\(\int \frac {-225-30 x-4 x^2-6 x^3-x^4+e^{12} (-625-250 x-25 x^2)+e^9 (-2000-525 x+50 x^2+15 x^3)+e^3 (-1200-215 x+40 x^2+4 x^3-2 x^4)+e^6 (-2350-500 x+115 x^2+25 x^3-x^4)}{225 x+240 x^2+94 x^3+16 x^4+x^5+e^{12} (625 x+250 x^2+25 x^3)+e^9 (2000 x+1150 x^2+200 x^3+10 x^4)+e^6 (2350 x+1750 x^2+435 x^3+40 x^4+x^5)+e^3 (1200 x+1090 x^2+350 x^3+46 x^4+2 x^5)} \, dx\) [1998]

Optimal result

Integrand size = 221, antiderivative size = 35 \[ \int \frac {-225-30 x-4 x^2-6 x^3-x^4+e^{12} \left (-625-250 x-25 x^2\right )+e^9 \left (-2000-525 x+50 x^2+15 x^3\right )+e^3 \left (-1200-215 x+40 x^2+4 x^3-2 x^4\right )+e^6 \left (-2350-500 x+115 x^2+25 x^3-x^4\right )}{225 x+240 x^2+94 x^3+16 x^4+x^5+e^{12} \left (625 x+250 x^2+25 x^3\right )+e^9 \left (2000 x+1150 x^2+200 x^3+10 x^4\right )+e^6 \left (2350 x+1750 x^2+435 x^3+40 x^4+x^5\right )+e^3 \left (1200 x+1090 x^2+350 x^3+46 x^4+2 x^5\right )} \, dx=\frac {2+x+\frac {x}{3+x+e^3 (5+x)}}{1+e^3+\frac {x}{5}}-\log (x) \] Output:

(x/(3+x+exp(3)*(5+x))+2+x)/(exp(3)+1+1/5*x)-ln(x)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.69 \[ \int \frac {-225-30 x-4 x^2-6 x^3-x^4+e^{12} \left (-625-250 x-25 x^2\right )+e^9 \left (-2000-525 x+50 x^2+15 x^3\right )+e^3 \left (-1200-215 x+40 x^2+4 x^3-2 x^4\right )+e^6 \left (-2350-500 x+115 x^2+25 x^3-x^4\right )}{225 x+240 x^2+94 x^3+16 x^4+x^5+e^{12} \left (625 x+250 x^2+25 x^3\right )+e^9 \left (2000 x+1150 x^2+200 x^3+10 x^4\right )+e^6 \left (2350 x+1750 x^2+435 x^3+40 x^4+x^5\right )+e^3 \left (1200 x+1090 x^2+350 x^3+46 x^4+2 x^5\right )} \, dx=-\frac {5 \left (9+2 x+5 e^6 (5+x)+e^3 (30+8 x)\right )}{15+8 x+x^2+5 e^6 (5+x)+e^3 \left (40+15 x+x^2\right )}-\log (x) \] Input:

Integrate[(-225 - 30*x - 4*x^2 - 6*x^3 - x^4 + E^12*(-625 - 250*x - 25*x^2 
) + E^9*(-2000 - 525*x + 50*x^2 + 15*x^3) + E^3*(-1200 - 215*x + 40*x^2 + 
4*x^3 - 2*x^4) + E^6*(-2350 - 500*x + 115*x^2 + 25*x^3 - x^4))/(225*x + 24 
0*x^2 + 94*x^3 + 16*x^4 + x^5 + E^12*(625*x + 250*x^2 + 25*x^3) + E^9*(200 
0*x + 1150*x^2 + 200*x^3 + 10*x^4) + E^6*(2350*x + 1750*x^2 + 435*x^3 + 40 
*x^4 + x^5) + E^3*(1200*x + 1090*x^2 + 350*x^3 + 46*x^4 + 2*x^5)),x]
 

Output:

(-5*(9 + 2*x + 5*E^6*(5 + x) + E^3*(30 + 8*x)))/(15 + 8*x + x^2 + 5*E^6*(5 
 + x) + E^3*(40 + 15*x + x^2)) - Log[x]
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(88\) vs. \(2(35)=70\).

Time = 0.72 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.51, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2026, 2462, 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.

\(\displaystyle \int \frac {-x^4-6 x^3-4 x^2+e^{12} \left (-25 x^2-250 x-625\right )+e^9 \left (15 x^3+50 x^2-525 x-2000\right )+e^3 \left (-2 x^4+4 x^3+40 x^2-215 x-1200\right )+e^6 \left (-x^4+25 x^3+115 x^2-500 x-2350\right )-30 x-225}{x^5+16 x^4+94 x^3+240 x^2+e^{12} \left (25 x^3+250 x^2+625 x\right )+e^9 \left (10 x^4+200 x^3+1150 x^2+2000 x\right )+e^6 \left (x^5+40 x^4+435 x^3+1750 x^2+2350 x\right )+e^3 \left (2 x^5+46 x^4+350 x^3+1090 x^2+1200 x\right )+225 x} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {-x^4-6 x^3-4 x^2+e^{12} \left (-25 x^2-250 x-625\right )+e^9 \left (15 x^3+50 x^2-525 x-2000\right )+e^3 \left (-2 x^4+4 x^3+40 x^2-215 x-1200\right )+e^6 \left (-x^4+25 x^3+115 x^2-500 x-2350\right )-30 x-225}{x \left (\left (1+e^3\right )^2 x^4+2 \left (8+23 e^3+20 e^6+5 e^9\right ) x^3+\left (94+350 e^3+435 e^6+200 e^9+25 e^{12}\right ) x^2+10 \left (24+109 e^3+175 e^6+115 e^9+25 e^{12}\right ) x+25 \left (3+8 e^3+5 e^6\right )^2\right )}dx\)

\(\Big \downarrow \) 2462

\(\displaystyle \int \left (\frac {5 \left (1+20 e^3+40 e^6+25 e^9\right )}{\left (2+5 e^3+5 e^6\right ) \left (x+5 e^3+5\right )^2}+\frac {5 \left (3+8 e^3+5 e^6\right )}{\left (2+5 e^3+5 e^6\right ) \left (\left (1+e^3\right ) x+5 e^3+3\right )^2}-\frac {1}{x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {5 \left (1+20 e^3+40 e^6+25 e^9\right )}{\left (2+5 e^3+5 e^6\right ) \left (x+5 \left (1+e^3\right )\right )}-\frac {5 \left (3+5 e^3\right )}{\left (2+5 e^3+5 e^6\right ) \left (\left (1+e^3\right ) x+5 e^3+3\right )}-\log (x)\)

Input:

Int[(-225 - 30*x - 4*x^2 - 6*x^3 - x^4 + E^12*(-625 - 250*x - 25*x^2) + E^ 
9*(-2000 - 525*x + 50*x^2 + 15*x^3) + E^3*(-1200 - 215*x + 40*x^2 + 4*x^3 
- 2*x^4) + E^6*(-2350 - 500*x + 115*x^2 + 25*x^3 - x^4))/(225*x + 240*x^2 
+ 94*x^3 + 16*x^4 + x^5 + E^12*(625*x + 250*x^2 + 25*x^3) + E^9*(2000*x + 
1150*x^2 + 200*x^3 + 10*x^4) + E^6*(2350*x + 1750*x^2 + 435*x^3 + 40*x^4 + 
 x^5) + E^3*(1200*x + 1090*x^2 + 350*x^3 + 46*x^4 + 2*x^5)),x]
 

Output:

(-5*(1 + 20*E^3 + 40*E^6 + 25*E^9))/((2 + 5*E^3 + 5*E^6)*(5*(1 + E^3) + x) 
) - (5*(3 + 5*E^3))/((2 + 5*E^3 + 5*E^6)*(3 + 5*E^3 + (1 + E^3)*x)) - Log[ 
x]
 

Defintions of rubi rules used

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 2462

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u*Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ 
[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 0 
] && RationalFunctionQ[u, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs. \(2(31)=62\).

Time = 1.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.86

method	result	size
risch	\(\frac {\left (-5 \,{\mathrm e}^{6}-8 \,{\mathrm e}^{3}-2\right ) x -25 \,{\mathrm e}^{6}-30 \,{\mathrm e}^{3}-9}{x \,{\mathrm e}^{6}+\frac {x^{2} {\mathrm e}^{3}}{5}+5 \,{\mathrm e}^{6}+3 x \,{\mathrm e}^{3}+\frac {x^{2}}{5}+8 \,{\mathrm e}^{3}+\frac {8 x}{5}+3}-\ln \left (x \right )\)	\(65\)
norman	\(\frac {-\frac {\left (25 \,{\mathrm e}^{9}+65 \,{\mathrm e}^{6}+50 \,{\mathrm e}^{3}+10\right ) x}{{\mathrm e}^{3}+1}-125 \,{\mathrm e}^{6}-45-150 \,{\mathrm e}^{3}}{x^{2} {\mathrm e}^{3}+5 x \,{\mathrm e}^{6}+15 x \,{\mathrm e}^{3}+x^{2}+25 \,{\mathrm e}^{6}+40 \,{\mathrm e}^{3}+8 x +15}-\ln \left (x \right )\)	\(84\)
parallelrisch	\(-\frac {45+10 x +5 x \,{\mathrm e}^{9} \ln \left (x \right )+{\mathrm e}^{6} \ln \left (x \right ) x^{2}+20 x \,{\mathrm e}^{6} \ln \left (x \right )+65 x \,{\mathrm e}^{6}+25 x \,{\mathrm e}^{9}+50 x \,{\mathrm e}^{3}+55 \,{\mathrm e}^{3} \ln \left (x \right )+23 x \,{\mathrm e}^{3} \ln \left (x \right )+125 \,{\mathrm e}^{9}+8 x \ln \left (x \right )+x^{2} \ln \left (x \right )+15 \ln \left (x \right )+195 \,{\mathrm e}^{3}+275 \,{\mathrm e}^{6}+65 \ln \left (x \right ) {\mathrm e}^{6}+25 \ln \left (x \right ) {\mathrm e}^{9}+2 \,{\mathrm e}^{3} \ln \left (x \right ) x^{2}}{\left ({\mathrm e}^{3}+1\right ) \left (x^{2} {\mathrm e}^{3}+5 x \,{\mathrm e}^{6}+15 x \,{\mathrm e}^{3}+x^{2}+25 \,{\mathrm e}^{6}+40 \,{\mathrm e}^{3}+8 x +15\right )}\)	\(168\)
default	\(-\ln \left (x \right )+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left ({\mathrm e}^{6}+2 \,{\mathrm e}^{3}+1\right ) \textit {\_Z}^{4}+\left (10 \,{\mathrm e}^{9}+40 \,{\mathrm e}^{6}+46 \,{\mathrm e}^{3}+16\right ) \textit {\_Z}^{3}+\left (25 \,{\mathrm e}^{12}+200 \,{\mathrm e}^{9}+435 \,{\mathrm e}^{6}+350 \,{\mathrm e}^{3}+94\right ) \textit {\_Z}^{2}+\left (250 \,{\mathrm e}^{12}+1150 \,{\mathrm e}^{9}+1750 \,{\mathrm e}^{6}+1090 \,{\mathrm e}^{3}+240\right ) \textit {\_Z} +625 \,{\mathrm e}^{12}+2000 \,{\mathrm e}^{9}+2350 \,{\mathrm e}^{6}+1200 \,{\mathrm e}^{3}+225\right )}{\sum }\frac {\left (42+\left (2+5 \,{\mathrm e}^{9}+13 \,{\mathrm e}^{6}+10 \,{\mathrm e}^{3}\right ) \textit {\_R}^{2}+2 \left (9+25 \,{\mathrm e}^{9}+55 \,{\mathrm e}^{6}+39 \,{\mathrm e}^{3}\right ) \textit {\_R} +125 \,{\mathrm e}^{9}+250 \,{\mathrm e}^{6}+175 \,{\mathrm e}^{3}\right ) \ln \left (x -\textit {\_R} \right )}{120+25 \textit {\_R} \,{\mathrm e}^{12}+15 \,{\mathrm e}^{9} \textit {\_R}^{2}+2 \textit {\_R}^{3} {\mathrm e}^{6}+125 \,{\mathrm e}^{12}+200 \textit {\_R} \,{\mathrm e}^{9}+60 \textit {\_R}^{2} {\mathrm e}^{6}+4 \textit {\_R}^{3} {\mathrm e}^{3}+575 \,{\mathrm e}^{9}+435 \textit {\_R} \,{\mathrm e}^{6}+69 \textit {\_R}^{2} {\mathrm e}^{3}+2 \textit {\_R}^{3}+875 \,{\mathrm e}^{6}+350 \textit {\_R} \,{\mathrm e}^{3}+24 \textit {\_R}^{2}+545 \,{\mathrm e}^{3}+94 \textit {\_R}}\right )}{2}\)	\(246\)

Input:

int(((-25*x^2-250*x-625)*exp(3)^4+(15*x^3+50*x^2-525*x-2000)*exp(3)^3+(-x^ 
4+25*x^3+115*x^2-500*x-2350)*exp(3)^2+(-2*x^4+4*x^3+40*x^2-215*x-1200)*exp 
(3)-x^4-6*x^3-4*x^2-30*x-225)/((25*x^3+250*x^2+625*x)*exp(3)^4+(10*x^4+200 
*x^3+1150*x^2+2000*x)*exp(3)^3+(x^5+40*x^4+435*x^3+1750*x^2+2350*x)*exp(3) 
^2+(2*x^5+46*x^4+350*x^3+1090*x^2+1200*x)*exp(3)+x^5+16*x^4+94*x^3+240*x^2 
+225*x),x,method=_RETURNVERBOSE)
 

Output:

((-5*exp(6)-8*exp(3)-2)*x-25*exp(6)-30*exp(3)-9)/(x*exp(6)+1/5*x^2*exp(3)+ 
5*exp(6)+3*x*exp(3)+1/5*x^2+8*exp(3)+8/5*x+3)-ln(x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (32) = 64\).

Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.29 \[ \int \frac {-225-30 x-4 x^2-6 x^3-x^4+e^{12} \left (-625-250 x-25 x^2\right )+e^9 \left (-2000-525 x+50 x^2+15 x^3\right )+e^3 \left (-1200-215 x+40 x^2+4 x^3-2 x^4\right )+e^6 \left (-2350-500 x+115 x^2+25 x^3-x^4\right )}{225 x+240 x^2+94 x^3+16 x^4+x^5+e^{12} \left (625 x+250 x^2+25 x^3\right )+e^9 \left (2000 x+1150 x^2+200 x^3+10 x^4\right )+e^6 \left (2350 x+1750 x^2+435 x^3+40 x^4+x^5\right )+e^3 \left (1200 x+1090 x^2+350 x^3+46 x^4+2 x^5\right )} \, dx=-\frac {25 \, {\left (x + 5\right )} e^{6} + 10 \, {\left (4 \, x + 15\right )} e^{3} + {\left (x^{2} + 5 \, {\left (x + 5\right )} e^{6} + {\left (x^{2} + 15 \, x + 40\right )} e^{3} + 8 \, x + 15\right )} \log \left (x\right ) + 10 \, x + 45}{x^{2} + 5 \, {\left (x + 5\right )} e^{6} + {\left (x^{2} + 15 \, x + 40\right )} e^{3} + 8 \, x + 15} \] Input:

integrate(((-25*x^2-250*x-625)*exp(3)^4+(15*x^3+50*x^2-525*x-2000)*exp(3)^ 
3+(-x^4+25*x^3+115*x^2-500*x-2350)*exp(3)^2+(-2*x^4+4*x^3+40*x^2-215*x-120 
0)*exp(3)-x^4-6*x^3-4*x^2-30*x-225)/((25*x^3+250*x^2+625*x)*exp(3)^4+(10*x 
^4+200*x^3+1150*x^2+2000*x)*exp(3)^3+(x^5+40*x^4+435*x^3+1750*x^2+2350*x)* 
exp(3)^2+(2*x^5+46*x^4+350*x^3+1090*x^2+1200*x)*exp(3)+x^5+16*x^4+94*x^3+2 
40*x^2+225*x),x, algorithm="fricas")
 

Output:

-(25*(x + 5)*e^6 + 10*(4*x + 15)*e^3 + (x^2 + 5*(x + 5)*e^6 + (x^2 + 15*x 
+ 40)*e^3 + 8*x + 15)*log(x) + 10*x + 45)/(x^2 + 5*(x + 5)*e^6 + (x^2 + 15 
*x + 40)*e^3 + 8*x + 15)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (26) = 52\).

Time = 3.86 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.80 \[ \int \frac {-225-30 x-4 x^2-6 x^3-x^4+e^{12} \left (-625-250 x-25 x^2\right )+e^9 \left (-2000-525 x+50 x^2+15 x^3\right )+e^3 \left (-1200-215 x+40 x^2+4 x^3-2 x^4\right )+e^6 \left (-2350-500 x+115 x^2+25 x^3-x^4\right )}{225 x+240 x^2+94 x^3+16 x^4+x^5+e^{12} \left (625 x+250 x^2+25 x^3\right )+e^9 \left (2000 x+1150 x^2+200 x^3+10 x^4\right )+e^6 \left (2350 x+1750 x^2+435 x^3+40 x^4+x^5\right )+e^3 \left (1200 x+1090 x^2+350 x^3+46 x^4+2 x^5\right )} \, dx=- \frac {x \left (10 + 40 e^{3} + 25 e^{6}\right ) + 45 + 150 e^{3} + 125 e^{6}}{x^{2} \cdot \left (1 + e^{3}\right ) + x \left (8 + 15 e^{3} + 5 e^{6}\right ) + 15 + 40 e^{3} + 25 e^{6}} - \log {\left (x \right )} \] Input:

integrate(((-25*x**2-250*x-625)*exp(3)**4+(15*x**3+50*x**2-525*x-2000)*exp 
(3)**3+(-x**4+25*x**3+115*x**2-500*x-2350)*exp(3)**2+(-2*x**4+4*x**3+40*x* 
*2-215*x-1200)*exp(3)-x**4-6*x**3-4*x**2-30*x-225)/((25*x**3+250*x**2+625* 
x)*exp(3)**4+(10*x**4+200*x**3+1150*x**2+2000*x)*exp(3)**3+(x**5+40*x**4+4 
35*x**3+1750*x**2+2350*x)*exp(3)**2+(2*x**5+46*x**4+350*x**3+1090*x**2+120 
0*x)*exp(3)+x**5+16*x**4+94*x**3+240*x**2+225*x),x)
 

Output:

-(x*(10 + 40*exp(3) + 25*exp(6)) + 45 + 150*exp(3) + 125*exp(6))/(x**2*(1 
+ exp(3)) + x*(8 + 15*exp(3) + 5*exp(6)) + 15 + 40*exp(3) + 25*exp(6)) - l 
og(x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.74 \[ \int \frac {-225-30 x-4 x^2-6 x^3-x^4+e^{12} \left (-625-250 x-25 x^2\right )+e^9 \left (-2000-525 x+50 x^2+15 x^3\right )+e^3 \left (-1200-215 x+40 x^2+4 x^3-2 x^4\right )+e^6 \left (-2350-500 x+115 x^2+25 x^3-x^4\right )}{225 x+240 x^2+94 x^3+16 x^4+x^5+e^{12} \left (625 x+250 x^2+25 x^3\right )+e^9 \left (2000 x+1150 x^2+200 x^3+10 x^4\right )+e^6 \left (2350 x+1750 x^2+435 x^3+40 x^4+x^5\right )+e^3 \left (1200 x+1090 x^2+350 x^3+46 x^4+2 x^5\right )} \, dx=-\frac {5 \, {\left (x {\left (5 \, e^{6} + 8 \, e^{3} + 2\right )} + 25 \, e^{6} + 30 \, e^{3} + 9\right )}}{x^{2} {\left (e^{3} + 1\right )} + x {\left (5 \, e^{6} + 15 \, e^{3} + 8\right )} + 25 \, e^{6} + 40 \, e^{3} + 15} - \log \left (x\right ) \] Input:

integrate(((-25*x^2-250*x-625)*exp(3)^4+(15*x^3+50*x^2-525*x-2000)*exp(3)^ 
3+(-x^4+25*x^3+115*x^2-500*x-2350)*exp(3)^2+(-2*x^4+4*x^3+40*x^2-215*x-120 
0)*exp(3)-x^4-6*x^3-4*x^2-30*x-225)/((25*x^3+250*x^2+625*x)*exp(3)^4+(10*x 
^4+200*x^3+1150*x^2+2000*x)*exp(3)^3+(x^5+40*x^4+435*x^3+1750*x^2+2350*x)* 
exp(3)^2+(2*x^5+46*x^4+350*x^3+1090*x^2+1200*x)*exp(3)+x^5+16*x^4+94*x^3+2 
40*x^2+225*x),x, algorithm="maxima")
 

Output:

-5*(x*(5*e^6 + 8*e^3 + 2) + 25*e^6 + 30*e^3 + 9)/(x^2*(e^3 + 1) + x*(5*e^6 
 + 15*e^3 + 8) + 25*e^6 + 40*e^3 + 15) - log(x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (32) = 64\).

Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.86 \[ \int \frac {-225-30 x-4 x^2-6 x^3-x^4+e^{12} \left (-625-250 x-25 x^2\right )+e^9 \left (-2000-525 x+50 x^2+15 x^3\right )+e^3 \left (-1200-215 x+40 x^2+4 x^3-2 x^4\right )+e^6 \left (-2350-500 x+115 x^2+25 x^3-x^4\right )}{225 x+240 x^2+94 x^3+16 x^4+x^5+e^{12} \left (625 x+250 x^2+25 x^3\right )+e^9 \left (2000 x+1150 x^2+200 x^3+10 x^4\right )+e^6 \left (2350 x+1750 x^2+435 x^3+40 x^4+x^5\right )+e^3 \left (1200 x+1090 x^2+350 x^3+46 x^4+2 x^5\right )} \, dx=-\frac {5 \, {\left (5 \, x e^{6} + 8 \, x e^{3} + 2 \, x + 25 \, e^{6} + 30 \, e^{3} + 9\right )}}{x^{2} e^{3} + x^{2} + 5 \, x e^{6} + 15 \, x e^{3} + 8 \, x + 25 \, e^{6} + 40 \, e^{3} + 15} - \log \left ({\left | x \right |}\right ) \] Input:

integrate(((-25*x^2-250*x-625)*exp(3)^4+(15*x^3+50*x^2-525*x-2000)*exp(3)^ 
3+(-x^4+25*x^3+115*x^2-500*x-2350)*exp(3)^2+(-2*x^4+4*x^3+40*x^2-215*x-120 
0)*exp(3)-x^4-6*x^3-4*x^2-30*x-225)/((25*x^3+250*x^2+625*x)*exp(3)^4+(10*x 
^4+200*x^3+1150*x^2+2000*x)*exp(3)^3+(x^5+40*x^4+435*x^3+1750*x^2+2350*x)* 
exp(3)^2+(2*x^5+46*x^4+350*x^3+1090*x^2+1200*x)*exp(3)+x^5+16*x^4+94*x^3+2 
40*x^2+225*x),x, algorithm="giac")
 

Output:

-5*(5*x*e^6 + 8*x*e^3 + 2*x + 25*e^6 + 30*e^3 + 9)/(x^2*e^3 + x^2 + 5*x*e^ 
6 + 15*x*e^3 + 8*x + 25*e^6 + 40*e^3 + 15) - log(abs(x))
 

Mupad [B] (verification not implemented)

Time = 4.58 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.74 \[ \int \frac {-225-30 x-4 x^2-6 x^3-x^4+e^{12} \left (-625-250 x-25 x^2\right )+e^9 \left (-2000-525 x+50 x^2+15 x^3\right )+e^3 \left (-1200-215 x+40 x^2+4 x^3-2 x^4\right )+e^6 \left (-2350-500 x+115 x^2+25 x^3-x^4\right )}{225 x+240 x^2+94 x^3+16 x^4+x^5+e^{12} \left (625 x+250 x^2+25 x^3\right )+e^9 \left (2000 x+1150 x^2+200 x^3+10 x^4\right )+e^6 \left (2350 x+1750 x^2+435 x^3+40 x^4+x^5\right )+e^3 \left (1200 x+1090 x^2+350 x^3+46 x^4+2 x^5\right )} \, dx=-\ln \left (x\right )-\frac {150\,{\mathrm {e}}^3+125\,{\mathrm {e}}^6+x\,\left (40\,{\mathrm {e}}^3+25\,{\mathrm {e}}^6+10\right )+45}{\left ({\mathrm {e}}^3+1\right )\,x^2+\left (15\,{\mathrm {e}}^3+5\,{\mathrm {e}}^6+8\right )\,x+40\,{\mathrm {e}}^3+25\,{\mathrm {e}}^6+15} \] Input:

int(-(30*x + exp(12)*(250*x + 25*x^2 + 625) + exp(6)*(500*x - 115*x^2 - 25 
*x^3 + x^4 + 2350) + exp(9)*(525*x - 50*x^2 - 15*x^3 + 2000) + exp(3)*(215 
*x - 40*x^2 - 4*x^3 + 2*x^4 + 1200) + 4*x^2 + 6*x^3 + x^4 + 225)/(225*x + 
exp(12)*(625*x + 250*x^2 + 25*x^3) + exp(6)*(2350*x + 1750*x^2 + 435*x^3 + 
 40*x^4 + x^5) + exp(9)*(2000*x + 1150*x^2 + 200*x^3 + 10*x^4) + exp(3)*(1 
200*x + 1090*x^2 + 350*x^3 + 46*x^4 + 2*x^5) + 240*x^2 + 94*x^3 + 16*x^4 + 
 x^5),x)
 

Output:

- log(x) - (150*exp(3) + 125*exp(6) + x*(40*exp(3) + 25*exp(6) + 10) + 45) 
/(40*exp(3) + 25*exp(6) + x*(15*exp(3) + 5*exp(6) + 8) + x^2*(exp(3) + 1) 
+ 15)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 233, normalized size of antiderivative = 6.66 \[ \int \frac {-225-30 x-4 x^2-6 x^3-x^4+e^{12} \left (-625-250 x-25 x^2\right )+e^9 \left (-2000-525 x+50 x^2+15 x^3\right )+e^3 \left (-1200-215 x+40 x^2+4 x^3-2 x^4\right )+e^6 \left (-2350-500 x+115 x^2+25 x^3-x^4\right )}{225 x+240 x^2+94 x^3+16 x^4+x^5+e^{12} \left (625 x+250 x^2+25 x^3\right )+e^9 \left (2000 x+1150 x^2+200 x^3+10 x^4\right )+e^6 \left (2350 x+1750 x^2+435 x^3+40 x^4+x^5\right )+e^3 \left (1200 x+1090 x^2+350 x^3+46 x^4+2 x^5\right )} \, dx=\frac {-25 \,\mathrm {log}\left (x \right ) e^{12} x -125 \,\mathrm {log}\left (x \right ) e^{12}-5 \,\mathrm {log}\left (x \right ) e^{9} x^{2}-150 \,\mathrm {log}\left (x \right ) e^{9} x -575 \,\mathrm {log}\left (x \right ) e^{9}-20 \,\mathrm {log}\left (x \right ) e^{6} x^{2}-305 \,\mathrm {log}\left (x \right ) e^{6} x -875 \,\mathrm {log}\left (x \right ) e^{6}-23 \,\mathrm {log}\left (x \right ) e^{3} x^{2}-240 \,\mathrm {log}\left (x \right ) e^{3} x -545 \,\mathrm {log}\left (x \right ) e^{3}-8 \,\mathrm {log}\left (x \right ) x^{2}-64 \,\mathrm {log}\left (x \right ) x -120 \,\mathrm {log}\left (x \right )+25 e^{9} x^{2}-625 e^{9}+65 e^{6} x^{2}-1250 e^{6}+50 e^{3} x^{2}-875 e^{3}+10 x^{2}-210}{25 e^{12} x +125 e^{12}+5 e^{9} x^{2}+150 e^{9} x +575 e^{9}+20 e^{6} x^{2}+305 e^{6} x +875 e^{6}+23 e^{3} x^{2}+240 e^{3} x +545 e^{3}+8 x^{2}+64 x +120} \] Input:

int(((-25*x^2-250*x-625)*exp(3)^4+(15*x^3+50*x^2-525*x-2000)*exp(3)^3+(-x^ 
4+25*x^3+115*x^2-500*x-2350)*exp(3)^2+(-2*x^4+4*x^3+40*x^2-215*x-1200)*exp 
(3)-x^4-6*x^3-4*x^2-30*x-225)/((25*x^3+250*x^2+625*x)*exp(3)^4+(10*x^4+200 
*x^3+1150*x^2+2000*x)*exp(3)^3+(x^5+40*x^4+435*x^3+1750*x^2+2350*x)*exp(3) 
^2+(2*x^5+46*x^4+350*x^3+1090*x^2+1200*x)*exp(3)+x^5+16*x^4+94*x^3+240*x^2 
+225*x),x)
 

Output:

( - 25*log(x)*e**12*x - 125*log(x)*e**12 - 5*log(x)*e**9*x**2 - 150*log(x) 
*e**9*x - 575*log(x)*e**9 - 20*log(x)*e**6*x**2 - 305*log(x)*e**6*x - 875* 
log(x)*e**6 - 23*log(x)*e**3*x**2 - 240*log(x)*e**3*x - 545*log(x)*e**3 - 
8*log(x)*x**2 - 64*log(x)*x - 120*log(x) + 25*e**9*x**2 - 625*e**9 + 65*e* 
*6*x**2 - 1250*e**6 + 50*e**3*x**2 - 875*e**3 + 10*x**2 - 210)/(25*e**12*x 
 + 125*e**12 + 5*e**9*x**2 + 150*e**9*x + 575*e**9 + 20*e**6*x**2 + 305*e* 
*6*x + 875*e**6 + 23*e**3*x**2 + 240*e**3*x + 545*e**3 + 8*x**2 + 64*x + 1 
20)