 32*x*log(2)^2*log(x) + 625*x^2*lo 
g(x)^2 + 5000*x^3 + 625*x^2*e^2 + 800*x*e*log(2) - 800*x*log(2)*log(x) + 5 
000*x*e + 64*log(2)^2 - 5000*x*log(x) + 1600*log(2) + 10000)/(x^6 + 2*x^5 
+ x^4)
 

Mupad [B] (verification not implemented)

Time = 0.87 (sec) , antiderivative size = 163, normalized size of antiderivative = 5.26 \[ \int \frac {-40000-65000 x-5000 x^2-5000 x^3-16250 x^4-12500 x^5+3750 x^7+3750 x^8+1250 x^9+e^2 \left (-1250 x^2-2500 x^3\right )+e \left (-15000 x-26250 x^2-1250 x^3-1250 x^5-1250 x^6\right )+\left (-3200-5200 x-400 x^2-400 x^3-1300 x^4-1000 x^5+300 x^7+300 x^8+100 x^9+e^2 \left (-100 x^2-200 x^3\right )+e \left (-1200 x-2100 x^2-100 x^3-100 x^5-100 x^6\right )\right ) \log (4)+\left (-64-104 x-8 x^2-8 x^3-26 x^4-20 x^5+6 x^7+6 x^8+2 x^9+e^2 \left (-2 x^2-4 x^3\right )+e \left (-24 x-42 x^2-2 x^3-2 x^5-2 x^6\right )\right ) \log ^2(4)+\left (15000 x+26250 x^2+1250 x^3+1250 x^5+1250 x^6+e \left (2500 x^2+5000 x^3\right )+\left (1200 x+2100 x^2+100 x^3+100 x^5+100 x^6+e \left (200 x^2+400 x^3\right )\right ) \log (4)+\left (24 x+42 x^2+2 x^3+2 x^5+2 x^6+e \left (4 x^2+8 x^3\right )\right ) \log ^2(4)\right ) \log (x)+\left (-1250 x^2-2500 x^3+\left (-100 x^2-200 x^3\right ) \log (4)+\left (-2 x^2-4 x^3\right ) \log ^2(4)\right ) \log ^2(x)}{x^5+3 x^6+3 x^7+x^8} \, dx=x^2\,{\left (\ln \left (4\right )+25\right )}^2+\frac {16\,{\left (\ln \left (4\right )+25\right )}^2}{x^4}-\frac {2\,{\left (\ln \left (4\right )+25\right )}^2\,\left ({\mathrm {e}}^2-12\,\mathrm {e}+12\,\ln \left (x\right )+{\ln \left (x\right )}^2-2\,\mathrm {e}\,\ln \left (x\right )+28\right )}{x}+\frac {{\left (\ln \left (4\right )+25\right )}^2\,\left ({\mathrm {e}}^2-16\,\mathrm {e}+16\,\ln \left (x\right )+{\ln \left (x\right )}^2-2\,\mathrm {e}\,\ln \left (x\right )+48\right )}{x^2}+\frac {{\left (\ln \left (4\right )+25\right )}^2\,{\left (\ln \left (x\right )-\mathrm {e}+4\right )}^2}{{\left (x+1\right )}^2}-\frac {8\,{\left (\ln \left (4\right )+25\right )}^2\,\left (\ln \left (x\right )-\mathrm {e}+4\right )}{x^3}+\frac {2\,{\left (\ln \left (4\right )+25\right )}^2\,\left ({\mathrm {e}}^2-11\,\mathrm {e}+11\,\ln \left (x\right )+{\ln \left (x\right )}^2-2\,\mathrm {e}\,\ln \left (x\right )+28\right )}{x+1} \] Input:

int(-(65000*x + log(x)^2*(2*log(2)*(100*x^2 + 200*x^3) + 1250*x^2 + 2500*x 
^3 + 4*log(2)^2*(2*x^2 + 4*x^3)) - log(x)*(15000*x + 2*log(2)*(1200*x + ex 
p(1)*(200*x^2 + 400*x^3) + 2100*x^2 + 100*x^3 + 100*x^5 + 100*x^6) + exp(1 
)*(2500*x^2 + 5000*x^3) + 4*log(2)^2*(24*x + exp(1)*(4*x^2 + 8*x^3) + 42*x 
^2 + 2*x^3 + 2*x^5 + 2*x^6) + 26250*x^2 + 1250*x^3 + 1250*x^5 + 1250*x^6) 
+ 4*log(2)^2*(104*x + exp(2)*(2*x^2 + 4*x^3) + exp(1)*(24*x + 42*x^2 + 2*x 
^3 + 2*x^5 + 2*x^6) + 8*x^2 + 8*x^3 + 26*x^4 + 20*x^5 - 6*x^7 - 6*x^8 - 2* 
x^9 + 64) + exp(2)*(1250*x^2 + 2500*x^3) + exp(1)*(15000*x + 26250*x^2 + 1 
250*x^3 + 1250*x^5 + 1250*x^6) + 5000*x^2 + 5000*x^3 + 16250*x^4 + 12500*x 
^5 - 3750*x^7 - 3750*x^8 - 1250*x^9 + 2*log(2)*(5200*x + exp(2)*(100*x^2 + 
 200*x^3) + exp(1)*(1200*x + 2100*x^2 + 100*x^3 + 100*x^5 + 100*x^6) + 400 
*x^2 + 400*x^3 + 1300*x^4 + 1000*x^5 - 300*x^7 - 300*x^8 - 100*x^9 + 3200) 
 + 40000)/(x^5 + 3*x^6 + 3*x^7 + x^8),x)
 

Output:

x^2*(log(4) + 25)^2 + (16*(log(4) + 25)^2)/x^4 - (2*(log(4) + 25)^2*(exp(2 
) - 12*exp(1) + 12*log(x) + log(x)^2 - 2*exp(1)*log(x) + 28))/x + ((log(4) 
 + 25)^2*(exp(2) - 16*exp(1) + 16*log(x) + log(x)^2 - 2*exp(1)*log(x) + 48 
))/x^2 + ((log(4) + 25)^2*(log(x) - exp(1) + 4)^2)/(x + 1)^2 - (8*(log(4) 
+ 25)^2*(log(x) - exp(1) + 4))/x^3 + (2*(log(4) + 25)^2*(exp(2) - 11*exp(1 
) + 11*log(x) + log(x)^2 - 2*exp(1)*log(x) + 28))/(x + 1)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 365, normalized size of antiderivative = 11.77 \[ \int \frac {-40000-65000 x-5000 x^2-5000 x^3-16250 x^4-12500 x^5+3750 x^7+3750 x^8+1250 x^9+e^2 \left (-1250 x^2-2500 x^3\right )+e \left (-15000 x-26250 x^2-1250 x^3-1250 x^5-1250 x^6\right )+\left (-3200-5200 x-400 x^2-400 x^3-1300 x^4-1000 x^5+300 x^7+300 x^8+100 x^9+e^2 \left (-100 x^2-200 x^3\right )+e \left (-1200 x-2100 x^2-100 x^3-100 x^5-100 x^6\right )\right ) \log (4)+\left (-64-104 x-8 x^2-8 x^3-26 x^4-20 x^5+6 x^7+6 x^8+2 x^9+e^2 \left (-2 x^2-4 x^3\right )+e \left (-24 x-42 x^2-2 x^3-2 x^5-2 x^6\right )\right ) \log ^2(4)+\left (15000 x+26250 x^2+1250 x^3+1250 x^5+1250 x^6+e \left (2500 x^2+5000 x^3\right )+\left (1200 x+2100 x^2+100 x^3+100 x^5+100 x^6+e \left (200 x^2+400 x^3\right )\right ) \log (4)+\left (24 x+42 x^2+2 x^3+2 x^5+2 x^6+e \left (4 x^2+8 x^3\right )\right ) \log ^2(4)\right ) \log (x)+\left (-1250 x^2-2500 x^3+\left (-100 x^2-200 x^3\right ) \log (4)+\left (-2 x^2-4 x^3\right ) \log ^2(4)\right ) \log ^2(x)}{x^5+3 x^6+3 x^7+x^8} \, dx=\frac {10000+625 e^{2} x^{2}+625 \mathrm {log}\left (x \right )^{2} x^{2}-1250 \,\mathrm {log}\left (x \right ) x^{4}+4 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (2\right )^{2} x^{2}+100 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (2\right ) x^{2}-8 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right )^{2} x^{5}-8 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right )^{2} x^{4}-32 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right )^{2} x -200 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x^{5}-200 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x^{4}-800 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x -1250 \,\mathrm {log}\left (x \right ) e \,x^{2}+4 \mathrm {log}\left (2\right )^{2} e^{2} x^{2}-4 \mathrm {log}\left (2\right )^{2} e \,x^{6}+4 \mathrm {log}\left (2\right )^{2} e \,x^{4}+32 \mathrm {log}\left (2\right )^{2} e x +100 \,\mathrm {log}\left (2\right ) e^{2} x^{2}-100 \,\mathrm {log}\left (2\right ) e \,x^{6}+100 \,\mathrm {log}\left (2\right ) e \,x^{4}+800 \,\mathrm {log}\left (2\right ) e x +64 \mathrm {log}\left (2\right )^{2}+1250 x^{7}-625 e \,x^{6}+625 e \,x^{4}+100 x^{6} \mathrm {log}\left (2\right )+5000 x^{3}+32 \mathrm {log}\left (2\right )^{2} x^{4}+32 \mathrm {log}\left (2\right )^{2} x^{3}+800 \,\mathrm {log}\left (2\right ) x^{4}+625 x^{8}+625 x^{6}+800 \,\mathrm {log}\left (2\right ) x^{3}+1600 \,\mathrm {log}\left (2\right )+5000 e x +4 \mathrm {log}\left (2\right )^{2} x^{8}+8 \mathrm {log}\left (2\right )^{2} x^{7}+4 \mathrm {log}\left (2\right )^{2} x^{6}+100 \,\mathrm {log}\left (2\right ) x^{8}+200 \,\mathrm {log}\left (2\right ) x^{7}-1250 \,\mathrm {log}\left (x \right ) x^{5}+5000 x^{4}-8 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right )^{2} e \,x^{2}-200 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) e \,x^{2}-5000 \,\mathrm {log}\left (x \right ) x}{x^{4} \left (x^{2}+2 x +1\right )} \] Input:

int(((4*(-4*x^3-2*x^2)*log(2)^2+2*(-200*x^3-100*x^2)*log(2)-2500*x^3-1250* 
x^2)*log(x)^2+(4*((8*x^3+4*x^2)*exp(1)+2*x^6+2*x^5+2*x^3+42*x^2+24*x)*log( 
2)^2+2*((400*x^3+200*x^2)*exp(1)+100*x^6+100*x^5+100*x^3+2100*x^2+1200*x)* 
log(2)+(5000*x^3+2500*x^2)*exp(1)+1250*x^6+1250*x^5+1250*x^3+26250*x^2+150 
00*x)*log(x)+4*((-4*x^3-2*x^2)*exp(1)^2+(-2*x^6-2*x^5-2*x^3-42*x^2-24*x)*e 
xp(1)+2*x^9+6*x^8+6*x^7-20*x^5-26*x^4-8*x^3-8*x^2-104*x-64)*log(2)^2+2*((- 
200*x^3-100*x^2)*exp(1)^2+(-100*x^6-100*x^5-100*x^3-2100*x^2-1200*x)*exp(1 
)+100*x^9+300*x^8+300*x^7-1000*x^5-1300*x^4-400*x^3-400*x^2-5200*x-3200)*l 
og(2)+(-2500*x^3-1250*x^2)*exp(1)^2+(-1250*x^6-1250*x^5-1250*x^3-26250*x^2 
-15000*x)*exp(1)+1250*x^9+3750*x^8+3750*x^7-12500*x^5-16250*x^4-5000*x^3-5 
000*x^2-65000*x-40000)/(x^8+3*x^7+3*x^6+x^5),x)
 

Output:

(4*log(x)**2*log(2)**2*x**2 + 100*log(x)**2*log(2)*x**2 + 625*log(x)**2*x* 
*2 - 8*log(x)*log(2)**2*e*x**2 - 8*log(x)*log(2)**2*x**5 - 8*log(x)*log(2) 
**2*x**4 - 32*log(x)*log(2)**2*x - 200*log(x)*log(2)*e*x**2 - 200*log(x)*l 
og(2)*x**5 - 200*log(x)*log(2)*x**4 - 800*log(x)*log(2)*x - 1250*log(x)*e* 
x**2 - 1250*log(x)*x**5 - 1250*log(x)*x**4 - 5000*log(x)*x + 4*log(2)**2*e 
**2*x**2 - 4*log(2)**2*e*x**6 + 4*log(2)**2*e*x**4 + 32*log(2)**2*e*x + 4* 
log(2)**2*x**8 + 8*log(2)**2*x**7 + 4*log(2)**2*x**6 + 32*log(2)**2*x**4 + 
 32*log(2)**2*x**3 + 64*log(2)**2 + 100*log(2)*e**2*x**2 - 100*log(2)*e*x* 
*6 + 100*log(2)*e*x**4 + 800*log(2)*e*x + 100*log(2)*x**8 + 200*log(2)*x** 
7 + 100*log(2)*x**6 + 800*log(2)*x**4 + 800*log(2)*x**3 + 1600*log(2) + 62 
5*e**2*x**2 - 625*e*x**6 + 625*e*x**4 + 5000*e*x + 625*x**8 + 1250*x**7 + 
625*x**6 + 5000*x**4 + 5000*x**3 + 10000)/(x**4*(x**2 + 2*x + 1))