\(\int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7)+(275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 (100 x^3-40 x^4+4 x^5)+e^2 (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7)) \log (x)+(550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 (-200 x-270 x^2+132 x^3-14 x^4)) \log ^2(x)+(550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 (-100 x-110 x^2+56 x^3-6 x^4)) \log ^3(x)+(275+150 x-90 x^2+10 x^3) \log ^4(x)+(55+30 x-18 x^2+2 x^3) \log ^5(x)}{25-10 x+x^2+(125-50 x+5 x^2) \log (x)+(250-100 x+10 x^2) \log ^2(x)+(250-100 x+10 x^2) \log ^3(x)+(125-50 x+5 x^2) \log ^4(x)+(25-10 x+x^2) \log ^5(x)} \, dx\) [1703]

Optimal result

Integrand size = 424, antiderivative size = 36 \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx=\frac {5}{5-x}+\left (-1-x+\frac {x^2 \left (e^2-x^2\right )}{(1+\log (x))^2}\right )^2 \] Output:

5/(5-x)+(-1+(exp(2)-x^2)/(1+ln(x))^2*x^2-x)^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.61 \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx=-\frac {5}{-5+x}+2 x+x^2+\frac {x^4 \left (e^2-x^2\right )^2}{(1+\log (x))^4}+\frac {2 x^2 (1+x) \left (-e^2+x^2\right )}{(1+\log (x))^2} \] Input:

Integrate[(55 + 30*x - 18*x^2 + 102*x^3 + 110*x^4 - 56*x^5 + 6*x^6 + 100*x 
^7 - 40*x^8 + 4*x^9 + E^2*(-50*x^2 + 20*x^3 - 2*x^4 - 100*x^5 + 40*x^6 - 4 
*x^7) + (275 + 150*x - 90*x^2 + 410*x^3 + 390*x^4 - 204*x^5 + 22*x^6 + 200 
*x^7 - 80*x^8 + 8*x^9 + E^4*(100*x^3 - 40*x^4 + 4*x^5) + E^2*(-100*x - 210 
*x^2 + 96*x^3 - 10*x^4 - 300*x^5 + 120*x^6 - 12*x^7))*Log[x] + (550 + 300* 
x - 180*x^2 + 520*x^3 + 450*x^4 - 240*x^5 + 26*x^6 + E^2*(-200*x - 270*x^2 
 + 132*x^3 - 14*x^4))*Log[x]^2 + (550 + 300*x - 180*x^2 + 220*x^3 + 170*x^ 
4 - 92*x^5 + 10*x^6 + E^2*(-100*x - 110*x^2 + 56*x^3 - 6*x^4))*Log[x]^3 + 
(275 + 150*x - 90*x^2 + 10*x^3)*Log[x]^4 + (55 + 30*x - 18*x^2 + 2*x^3)*Lo 
g[x]^5)/(25 - 10*x + x^2 + (125 - 50*x + 5*x^2)*Log[x] + (250 - 100*x + 10 
*x^2)*Log[x]^2 + (250 - 100*x + 10*x^2)*Log[x]^3 + (125 - 50*x + 5*x^2)*Lo 
g[x]^4 + (25 - 10*x + x^2)*Log[x]^5),x]
 

Output:

-5/(-5 + x) + 2*x + x^2 + (x^4*(E^2 - x^2)^2)/(1 + Log[x])^4 + (2*x^2*(1 + 
 x)*(-E^2 + x^2))/(1 + Log[x])^2
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(103\) vs. \(2(36)=72\).

Time = 6.78 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.86, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.007, Rules used = {7292, 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.

\(\displaystyle \int \frac {4 x^9-40 x^8+100 x^7+6 x^6-56 x^5+110 x^4+102 x^3-18 x^2+\left (2 x^3-18 x^2+30 x+55\right ) \log ^5(x)+\left (10 x^3-90 x^2+150 x+275\right ) \log ^4(x)+\left (10 x^6-92 x^5+170 x^4+220 x^3-180 x^2+e^2 \left (-6 x^4+56 x^3-110 x^2-100 x\right )+300 x+550\right ) \log ^3(x)+\left (26 x^6-240 x^5+450 x^4+520 x^3-180 x^2+e^2 \left (-14 x^4+132 x^3-270 x^2-200 x\right )+300 x+550\right ) \log ^2(x)+e^2 \left (-4 x^7+40 x^6-100 x^5-2 x^4+20 x^3-50 x^2\right )+\left (8 x^9-80 x^8+200 x^7+22 x^6-204 x^5+390 x^4+410 x^3-90 x^2+e^4 \left (4 x^5-40 x^4+100 x^3\right )+e^2 \left (-12 x^7+120 x^6-300 x^5-10 x^4+96 x^3-210 x^2-100 x\right )+150 x+275\right ) \log (x)+30 x+55}{x^2+\left (x^2-10 x+25\right ) \log ^5(x)+\left (5 x^2-50 x+125\right ) \log ^4(x)+\left (10 x^2-100 x+250\right ) \log ^3(x)+\left (10 x^2-100 x+250\right ) \log ^2(x)+\left (5 x^2-50 x+125\right ) \log (x)-10 x+25} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {4 x^9-40 x^8+100 x^7+6 x^6-56 x^5+110 x^4+102 x^3-18 x^2+\left (2 x^3-18 x^2+30 x+55\right ) \log ^5(x)+\left (10 x^3-90 x^2+150 x+275\right ) \log ^4(x)+\left (10 x^6-92 x^5+170 x^4+220 x^3-180 x^2+e^2 \left (-6 x^4+56 x^3-110 x^2-100 x\right )+300 x+550\right ) \log ^3(x)+\left (26 x^6-240 x^5+450 x^4+520 x^3-180 x^2+e^2 \left (-14 x^4+132 x^3-270 x^2-200 x\right )+300 x+550\right ) \log ^2(x)+e^2 \left (-4 x^7+40 x^6-100 x^5-2 x^4+20 x^3-50 x^2\right )+\left (8 x^9-80 x^8+200 x^7+22 x^6-204 x^5+390 x^4+410 x^3-90 x^2+e^4 \left (4 x^5-40 x^4+100 x^3\right )+e^2 \left (-12 x^7+120 x^6-300 x^5-10 x^4+96 x^3-210 x^2-100 x\right )+150 x+275\right ) \log (x)+30 x+55}{(5-x)^2 (\log (x)+1)^5}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {4 (x+1) \left (x^2-e^2\right ) x}{(\log (x)+1)^3}+\frac {2 x^3-18 x^2+30 x+55}{(x-5)^2}-\frac {4 \left (x^2-e^2\right )^2 x^3}{(\log (x)+1)^5}+\frac {2 \left (5 x^3+4 x^2-3 e^2 x-2 e^2\right ) x}{(\log (x)+1)^2}+\frac {4 \left (2 x^4-3 e^2 x^2+e^4\right ) x^3}{(\log (x)+1)^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^8}{(\log (x)+1)^4}-\frac {2 e^2 x^6}{(\log (x)+1)^4}+\frac {2 x^5}{(\log (x)+1)^2}+\frac {2 x^4}{(\log (x)+1)^2}+\frac {e^4 x^4}{(\log (x)+1)^4}-\frac {2 e^2 x^3}{(\log (x)+1)^2}+x^2-\frac {2 e^2 x^2}{(\log (x)+1)^2}+2 x+\frac {5}{5-x}\)

Input:

Int[(55 + 30*x - 18*x^2 + 102*x^3 + 110*x^4 - 56*x^5 + 6*x^6 + 100*x^7 - 4 
0*x^8 + 4*x^9 + E^2*(-50*x^2 + 20*x^3 - 2*x^4 - 100*x^5 + 40*x^6 - 4*x^7) 
+ (275 + 150*x - 90*x^2 + 410*x^3 + 390*x^4 - 204*x^5 + 22*x^6 + 200*x^7 - 
 80*x^8 + 8*x^9 + E^4*(100*x^3 - 40*x^4 + 4*x^5) + E^2*(-100*x - 210*x^2 + 
 96*x^3 - 10*x^4 - 300*x^5 + 120*x^6 - 12*x^7))*Log[x] + (550 + 300*x - 18 
0*x^2 + 520*x^3 + 450*x^4 - 240*x^5 + 26*x^6 + E^2*(-200*x - 270*x^2 + 132 
*x^3 - 14*x^4))*Log[x]^2 + (550 + 300*x - 180*x^2 + 220*x^3 + 170*x^4 - 92 
*x^5 + 10*x^6 + E^2*(-100*x - 110*x^2 + 56*x^3 - 6*x^4))*Log[x]^3 + (275 + 
 150*x - 90*x^2 + 10*x^3)*Log[x]^4 + (55 + 30*x - 18*x^2 + 2*x^3)*Log[x]^5 
)/(25 - 10*x + x^2 + (125 - 50*x + 5*x^2)*Log[x] + (250 - 100*x + 10*x^2)* 
Log[x]^2 + (250 - 100*x + 10*x^2)*Log[x]^3 + (125 - 50*x + 5*x^2)*Log[x]^4 
 + (25 - 10*x + x^2)*Log[x]^5),x]
 

Output:

5/(5 - x) + 2*x + x^2 + (E^4*x^4)/(1 + Log[x])^4 - (2*E^2*x^6)/(1 + Log[x] 
)^4 + x^8/(1 + Log[x])^4 - (2*E^2*x^2)/(1 + Log[x])^2 - (2*E^2*x^3)/(1 + L 
og[x])^2 + (2*x^4)/(1 + Log[x])^2 + (2*x^5)/(1 + Log[x])^2
 

Defintions of rubi rules used

rule 2009

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

rule 7292

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

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. \(111\) vs. \(2(35)=70\).

Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.11

\[\frac {x^{8}}{\left (\ln \left (x \right )+1\right )^{4}}-\frac {2 \,{\mathrm e}^{2} x^{6}}{\left (\ln \left (x \right )+1\right )^{4}}+\frac {2 x^{5}}{\left (\ln \left (x \right )+1\right )^{2}}+\frac {\left ({\mathrm e}^{4}+2 \ln \left (x \right )^{2}+4 \ln \left (x \right )+2\right ) x^{4}}{\left (\ln \left (x \right )+1\right )^{4}}-\frac {2 \,{\mathrm e}^{2} x^{3}}{\left (\ln \left (x \right )+1\right )^{2}}-\frac {\left (-\ln \left (x \right )^{2}+2 \,{\mathrm e}^{2}-2 \ln \left (x \right )-1\right ) x^{2}}{\left (\ln \left (x \right )+1\right )^{2}}+2 x -\frac {5}{-5+x}\]

Input:

int(((2*x^3-18*x^2+30*x+55)*ln(x)^5+(10*x^3-90*x^2+150*x+275)*ln(x)^4+((-6 
*x^4+56*x^3-110*x^2-100*x)*exp(2)+10*x^6-92*x^5+170*x^4+220*x^3-180*x^2+30 
0*x+550)*ln(x)^3+((-14*x^4+132*x^3-270*x^2-200*x)*exp(2)+26*x^6-240*x^5+45 
0*x^4+520*x^3-180*x^2+300*x+550)*ln(x)^2+((4*x^5-40*x^4+100*x^3)*exp(2)^2+ 
(-12*x^7+120*x^6-300*x^5-10*x^4+96*x^3-210*x^2-100*x)*exp(2)+8*x^9-80*x^8+ 
200*x^7+22*x^6-204*x^5+390*x^4+410*x^3-90*x^2+150*x+275)*ln(x)+(-4*x^7+40* 
x^6-100*x^5-2*x^4+20*x^3-50*x^2)*exp(2)+4*x^9-40*x^8+100*x^7+6*x^6-56*x^5+ 
110*x^4+102*x^3-18*x^2+30*x+55)/((x^2-10*x+25)*ln(x)^5+(5*x^2-50*x+125)*ln 
(x)^4+(10*x^2-100*x+250)*ln(x)^3+(10*x^2-100*x+250)*ln(x)^2+(5*x^2-50*x+12 
5)*ln(x)+x^2-10*x+25),x)
 

Output:

1/(ln(x)+1)^4*x^8-2*exp(2)/(ln(x)+1)^4*x^6+2/(ln(x)+1)^2*x^5+(exp(2)^2+2*l 
n(x)^2+4*ln(x)+2)/(ln(x)+1)^4*x^4-2*exp(2)/(ln(x)+1)^2*x^3-(-ln(x)^2+2*exp 
(2)-2*ln(x)-1)/(ln(x)+1)^2*x^2+2*x-5/(-5+x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (31) = 62\).

Time = 0.10 (sec) , antiderivative size = 250, normalized size of antiderivative = 6.94 \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx=\frac {x^{9} - 5 \, x^{8} + 2 \, x^{6} - 8 \, x^{5} + {\left (x^{3} - 3 \, x^{2} - 10 \, x - 5\right )} \log \left (x\right )^{4} - 10 \, x^{4} + 4 \, {\left (x^{3} - 3 \, x^{2} - 10 \, x - 5\right )} \log \left (x\right )^{3} + x^{3} + 2 \, {\left (x^{6} - 4 \, x^{5} - 5 \, x^{4} + 3 \, x^{3} - 9 \, x^{2} - {\left (x^{4} - 4 \, x^{3} - 5 \, x^{2}\right )} e^{2} - 30 \, x - 15\right )} \log \left (x\right )^{2} - 3 \, x^{2} + {\left (x^{5} - 5 \, x^{4}\right )} e^{4} - 2 \, {\left (x^{7} - 5 \, x^{6} + x^{4} - 4 \, x^{3} - 5 \, x^{2}\right )} e^{2} + 4 \, {\left (x^{6} - 4 \, x^{5} - 5 \, x^{4} + x^{3} - 3 \, x^{2} - {\left (x^{4} - 4 \, x^{3} - 5 \, x^{2}\right )} e^{2} - 10 \, x - 5\right )} \log \left (x\right ) - 10 \, x - 5}{{\left (x - 5\right )} \log \left (x\right )^{4} + 4 \, {\left (x - 5\right )} \log \left (x\right )^{3} + 6 \, {\left (x - 5\right )} \log \left (x\right )^{2} + 4 \, {\left (x - 5\right )} \log \left (x\right ) + x - 5} \] Input:

integrate(((2*x^3-18*x^2+30*x+55)*log(x)^5+(10*x^3-90*x^2+150*x+275)*log(x 
)^4+((-6*x^4+56*x^3-110*x^2-100*x)*exp(2)+10*x^6-92*x^5+170*x^4+220*x^3-18 
0*x^2+300*x+550)*log(x)^3+((-14*x^4+132*x^3-270*x^2-200*x)*exp(2)+26*x^6-2 
40*x^5+450*x^4+520*x^3-180*x^2+300*x+550)*log(x)^2+((4*x^5-40*x^4+100*x^3) 
*exp(2)^2+(-12*x^7+120*x^6-300*x^5-10*x^4+96*x^3-210*x^2-100*x)*exp(2)+8*x 
^9-80*x^8+200*x^7+22*x^6-204*x^5+390*x^4+410*x^3-90*x^2+150*x+275)*log(x)+ 
(-4*x^7+40*x^6-100*x^5-2*x^4+20*x^3-50*x^2)*exp(2)+4*x^9-40*x^8+100*x^7+6* 
x^6-56*x^5+110*x^4+102*x^3-18*x^2+30*x+55)/((x^2-10*x+25)*log(x)^5+(5*x^2- 
50*x+125)*log(x)^4+(10*x^2-100*x+250)*log(x)^3+(10*x^2-100*x+250)*log(x)^2 
+(5*x^2-50*x+125)*log(x)+x^2-10*x+25),x, algorithm="fricas")
 

Output:

(x^9 - 5*x^8 + 2*x^6 - 8*x^5 + (x^3 - 3*x^2 - 10*x - 5)*log(x)^4 - 10*x^4 
+ 4*(x^3 - 3*x^2 - 10*x - 5)*log(x)^3 + x^3 + 2*(x^6 - 4*x^5 - 5*x^4 + 3*x 
^3 - 9*x^2 - (x^4 - 4*x^3 - 5*x^2)*e^2 - 30*x - 15)*log(x)^2 - 3*x^2 + (x^ 
5 - 5*x^4)*e^4 - 2*(x^7 - 5*x^6 + x^4 - 4*x^3 - 5*x^2)*e^2 + 4*(x^6 - 4*x^ 
5 - 5*x^4 + x^3 - 3*x^2 - (x^4 - 4*x^3 - 5*x^2)*e^2 - 10*x - 5)*log(x) - 1 
0*x - 5)/((x - 5)*log(x)^4 + 4*(x - 5)*log(x)^3 + 6*(x - 5)*log(x)^2 + 4*( 
x - 5)*log(x) + x - 5)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (26) = 52\).

Time = 0.28 (sec) , antiderivative size = 144, normalized size of antiderivative = 4.00 \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx=x^{2} + 2 x + \frac {x^{8} - 2 x^{6} e^{2} + 2 x^{5} + 2 x^{4} + x^{4} e^{4} - 2 x^{3} e^{2} - 2 x^{2} e^{2} + \left (2 x^{5} + 2 x^{4} - 2 x^{3} e^{2} - 2 x^{2} e^{2}\right ) \log {\left (x \right )}^{2} + \left (4 x^{5} + 4 x^{4} - 4 x^{3} e^{2} - 4 x^{2} e^{2}\right ) \log {\left (x \right )}}{\log {\left (x \right )}^{4} + 4 \log {\left (x \right )}^{3} + 6 \log {\left (x \right )}^{2} + 4 \log {\left (x \right )} + 1} - \frac {5}{x - 5} \] Input:

integrate(((2*x**3-18*x**2+30*x+55)*ln(x)**5+(10*x**3-90*x**2+150*x+275)*l 
n(x)**4+((-6*x**4+56*x**3-110*x**2-100*x)*exp(2)+10*x**6-92*x**5+170*x**4+ 
220*x**3-180*x**2+300*x+550)*ln(x)**3+((-14*x**4+132*x**3-270*x**2-200*x)* 
exp(2)+26*x**6-240*x**5+450*x**4+520*x**3-180*x**2+300*x+550)*ln(x)**2+((4 
*x**5-40*x**4+100*x**3)*exp(2)**2+(-12*x**7+120*x**6-300*x**5-10*x**4+96*x 
**3-210*x**2-100*x)*exp(2)+8*x**9-80*x**8+200*x**7+22*x**6-204*x**5+390*x* 
*4+410*x**3-90*x**2+150*x+275)*ln(x)+(-4*x**7+40*x**6-100*x**5-2*x**4+20*x 
**3-50*x**2)*exp(2)+4*x**9-40*x**8+100*x**7+6*x**6-56*x**5+110*x**4+102*x* 
*3-18*x**2+30*x+55)/((x**2-10*x+25)*ln(x)**5+(5*x**2-50*x+125)*ln(x)**4+(1 
0*x**2-100*x+250)*ln(x)**3+(10*x**2-100*x+250)*ln(x)**2+(5*x**2-50*x+125)* 
ln(x)+x**2-10*x+25),x)
 

Output:

x**2 + 2*x + (x**8 - 2*x**6*exp(2) + 2*x**5 + 2*x**4 + x**4*exp(4) - 2*x** 
3*exp(2) - 2*x**2*exp(2) + (2*x**5 + 2*x**4 - 2*x**3*exp(2) - 2*x**2*exp(2 
))*log(x)**2 + (4*x**5 + 4*x**4 - 4*x**3*exp(2) - 4*x**2*exp(2))*log(x))/( 
log(x)**4 + 4*log(x)**3 + 6*log(x)**2 + 4*log(x) + 1) - 5/(x - 5)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (31) = 62\).

Time = 0.13 (sec) , antiderivative size = 244, normalized size of antiderivative = 6.78 \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx=\frac {x^{9} - 5 \, x^{8} - 2 \, x^{7} e^{2} + 2 \, x^{6} {\left (5 \, e^{2} + 1\right )} + x^{5} {\left (e^{4} - 8\right )} - x^{4} {\left (5 \, e^{4} + 2 \, e^{2} + 10\right )} + {\left (x^{3} - 3 \, x^{2} - 10 \, x - 5\right )} \log \left (x\right )^{4} + x^{3} {\left (8 \, e^{2} + 1\right )} + 4 \, {\left (x^{3} - 3 \, x^{2} - 10 \, x - 5\right )} \log \left (x\right )^{3} + x^{2} {\left (10 \, e^{2} - 3\right )} + 2 \, {\left (x^{6} - 4 \, x^{5} - x^{4} {\left (e^{2} + 5\right )} + x^{3} {\left (4 \, e^{2} + 3\right )} + x^{2} {\left (5 \, e^{2} - 9\right )} - 30 \, x - 15\right )} \log \left (x\right )^{2} + 4 \, {\left (x^{6} - 4 \, x^{5} - x^{4} {\left (e^{2} + 5\right )} + x^{3} {\left (4 \, e^{2} + 1\right )} + x^{2} {\left (5 \, e^{2} - 3\right )} - 10 \, x - 5\right )} \log \left (x\right ) - 10 \, x - 5}{{\left (x - 5\right )} \log \left (x\right )^{4} + 4 \, {\left (x - 5\right )} \log \left (x\right )^{3} + 6 \, {\left (x - 5\right )} \log \left (x\right )^{2} + 4 \, {\left (x - 5\right )} \log \left (x\right ) + x - 5} \] Input:

integrate(((2*x^3-18*x^2+30*x+55)*log(x)^5+(10*x^3-90*x^2+150*x+275)*log(x 
)^4+((-6*x^4+56*x^3-110*x^2-100*x)*exp(2)+10*x^6-92*x^5+170*x^4+220*x^3-18 
0*x^2+300*x+550)*log(x)^3+((-14*x^4+132*x^3-270*x^2-200*x)*exp(2)+26*x^6-2 
40*x^5+450*x^4+520*x^3-180*x^2+300*x+550)*log(x)^2+((4*x^5-40*x^4+100*x^3) 
*exp(2)^2+(-12*x^7+120*x^6-300*x^5-10*x^4+96*x^3-210*x^2-100*x)*exp(2)+8*x 
^9-80*x^8+200*x^7+22*x^6-204*x^5+390*x^4+410*x^3-90*x^2+150*x+275)*log(x)+ 
(-4*x^7+40*x^6-100*x^5-2*x^4+20*x^3-50*x^2)*exp(2)+4*x^9-40*x^8+100*x^7+6* 
x^6-56*x^5+110*x^4+102*x^3-18*x^2+30*x+55)/((x^2-10*x+25)*log(x)^5+(5*x^2- 
50*x+125)*log(x)^4+(10*x^2-100*x+250)*log(x)^3+(10*x^2-100*x+250)*log(x)^2 
+(5*x^2-50*x+125)*log(x)+x^2-10*x+25),x, algorithm="maxima")
 

Output:

(x^9 - 5*x^8 - 2*x^7*e^2 + 2*x^6*(5*e^2 + 1) + x^5*(e^4 - 8) - x^4*(5*e^4 
+ 2*e^2 + 10) + (x^3 - 3*x^2 - 10*x - 5)*log(x)^4 + x^3*(8*e^2 + 1) + 4*(x 
^3 - 3*x^2 - 10*x - 5)*log(x)^3 + x^2*(10*e^2 - 3) + 2*(x^6 - 4*x^5 - x^4* 
(e^2 + 5) + x^3*(4*e^2 + 3) + x^2*(5*e^2 - 9) - 30*x - 15)*log(x)^2 + 4*(x 
^6 - 4*x^5 - x^4*(e^2 + 5) + x^3*(4*e^2 + 1) + x^2*(5*e^2 - 3) - 10*x - 5) 
*log(x) - 10*x - 5)/((x - 5)*log(x)^4 + 4*(x - 5)*log(x)^3 + 6*(x - 5)*log 
(x)^2 + 4*(x - 5)*log(x) + x - 5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (31) = 62\).

Time = 0.19 (sec) , antiderivative size = 360, normalized size of antiderivative = 10.00 \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx=\frac {x^{9} - 5 \, x^{8} - 2 \, x^{7} e^{2} + 2 \, x^{6} \log \left (x\right )^{2} + 10 \, x^{6} e^{2} + 4 \, x^{6} \log \left (x\right ) - 8 \, x^{5} \log \left (x\right )^{2} - 2 \, x^{4} e^{2} \log \left (x\right )^{2} + x^{3} \log \left (x\right )^{4} + 2 \, x^{6} + x^{5} e^{4} - 16 \, x^{5} \log \left (x\right ) - 4 \, x^{4} e^{2} \log \left (x\right ) - 10 \, x^{4} \log \left (x\right )^{2} + 8 \, x^{3} e^{2} \log \left (x\right )^{2} + 4 \, x^{3} \log \left (x\right )^{3} - 3 \, x^{2} \log \left (x\right )^{4} - 8 \, x^{5} - 5 \, x^{4} e^{4} - 2 \, x^{4} e^{2} - 20 \, x^{4} \log \left (x\right ) + 16 \, x^{3} e^{2} \log \left (x\right ) + 6 \, x^{3} \log \left (x\right )^{2} + 10 \, x^{2} e^{2} \log \left (x\right )^{2} - 12 \, x^{2} \log \left (x\right )^{3} - 10 \, x \log \left (x\right )^{4} - 10 \, x^{4} + 8 \, x^{3} e^{2} + 4 \, x^{3} \log \left (x\right ) + 20 \, x^{2} e^{2} \log \left (x\right ) - 18 \, x^{2} \log \left (x\right )^{2} - 40 \, x \log \left (x\right )^{3} - 5 \, \log \left (x\right )^{4} + x^{3} + 10 \, x^{2} e^{2} - 12 \, x^{2} \log \left (x\right ) - 60 \, x \log \left (x\right )^{2} - 20 \, \log \left (x\right )^{3} - 3 \, x^{2} - 40 \, x \log \left (x\right ) - 30 \, \log \left (x\right )^{2} - 10 \, x - 20 \, \log \left (x\right ) - 5}{x \log \left (x\right )^{4} + 4 \, x \log \left (x\right )^{3} - 5 \, \log \left (x\right )^{4} + 6 \, x \log \left (x\right )^{2} - 20 \, \log \left (x\right )^{3} + 4 \, x \log \left (x\right ) - 30 \, \log \left (x\right )^{2} + x - 20 \, \log \left (x\right ) - 5} \] Input:

integrate(((2*x^3-18*x^2+30*x+55)*log(x)^5+(10*x^3-90*x^2+150*x+275)*log(x 
)^4+((-6*x^4+56*x^3-110*x^2-100*x)*exp(2)+10*x^6-92*x^5+170*x^4+220*x^3-18 
0*x^2+300*x+550)*log(x)^3+((-14*x^4+132*x^3-270*x^2-200*x)*exp(2)+26*x^6-2 
40*x^5+450*x^4+520*x^3-180*x^2+300*x+550)*log(x)^2+((4*x^5-40*x^4+100*x^3) 
*exp(2)^2+(-12*x^7+120*x^6-300*x^5-10*x^4+96*x^3-210*x^2-100*x)*exp(2)+8*x 
^9-80*x^8+200*x^7+22*x^6-204*x^5+390*x^4+410*x^3-90*x^2+150*x+275)*log(x)+ 
(-4*x^7+40*x^6-100*x^5-2*x^4+20*x^3-50*x^2)*exp(2)+4*x^9-40*x^8+100*x^7+6* 
x^6-56*x^5+110*x^4+102*x^3-18*x^2+30*x+55)/((x^2-10*x+25)*log(x)^5+(5*x^2- 
50*x+125)*log(x)^4+(10*x^2-100*x+250)*log(x)^3+(10*x^2-100*x+250)*log(x)^2 
+(5*x^2-50*x+125)*log(x)+x^2-10*x+25),x, algorithm="giac")
 

Output:

(x^9 - 5*x^8 - 2*x^7*e^2 + 2*x^6*log(x)^2 + 10*x^6*e^2 + 4*x^6*log(x) - 8* 
x^5*log(x)^2 - 2*x^4*e^2*log(x)^2 + x^3*log(x)^4 + 2*x^6 + x^5*e^4 - 16*x^ 
5*log(x) - 4*x^4*e^2*log(x) - 10*x^4*log(x)^2 + 8*x^3*e^2*log(x)^2 + 4*x^3 
*log(x)^3 - 3*x^2*log(x)^4 - 8*x^5 - 5*x^4*e^4 - 2*x^4*e^2 - 20*x^4*log(x) 
 + 16*x^3*e^2*log(x) + 6*x^3*log(x)^2 + 10*x^2*e^2*log(x)^2 - 12*x^2*log(x 
)^3 - 10*x*log(x)^4 - 10*x^4 + 8*x^3*e^2 + 4*x^3*log(x) + 20*x^2*e^2*log(x 
) - 18*x^2*log(x)^2 - 40*x*log(x)^3 - 5*log(x)^4 + x^3 + 10*x^2*e^2 - 12*x 
^2*log(x) - 60*x*log(x)^2 - 20*log(x)^3 - 3*x^2 - 40*x*log(x) - 30*log(x)^ 
2 - 10*x - 20*log(x) - 5)/(x*log(x)^4 + 4*x*log(x)^3 - 5*log(x)^4 + 6*x*lo 
g(x)^2 - 20*log(x)^3 + 4*x*log(x) - 30*log(x)^2 + x - 20*log(x) - 5)
 

Mupad [B] (verification not implemented)

Time = 4.57 (sec) , antiderivative size = 783, normalized size of antiderivative = 21.75 \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx =\text {Too large to display} \] Input:

int((30*x + log(x)*(150*x - exp(2)*(100*x + 210*x^2 - 96*x^3 + 10*x^4 + 30 
0*x^5 - 120*x^6 + 12*x^7) + exp(4)*(100*x^3 - 40*x^4 + 4*x^5) - 90*x^2 + 4 
10*x^3 + 390*x^4 - 204*x^5 + 22*x^6 + 200*x^7 - 80*x^8 + 8*x^9 + 275) + lo 
g(x)^5*(30*x - 18*x^2 + 2*x^3 + 55) + log(x)^4*(150*x - 90*x^2 + 10*x^3 + 
275) - exp(2)*(50*x^2 - 20*x^3 + 2*x^4 + 100*x^5 - 40*x^6 + 4*x^7) + log(x 
)^3*(300*x - exp(2)*(100*x + 110*x^2 - 56*x^3 + 6*x^4) - 180*x^2 + 220*x^3 
 + 170*x^4 - 92*x^5 + 10*x^6 + 550) + log(x)^2*(300*x - exp(2)*(200*x + 27 
0*x^2 - 132*x^3 + 14*x^4) - 180*x^2 + 520*x^3 + 450*x^4 - 240*x^5 + 26*x^6 
 + 550) - 18*x^2 + 102*x^3 + 110*x^4 - 56*x^5 + 6*x^6 + 100*x^7 - 40*x^8 + 
 4*x^9 + 55)/(log(x)^4*(5*x^2 - 50*x + 125) - 10*x + log(x)^2*(10*x^2 - 10 
0*x + 250) + log(x)^3*(10*x^2 - 100*x + 250) + log(x)*(5*x^2 - 50*x + 125) 
 + log(x)^5*(x^2 - 10*x + 25) + x^2 + 25),x)
 

Output:

2*x - 5/(x - 5) - ((x*(16*x^3*exp(4) - 53*x^2*exp(2) - 16*x*exp(2) - 144*x 
^5*exp(2) + 116*x^3 + 211*x^4 + 192*x^7))/12 - (x*log(x)^3*(8*x*exp(2) + 2 
7*x^2*exp(2) - 64*x^3 - 125*x^4))/12 - (x*log(x)^2*(40*x*exp(2) + 117*x^2* 
exp(2) - 256*x^3 - 475*x^4))/12 + (x*log(x)*(32*x^3*exp(4) - 147*x^2*exp(2 
) - 52*x*exp(2) - 216*x^5*exp(2) + 312*x^3 + 565*x^4 + 256*x^7))/12)/(2*lo 
g(x) + log(x)^2 + 1) - log(x)*(8*x^2*exp(2) + (63*x^3*exp(2))/2 - (256*x^4 
)/3 - (375*x^5)/2) - x^2*((32*exp(2))/3 - 1) + x^4*((32*exp(4))/3 + 80) - 
(135*x^3*exp(2))/4 - 108*x^6*exp(2) - ((x*(48*x^3*exp(4) - 153*x^2*exp(2) 
- 42*x*exp(2) - 540*x^5*exp(2) + 388*x^3 + 810*x^4 + 896*x^7))/6 - (x*log( 
x)^3*(16*x*exp(2) + 81*x^2*exp(2) - 256*x^3 - 625*x^4))/12 - (x*log(x)^2*( 
52*x*exp(2) + 216*x^2*exp(2) - 608*x^3 - 1375*x^4))/6 + (x*log(x)*(128*x^3 
*exp(4) - 675*x^2*exp(2) - 184*x*exp(2) - 1296*x^5*exp(2) + 1760*x^3 + 377 
5*x^4 + 2048*x^7))/12)/(log(x) + 1) - log(x)^2*((4*x^2*exp(2))/3 + (27*x^3 
*exp(2))/4 - (64*x^4)/3 - (625*x^5)/12) - ((x*(2*x^3 - 2*x^5*exp(2) - x^2* 
exp(2) + 3*x^4 + 2*x^7))/2 - (x*log(x)^3*(2*x*exp(2) + 3*x^2*exp(2) - 4*x^ 
3 - 5*x^4))/2 - (x*log(x)^2*(4*x*exp(2) + 7*x^2*exp(2) - 10*x^3 - 13*x^4)) 
/2 + (x*log(x)*(2*x^3*exp(4) - 5*x^2*exp(2) - 2*x*exp(2) - 6*x^5*exp(2) + 
8*x^3 + 11*x^4 + 4*x^7))/2)/(4*log(x) + 6*log(x)^2 + 4*log(x)^3 + log(x)^4 
 + 1) + (1925*x^5)/12 + (512*x^8)/3 - ((x*(x^3*exp(4) - 4*x^2*exp(2) - x*e 
xp(2) - 9*x^5*exp(2) + 8*x^3 + 13*x^4 + 10*x^7))/3 - (x*log(x)^3*(4*x*e...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 329, normalized size of antiderivative = 9.14 \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx=\frac {x \left (-11-3 x +8 e^{2} x^{2}-2 e^{2} x^{3}-8 \mathrm {log}\left (x \right )^{2} x^{4}-10 \mathrm {log}\left (x \right )^{2} x^{3}+10 e^{2} x +2 \mathrm {log}\left (x \right )^{2} x^{5}-2 \mathrm {log}\left (x \right )^{2} e^{2} x^{3}+8 \mathrm {log}\left (x \right )^{2} e^{2} x^{2}+10 \mathrm {log}\left (x \right )^{2} e^{2} x -4 \,\mathrm {log}\left (x \right ) e^{2} x^{3}+16 \,\mathrm {log}\left (x \right ) e^{2} x^{2}+20 \,\mathrm {log}\left (x \right ) e^{2} x +6 \mathrm {log}\left (x \right )^{2} x^{2}-16 \,\mathrm {log}\left (x \right ) x^{4}+x^{2}-5 e^{4} x^{3}+2 x^{5}-5 x^{7}-10 x^{3}+\mathrm {log}\left (x \right )^{4} x^{2}+x^{8}-44 \,\mathrm {log}\left (x \right )+4 \,\mathrm {log}\left (x \right ) x^{2}+4 \,\mathrm {log}\left (x \right ) x^{5}-3 \mathrm {log}\left (x \right )^{4} x +e^{4} x^{4}-20 \,\mathrm {log}\left (x \right ) x^{3}-12 \mathrm {log}\left (x \right )^{3} x -11 \mathrm {log}\left (x \right )^{4}-8 x^{4}-66 \mathrm {log}\left (x \right )^{2}-12 \,\mathrm {log}\left (x \right ) x -18 \mathrm {log}\left (x \right )^{2} x +4 \mathrm {log}\left (x \right )^{3} x^{2}-44 \mathrm {log}\left (x \right )^{3}+10 e^{2} x^{5}-2 e^{2} x^{6}\right )}{\mathrm {log}\left (x \right )^{4} x -5 \mathrm {log}\left (x \right )^{4}+4 \mathrm {log}\left (x \right )^{3} x -20 \mathrm {log}\left (x \right )^{3}+6 \mathrm {log}\left (x \right )^{2} x -30 \mathrm {log}\left (x \right )^{2}+4 \,\mathrm {log}\left (x \right ) x -20 \,\mathrm {log}\left (x \right )+x -5} \] Input:

int(((2*x^3-18*x^2+30*x+55)*log(x)^5+(10*x^3-90*x^2+150*x+275)*log(x)^4+(( 
-6*x^4+56*x^3-110*x^2-100*x)*exp(2)+10*x^6-92*x^5+170*x^4+220*x^3-180*x^2+ 
300*x+550)*log(x)^3+((-14*x^4+132*x^3-270*x^2-200*x)*exp(2)+26*x^6-240*x^5 
+450*x^4+520*x^3-180*x^2+300*x+550)*log(x)^2+((4*x^5-40*x^4+100*x^3)*exp(2 
)^2+(-12*x^7+120*x^6-300*x^5-10*x^4+96*x^3-210*x^2-100*x)*exp(2)+8*x^9-80* 
x^8+200*x^7+22*x^6-204*x^5+390*x^4+410*x^3-90*x^2+150*x+275)*log(x)+(-4*x^ 
7+40*x^6-100*x^5-2*x^4+20*x^3-50*x^2)*exp(2)+4*x^9-40*x^8+100*x^7+6*x^6-56 
*x^5+110*x^4+102*x^3-18*x^2+30*x+55)/((x^2-10*x+25)*log(x)^5+(5*x^2-50*x+1 
25)*log(x)^4+(10*x^2-100*x+250)*log(x)^3+(10*x^2-100*x+250)*log(x)^2+(5*x^ 
2-50*x+125)*log(x)+x^2-10*x+25),x)
                                                                                    
                                                                                    
 

Output:

(x*(log(x)**4*x**2 - 3*log(x)**4*x - 11*log(x)**4 + 4*log(x)**3*x**2 - 12* 
log(x)**3*x - 44*log(x)**3 - 2*log(x)**2*e**2*x**3 + 8*log(x)**2*e**2*x**2 
 + 10*log(x)**2*e**2*x + 2*log(x)**2*x**5 - 8*log(x)**2*x**4 - 10*log(x)** 
2*x**3 + 6*log(x)**2*x**2 - 18*log(x)**2*x - 66*log(x)**2 - 4*log(x)*e**2* 
x**3 + 16*log(x)*e**2*x**2 + 20*log(x)*e**2*x + 4*log(x)*x**5 - 16*log(x)* 
x**4 - 20*log(x)*x**3 + 4*log(x)*x**2 - 12*log(x)*x - 44*log(x) + e**4*x** 
4 - 5*e**4*x**3 - 2*e**2*x**6 + 10*e**2*x**5 - 2*e**2*x**3 + 8*e**2*x**2 + 
 10*e**2*x + x**8 - 5*x**7 + 2*x**5 - 8*x**4 - 10*x**3 + x**2 - 3*x - 11)) 
/(log(x)**4*x - 5*log(x)**4 + 4*log(x)**3*x - 20*log(x)**3 + 6*log(x)**2*x 
 - 30*log(x)**2 + 4*log(x)*x - 20*log(x) + x - 5)