\(\int \frac {50625+54000 x+20990 x^2+3560 x^3+225 x^4+e^4 (81+54 x+12 x^2)+e^2 (-4050-3510 x-1106 x^2-120 x^3)+(-16875-18000 x-7050 x^2-1200 x^3-75 x^4+e^4 (-27-18 x-3 x^2)+e^2 (1350+1170 x+330 x^2+30 x^3)) \log (x)}{16875 x^2+18000 x^3+7050 x^4+1200 x^5+75 x^6+e^4 (27 x^2+18 x^3+3 x^4)+e^2 (-1350 x^2-1170 x^3-330 x^4-30 x^5)} \, dx\) [1142]

Optimal result

Integrand size = 183, antiderivative size = 35 \[ \int \frac {50625+54000 x+20990 x^2+3560 x^3+225 x^4+e^4 \left (81+54 x+12 x^2\right )+e^2 \left (-4050-3510 x-1106 x^2-120 x^3\right )+\left (-16875-18000 x-7050 x^2-1200 x^3-75 x^4+e^4 \left (-27-18 x-3 x^2\right )+e^2 \left (1350+1170 x+330 x^2+30 x^3\right )\right ) \log (x)}{16875 x^2+18000 x^3+7050 x^4+1200 x^5+75 x^6+e^4 \left (27 x^2+18 x^3+3 x^4\right )+e^2 \left (-1350 x^2-1170 x^3-330 x^4-30 x^5\right )} \, dx=\frac {-2+\frac {\left (\frac {4}{3}+e^2\right ) x}{(3+x) \left (-e^2+5 (5+x)\right )}+\log (x)}{x} \] Output:

(ln(x)-2+(exp(2)+4/3)/(3+x)/(25+5*x-exp(2))*x)/x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.86 \[ \int \frac {50625+54000 x+20990 x^2+3560 x^3+225 x^4+e^4 \left (81+54 x+12 x^2\right )+e^2 \left (-4050-3510 x-1106 x^2-120 x^3\right )+\left (-16875-18000 x-7050 x^2-1200 x^3-75 x^4+e^4 \left (-27-18 x-3 x^2\right )+e^2 \left (1350+1170 x+330 x^2+30 x^3\right )\right ) \log (x)}{16875 x^2+18000 x^3+7050 x^4+1200 x^5+75 x^6+e^4 \left (27 x^2+18 x^3+3 x^4\right )+e^2 \left (-1350 x^2-1170 x^3-330 x^4-30 x^5\right )} \, dx=\frac {1}{3} \left (-\frac {6}{x}+\frac {-4-3 e^2}{\left (-10+e^2\right ) (3+x)}+\frac {5 \left (4+3 e^2\right )}{\left (-10+e^2\right ) \left (25-e^2+5 x\right )}+\frac {3 \log (x)}{x}\right ) \] Input:

Integrate[(50625 + 54000*x + 20990*x^2 + 3560*x^3 + 225*x^4 + E^4*(81 + 54 
*x + 12*x^2) + E^2*(-4050 - 3510*x - 1106*x^2 - 120*x^3) + (-16875 - 18000 
*x - 7050*x^2 - 1200*x^3 - 75*x^4 + E^4*(-27 - 18*x - 3*x^2) + E^2*(1350 + 
 1170*x + 330*x^2 + 30*x^3))*Log[x])/(16875*x^2 + 18000*x^3 + 7050*x^4 + 1 
200*x^5 + 75*x^6 + E^4*(27*x^2 + 18*x^3 + 3*x^4) + E^2*(-1350*x^2 - 1170*x 
^3 - 330*x^4 - 30*x^5)),x]
 

Output:

(-6/x + (-4 - 3*E^2)/((-10 + E^2)*(3 + x)) + (5*(4 + 3*E^2))/((-10 + E^2)* 
(25 - E^2 + 5*x)) + (3*Log[x])/x)/3
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1706\) vs. \(2(35)=70\).

Time = 4.13 (sec) , antiderivative size = 1706, normalized size of antiderivative = 48.74, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {2026, 2463, 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.

\(\displaystyle \int \frac {225 x^4+3560 x^3+20990 x^2+e^4 \left (12 x^2+54 x+81\right )+e^2 \left (-120 x^3-1106 x^2-3510 x-4050\right )+\left (-75 x^4-1200 x^3-7050 x^2+e^4 \left (-3 x^2-18 x-27\right )+e^2 \left (30 x^3+330 x^2+1170 x+1350\right )-18000 x-16875\right ) \log (x)+54000 x+50625}{75 x^6+1200 x^5+7050 x^4+18000 x^3+16875 x^2+e^4 \left (3 x^4+18 x^3+27 x^2\right )+e^2 \left (-30 x^5-330 x^4-1170 x^3-1350 x^2\right )} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {225 x^4+3560 x^3+20990 x^2+e^4 \left (12 x^2+54 x+81\right )+e^2 \left (-120 x^3-1106 x^2-3510 x-4050\right )+\left (-75 x^4-1200 x^3-7050 x^2+e^4 \left (-3 x^2-18 x-27\right )+e^2 \left (30 x^3+330 x^2+1170 x+1350\right )-18000 x-16875\right ) \log (x)+54000 x+50625}{x^2 \left (75 x^4+30 \left (40-e^2\right ) x^3+3 \left (2350-110 e^2+e^4\right ) x^2+18 \left (1000-65 e^2+e^4\right ) x+27 \left (25-e^2\right )^2\right )}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {10 \left (225 x^4+3560 x^3+20990 x^2+e^4 \left (12 x^2+54 x+81\right )+e^2 \left (-120 x^3-1106 x^2-3510 x-4050\right )+\left (-75 x^4-1200 x^3-7050 x^2+e^4 \left (-3 x^2-18 x-27\right )+e^2 \left (30 x^3+330 x^2+1170 x+1350\right )-18000 x-16875\right ) \log (x)+54000 x+50625\right )}{3 \left (e^2-10\right )^3 x^2 (x+3)}+\frac {50 \left (225 x^4+3560 x^3+20990 x^2+e^4 \left (12 x^2+54 x+81\right )+e^2 \left (-120 x^3-1106 x^2-3510 x-4050\right )+\left (-75 x^4-1200 x^3-7050 x^2+e^4 \left (-3 x^2-18 x-27\right )+e^2 \left (30 x^3+330 x^2+1170 x+1350\right )-18000 x-16875\right ) \log (x)+54000 x+50625\right )}{3 \left (e^2-10\right )^3 \left (-5 x+e^2-25\right ) x^2}+\frac {25 \left (225 x^4+3560 x^3+20990 x^2+e^4 \left (12 x^2+54 x+81\right )+e^2 \left (-120 x^3-1106 x^2-3510 x-4050\right )+\left (-75 x^4-1200 x^3-7050 x^2+e^4 \left (-3 x^2-18 x-27\right )+e^2 \left (30 x^3+330 x^2+1170 x+1350\right )-18000 x-16875\right ) \log (x)+54000 x+50625\right )}{3 \left (e^2-10\right )^2 \left (-5 x+e^2-25\right )^2 x^2}+\frac {225 x^4+3560 x^3+20990 x^2+e^4 \left (12 x^2+54 x+81\right )+e^2 \left (-120 x^3-1106 x^2-3510 x-4050\right )+\left (-75 x^4-1200 x^3-7050 x^2+e^4 \left (-3 x^2-18 x-27\right )+e^2 \left (30 x^3+330 x^2+1170 x+1350\right )-18000 x-16875\right ) \log (x)+54000 x+50625}{3 \left (e^2-10\right )^2 x^2 (x+3)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5 \left (1375-80 e^2+e^4\right ) \log ^2(x)}{\left (10-e^2\right )^3}-\frac {5 \left (25-e^2\right ) \log ^2(x)}{\left (10-e^2\right )^2}-\frac {75 \log ^2(x)}{\left (10-e^2\right )^2}-\frac {75 \left (65-2 e^2\right ) \log ^2(x)}{\left (10-e^2\right )^3}-\frac {50 \left (55-e^2\right ) x \log (x)}{\left (10-e^2\right )^3}-\frac {50 x \log (x)}{\left (10-e^2\right )^2}+\frac {50 \left (65-2 e^2\right ) x \log (x)}{\left (10-e^2\right )^3}+\frac {\left (25-e^2\right )^2 \log (x)}{\left (10-e^2\right )^2 x}-\frac {30 \left (25-e^2\right )^2 \log (x)}{\left (10-e^2\right )^3 x}+\frac {450 \left (25-e^2\right ) \log (x)}{\left (10-e^2\right )^3 x}+\frac {225 \log (x)}{\left (10-e^2\right )^2 x}+\frac {450000 \log (x)}{\left (250-35 e^2+e^4\right )^2}+\frac {900000 \log (x)}{\left (10-e^2\right )^3 \left (25-e^2\right )}+\frac {450 e^4 \left (35-2 e^2\right ) \log (x)}{\left (10-e^2\right )^3 \left (25-e^2\right )^2}-\frac {4500 e^2 \left (250-13 e^2\right ) \log (x)}{\left (10-e^2\right )^3 \left (25-e^2\right )^2}-\frac {4218750 \log (x)}{\left (10-e^2\right )^3 \left (25-e^2\right )^2}+\frac {450 e^4 \log (x)}{\left (10-e^2\right ) \left (25-e^2\right )^3}-\frac {2250 e^2 \left (175-13 e^2\right ) \log (x)}{\left (10-e^2\right )^2 \left (25-e^2\right )^3}-\frac {4218750 \log (x)}{\left (10-e^2\right )^2 \left (25-e^2\right )^3}-\frac {30 e^2 \log (x)}{\left (10-e^2\right )^2}+\frac {750 \log (x)}{\left (10-e^2\right )^2}-\frac {30 e^4 \log (x)}{\left (10-e^2\right )^3}+\frac {2400 e^2 \log (x)}{\left (10-e^2\right )^3}-\frac {41250 \log (x)}{\left (10-e^2\right )^3}+\frac {50 \left (55-e^2\right ) x}{\left (10-e^2\right )^3}-\frac {150 \left (25-e^2\right ) x}{\left (10-e^2\right )^3}+\frac {200 x}{\left (10-e^2\right )^2}-\frac {50 \left (65-2 e^2\right ) x}{\left (10-e^2\right )^3}+\frac {2250 x}{\left (10-e^2\right )^3}-\frac {10 e^2 \log (x+3)}{\left (10-e^2\right )^2}-\frac {40 \log (x+3)}{3 \left (10-e^2\right )^2}-\frac {10 e^4 \log (x+3)}{\left (10-e^2\right )^3}+\frac {260 e^2 \log (x+3)}{3 \left (10-e^2\right )^3}+\frac {400 \log (x+3)}{3 \left (10-e^2\right )^3}-\frac {10 e^2 \left (23125-4575 e^2+300 e^4-4 e^6\right ) \log \left (5 x-e^2+25\right )}{\left (10-e^2\right )^2 \left (25-e^2\right )^3}+\frac {20 e^2 \left (10625-3625 e^2+347 e^4-12 e^6\right ) \log \left (5 x-e^2+25\right )}{3 \left (10-e^2\right )^3 \left (25-e^2\right )^2}+\frac {10 e^4 \left (925-110 e^2+4 e^4\right ) \log \left (5 x-e^2+25\right )}{\left (10-e^2\right )^3 \left (25-e^2\right )^2}-\frac {450000 \log \left (5 x-e^2+25\right )}{\left (250-35 e^2+e^4\right )^2}+\frac {30 \left (25-e^2\right )^2 \log \left (5 x-e^2+25\right )}{\left (10-e^2\right )^3}-\frac {30 \left (25-e^2\right ) \log \left (5 x-e^2+25\right )}{\left (10-e^2\right )^2}-\frac {7120 \left (25-e^2\right ) \log \left (5 x-e^2+25\right )}{3 \left (10-e^2\right )^3}-\frac {900000 \log \left (5 x-e^2+25\right )}{\left (10-e^2\right )^3 \left (25-e^2\right )}+\frac {4218750 \log \left (5 x-e^2+25\right )}{\left (10-e^2\right )^3 \left (25-e^2\right )^2}-\frac {450 e^4 \log \left (5 x-e^2+25\right )}{\left (10-e^2\right ) \left (25-e^2\right )^3}+\frac {4218750 \log \left (5 x-e^2+25\right )}{\left (10-e^2\right )^2 \left (25-e^2\right )^3}+\frac {3560 \log \left (5 x-e^2+25\right )}{3 \left (10-e^2\right )^2}+\frac {209900 \log \left (5 x-e^2+25\right )}{3 \left (10-e^2\right )^3}-\frac {675 e^4}{\left (250-35 e^2+e^4\right )^2 x}+\frac {33750 e^2}{\left (250-35 e^2+e^4\right )^2 x}-\frac {421875}{\left (250-35 e^2+e^4\right )^2 x}+\frac {\left (25-e^2\right )^2}{\left (10-e^2\right )^2 x}-\frac {30 \left (25-e^2\right )^2}{\left (10-e^2\right )^3 x}+\frac {450 \left (25-e^2\right )}{\left (10-e^2\right )^3 x}-\frac {1350 e^4}{\left (10-e^2\right )^3 \left (25-e^2\right ) x}+\frac {67500 e^2}{\left (10-e^2\right )^3 \left (25-e^2\right ) x}-\frac {843750}{\left (10-e^2\right )^3 \left (25-e^2\right ) x}-\frac {3 e^4}{\left (10-e^2\right )^2 x}+\frac {150 e^2}{\left (10-e^2\right )^2 x}-\frac {1650}{\left (10-e^2\right )^2 x}+\frac {90 e^4}{\left (10-e^2\right )^3 x}-\frac {4500 e^2}{\left (10-e^2\right )^3 x}+\frac {56250}{\left (10-e^2\right )^3 x}-\frac {e^4}{\left (10-e^2\right )^2 (x+3)}+\frac {26 e^2}{3 \left (10-e^2\right )^2 (x+3)}+\frac {40}{3 \left (10-e^2\right )^2 (x+3)}-\frac {10 e^2 \left (10625-3625 e^2+347 e^4-12 e^6\right )}{3 \left (250-35 e^2+e^4\right )^2 \left (5 x-e^2+25\right )}-\frac {5 e^4 \left (925-110 e^2+4 e^4\right )}{\left (250-35 e^2+e^4\right )^2 \left (5 x-e^2+25\right )}-\frac {2109375}{\left (250-35 e^2+e^4\right )^2 \left (5 x-e^2+25\right )}-\frac {15 \left (25-e^2\right )^2}{\left (10-e^2\right )^2 \left (5 x-e^2+25\right )}+\frac {3560 \left (25-e^2\right )}{3 \left (10-e^2\right )^2 \left (5 x-e^2+25\right )}+\frac {450000}{\left (10-e^2\right )^2 \left (25-e^2\right ) \left (5 x-e^2+25\right )}-\frac {104950}{3 \left (10-e^2\right )^2 \left (5 x-e^2+25\right )}\)

Input:

Int[(50625 + 54000*x + 20990*x^2 + 3560*x^3 + 225*x^4 + E^4*(81 + 54*x + 1 
2*x^2) + E^2*(-4050 - 3510*x - 1106*x^2 - 120*x^3) + (-16875 - 18000*x - 7 
050*x^2 - 1200*x^3 - 75*x^4 + E^4*(-27 - 18*x - 3*x^2) + E^2*(1350 + 1170* 
x + 330*x^2 + 30*x^3))*Log[x])/(16875*x^2 + 18000*x^3 + 7050*x^4 + 1200*x^ 
5 + 75*x^6 + E^4*(27*x^2 + 18*x^3 + 3*x^4) + E^2*(-1350*x^2 - 1170*x^3 - 3 
30*x^4 - 30*x^5)),x]
 

Output:

56250/((10 - E^2)^3*x) - (4500*E^2)/((10 - E^2)^3*x) + (90*E^4)/((10 - E^2 
)^3*x) - 1650/((10 - E^2)^2*x) + (150*E^2)/((10 - E^2)^2*x) - (3*E^4)/((10 
 - E^2)^2*x) - 843750/((10 - E^2)^3*(25 - E^2)*x) + (67500*E^2)/((10 - E^2 
)^3*(25 - E^2)*x) - (1350*E^4)/((10 - E^2)^3*(25 - E^2)*x) + (450*(25 - E^ 
2))/((10 - E^2)^3*x) - (30*(25 - E^2)^2)/((10 - E^2)^3*x) + (25 - E^2)^2/( 
(10 - E^2)^2*x) - 421875/((250 - 35*E^2 + E^4)^2*x) + (33750*E^2)/((250 - 
35*E^2 + E^4)^2*x) - (675*E^4)/((250 - 35*E^2 + E^4)^2*x) + (2250*x)/(10 - 
 E^2)^3 - (50*(65 - 2*E^2)*x)/(10 - E^2)^3 + (200*x)/(10 - E^2)^2 - (150*( 
25 - E^2)*x)/(10 - E^2)^3 + (50*(55 - E^2)*x)/(10 - E^2)^3 + 40/(3*(10 - E 
^2)^2*(3 + x)) + (26*E^2)/(3*(10 - E^2)^2*(3 + x)) - E^4/((10 - E^2)^2*(3 
+ x)) - 104950/(3*(10 - E^2)^2*(25 - E^2 + 5*x)) + 450000/((10 - E^2)^2*(2 
5 - E^2)*(25 - E^2 + 5*x)) + (3560*(25 - E^2))/(3*(10 - E^2)^2*(25 - E^2 + 
 5*x)) - (15*(25 - E^2)^2)/((10 - E^2)^2*(25 - E^2 + 5*x)) - 2109375/((250 
 - 35*E^2 + E^4)^2*(25 - E^2 + 5*x)) - (5*E^4*(925 - 110*E^2 + 4*E^4))/((2 
50 - 35*E^2 + E^4)^2*(25 - E^2 + 5*x)) - (10*E^2*(10625 - 3625*E^2 + 347*E 
^4 - 12*E^6))/(3*(250 - 35*E^2 + E^4)^2*(25 - E^2 + 5*x)) - (41250*Log[x]) 
/(10 - E^2)^3 + (2400*E^2*Log[x])/(10 - E^2)^3 - (30*E^4*Log[x])/(10 - E^2 
)^3 + (750*Log[x])/(10 - E^2)^2 - (30*E^2*Log[x])/(10 - E^2)^2 - (4218750* 
Log[x])/((10 - E^2)^2*(25 - E^2)^3) - (2250*E^2*(175 - 13*E^2)*Log[x])/((1 
0 - E^2)^2*(25 - E^2)^3) + (450*E^4*Log[x])/((10 - E^2)*(25 - E^2)^3) -...
 

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 2463

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] && Gt 
Q[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 
0]
 

Maple [A] (verified)

Time = 3.32 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.37

method	result	size
default	\(\frac {\ln \left (x \right )}{x}+\frac {1}{x}+\frac {15 x^{2}+\left (-4 \,{\mathrm e}^{2}+\frac {356}{3}\right ) x -9 \,{\mathrm e}^{2}+225}{x \left (3+x \right ) \left ({\mathrm e}^{2}-5 x -25\right )}\)	\(48\)
parts	\(\frac {\ln \left (x \right )}{x}+\frac {1}{x}+\frac {15 x^{2}+\left (-4 \,{\mathrm e}^{2}+\frac {356}{3}\right ) x -9 \,{\mathrm e}^{2}+225}{x \left (3+x \right ) \left ({\mathrm e}^{2}-5 x -25\right )}\)	\(48\)
risch	\(\frac {\ln \left (x \right )}{x}-\frac {9 \,{\mathrm e}^{2} x -30 x^{2}+18 \,{\mathrm e}^{2}-236 x -450}{3 \left ({\mathrm e}^{2} x -5 x^{2}+3 \,{\mathrm e}^{2}-40 x -75\right ) x}\)	\(52\)
norman	\(\frac {\left (3 \,{\mathrm e}^{2}-75\right ) \ln \left (x \right )+10 x^{2}+\left (-3 \,{\mathrm e}^{2}+\frac {236}{3}\right ) x -5 x^{2} \ln \left (x \right )+\left ({\mathrm e}^{2}-40\right ) x \ln \left (x \right )+150-6 \,{\mathrm e}^{2}}{x \left (3+x \right ) \left ({\mathrm e}^{2}-5 x -25\right )}\)	\(62\)
parallelrisch	\(\frac {2250+1180 x +45 \,{\mathrm e}^{2} \ln \left (x \right )-600 x \ln \left (x \right )-75 x^{2} \ln \left (x \right )-45 \,{\mathrm e}^{2} x -90 \,{\mathrm e}^{2}-1125 \ln \left (x \right )+150 x^{2}+15 x \,{\mathrm e}^{2} \ln \left (x \right )}{15 x \left ({\mathrm e}^{2} x -5 x^{2}+3 \,{\mathrm e}^{2}-40 x -75\right )}\)	\(74\)

Input:

int((((-3*x^2-18*x-27)*exp(2)^2+(30*x^3+330*x^2+1170*x+1350)*exp(2)-75*x^4 
-1200*x^3-7050*x^2-18000*x-16875)*ln(x)+(12*x^2+54*x+81)*exp(2)^2+(-120*x^ 
3-1106*x^2-3510*x-4050)*exp(2)+225*x^4+3560*x^3+20990*x^2+54000*x+50625)/( 
(3*x^4+18*x^3+27*x^2)*exp(2)^2+(-30*x^5-330*x^4-1170*x^3-1350*x^2)*exp(2)+ 
75*x^6+1200*x^5+7050*x^4+18000*x^3+16875*x^2),x,method=_RETURNVERBOSE)
 

Output:

ln(x)/x+1/x+(15*x^2+(-4*exp(2)+356/3)*x-9*exp(2)+225)/x/(3+x)/(exp(2)-5*x- 
25)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.91 \[ \int \frac {50625+54000 x+20990 x^2+3560 x^3+225 x^4+e^4 \left (81+54 x+12 x^2\right )+e^2 \left (-4050-3510 x-1106 x^2-120 x^3\right )+\left (-16875-18000 x-7050 x^2-1200 x^3-75 x^4+e^4 \left (-27-18 x-3 x^2\right )+e^2 \left (1350+1170 x+330 x^2+30 x^3\right )\right ) \log (x)}{16875 x^2+18000 x^3+7050 x^4+1200 x^5+75 x^6+e^4 \left (27 x^2+18 x^3+3 x^4\right )+e^2 \left (-1350 x^2-1170 x^3-330 x^4-30 x^5\right )} \, dx=-\frac {30 \, x^{2} - 9 \, {\left (x + 2\right )} e^{2} - 3 \, {\left (5 \, x^{2} - {\left (x + 3\right )} e^{2} + 40 \, x + 75\right )} \log \left (x\right ) + 236 \, x + 450}{3 \, {\left (5 \, x^{3} + 40 \, x^{2} - {\left (x^{2} + 3 \, x\right )} e^{2} + 75 \, x\right )}} \] Input:

integrate((((-3*x^2-18*x-27)*exp(2)^2+(30*x^3+330*x^2+1170*x+1350)*exp(2)- 
75*x^4-1200*x^3-7050*x^2-18000*x-16875)*log(x)+(12*x^2+54*x+81)*exp(2)^2+( 
-120*x^3-1106*x^2-3510*x-4050)*exp(2)+225*x^4+3560*x^3+20990*x^2+54000*x+5 
0625)/((3*x^4+18*x^3+27*x^2)*exp(2)^2+(-30*x^5-330*x^4-1170*x^3-1350*x^2)* 
exp(2)+75*x^6+1200*x^5+7050*x^4+18000*x^3+16875*x^2),x, algorithm="fricas" 
)
 

Output:

-1/3*(30*x^2 - 9*(x + 2)*e^2 - 3*(5*x^2 - (x + 3)*e^2 + 40*x + 75)*log(x) 
+ 236*x + 450)/(5*x^3 + 40*x^2 - (x^2 + 3*x)*e^2 + 75*x)
 

Sympy [A] (verification not implemented)

Time = 1.67 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.37 \[ \int \frac {50625+54000 x+20990 x^2+3560 x^3+225 x^4+e^4 \left (81+54 x+12 x^2\right )+e^2 \left (-4050-3510 x-1106 x^2-120 x^3\right )+\left (-16875-18000 x-7050 x^2-1200 x^3-75 x^4+e^4 \left (-27-18 x-3 x^2\right )+e^2 \left (1350+1170 x+330 x^2+30 x^3\right )\right ) \log (x)}{16875 x^2+18000 x^3+7050 x^4+1200 x^5+75 x^6+e^4 \left (27 x^2+18 x^3+3 x^4\right )+e^2 \left (-1350 x^2-1170 x^3-330 x^4-30 x^5\right )} \, dx=\frac {- 30 x^{2} + x \left (-236 + 9 e^{2}\right ) - 450 + 18 e^{2}}{15 x^{3} + x^{2} \cdot \left (120 - 3 e^{2}\right ) + x \left (225 - 9 e^{2}\right )} + \frac {\log {\left (x \right )}}{x} \] Input:

integrate((((-3*x**2-18*x-27)*exp(2)**2+(30*x**3+330*x**2+1170*x+1350)*exp 
(2)-75*x**4-1200*x**3-7050*x**2-18000*x-16875)*ln(x)+(12*x**2+54*x+81)*exp 
(2)**2+(-120*x**3-1106*x**2-3510*x-4050)*exp(2)+225*x**4+3560*x**3+20990*x 
**2+54000*x+50625)/((3*x**4+18*x**3+27*x**2)*exp(2)**2+(-30*x**5-330*x**4- 
1170*x**3-1350*x**2)*exp(2)+75*x**6+1200*x**5+7050*x**4+18000*x**3+16875*x 
**2),x)
 

Output:

(-30*x**2 + x*(-236 + 9*exp(2)) - 450 + 18*exp(2))/(15*x**3 + x**2*(120 - 
3*exp(2)) + x*(225 - 9*exp(2))) + log(x)/x
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1762 vs. \(2 (35) = 70\).

Time = 0.16 (sec) , antiderivative size = 1762, normalized size of antiderivative = 50.34 \[ \int \frac {50625+54000 x+20990 x^2+3560 x^3+225 x^4+e^4 \left (81+54 x+12 x^2\right )+e^2 \left (-4050-3510 x-1106 x^2-120 x^3\right )+\left (-16875-18000 x-7050 x^2-1200 x^3-75 x^4+e^4 \left (-27-18 x-3 x^2\right )+e^2 \left (1350+1170 x+330 x^2+30 x^3\right )\right ) \log (x)}{16875 x^2+18000 x^3+7050 x^4+1200 x^5+75 x^6+e^4 \left (27 x^2+18 x^3+3 x^4\right )+e^2 \left (-1350 x^2-1170 x^3-330 x^4-30 x^5\right )} \, dx=\text {Too large to display} \] Input:

integrate((((-3*x^2-18*x-27)*exp(2)^2+(30*x^3+330*x^2+1170*x+1350)*exp(2)- 
75*x^4-1200*x^3-7050*x^2-18000*x-16875)*log(x)+(12*x^2+54*x+81)*exp(2)^2+( 
-120*x^3-1106*x^2-3510*x-4050)*exp(2)+225*x^4+3560*x^3+20990*x^2+54000*x+5 
0625)/((3*x^4+18*x^3+27*x^2)*exp(2)^2+(-30*x^5-330*x^4-1170*x^3-1350*x^2)* 
exp(2)+75*x^6+1200*x^5+7050*x^4+18000*x^3+16875*x^2),x, algorithm="maxima" 
)
 

Output:

-(6750*(2*e^2 - 35)*log(5*x - e^2 + 25)/(e^12 - 105*e^10 + 4425*e^8 - 9537 
5*e^6 + 1106250*e^4 - 6562500*e^2 + 15625000) - 2*(e^2 + 5)*log(x + 3)/(e^ 
6 - 30*e^4 + 300*e^2 - 1000) + 2*(e^2 - 40)*log(x)/(e^6 - 75*e^4 + 1875*e^ 
2 - 15625) + 3*(10*x^2*(e^4 - 35*e^2 + 475) - x*(2*e^6 - 135*e^4 + 2775*e^ 
2 - 23000) - 3*e^6 + 135*e^4 - 1800*e^2 + 7500)/(5*x^3*(e^8 - 70*e^6 + 172 
5*e^4 - 17500*e^2 + 62500) - x^2*(e^10 - 110*e^8 + 4525*e^6 - 86500*e^4 + 
762500*e^2 - 2500000) - 3*x*(e^10 - 95*e^8 + 3475*e^6 - 60625*e^4 + 500000 
*e^2 - 1562500)))*e^4 - 2*(675*(e^2 - 20)*log(5*x - e^2 + 25)/(e^10 - 80*e 
^8 + 2425*e^6 - 34750*e^4 + 237500*e^2 - 625000) + (e^2 + 20)*log(x + 3)/( 
e^6 - 30*e^4 + 300*e^2 - 1000) - 3*(5*x*(e^2 - 40) - e^4 + 50*e^2 - 850)/( 
5*x^2*(e^6 - 45*e^4 + 600*e^2 - 2500) - x*(e^8 - 85*e^6 + 2400*e^4 - 26500 
*e^2 + 100000) - 3*e^8 + 210*e^6 - 5175*e^4 + 52500*e^2 - 187500) - log(x) 
/(e^4 - 50*e^2 + 625))*e^4 - 4*((10*x - e^2 + 40)/(5*x^2*(e^4 - 20*e^2 + 1 
00) - x*(e^6 - 60*e^4 + 900*e^2 - 4000) - 3*e^6 + 135*e^4 - 1800*e^2 + 750 
0) + 10*log(5*x - e^2 + 25)/(e^6 - 30*e^4 + 300*e^2 - 1000) - 10*log(x + 3 
)/(e^6 - 30*e^4 + 300*e^2 - 1000))*e^4 + 50*(6750*(2*e^2 - 35)*log(5*x - e 
^2 + 25)/(e^12 - 105*e^10 + 4425*e^8 - 95375*e^6 + 1106250*e^4 - 6562500*e 
^2 + 15625000) - 2*(e^2 + 5)*log(x + 3)/(e^6 - 30*e^4 + 300*e^2 - 1000) + 
2*(e^2 - 40)*log(x)/(e^6 - 75*e^4 + 1875*e^2 - 15625) + 3*(10*x^2*(e^4 - 3 
5*e^2 + 475) - x*(2*e^6 - 135*e^4 + 2775*e^2 - 23000) - 3*e^6 + 135*e^4...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (35) = 70\).

Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.23 \[ \int \frac {50625+54000 x+20990 x^2+3560 x^3+225 x^4+e^4 \left (81+54 x+12 x^2\right )+e^2 \left (-4050-3510 x-1106 x^2-120 x^3\right )+\left (-16875-18000 x-7050 x^2-1200 x^3-75 x^4+e^4 \left (-27-18 x-3 x^2\right )+e^2 \left (1350+1170 x+330 x^2+30 x^3\right )\right ) \log (x)}{16875 x^2+18000 x^3+7050 x^4+1200 x^5+75 x^6+e^4 \left (27 x^2+18 x^3+3 x^4\right )+e^2 \left (-1350 x^2-1170 x^3-330 x^4-30 x^5\right )} \, dx=\frac {15 \, x^{2} \log \left (x\right ) - 3 \, x e^{2} \log \left (x\right ) - 30 \, x^{2} + 9 \, x e^{2} + 120 \, x \log \left (x\right ) - 9 \, e^{2} \log \left (x\right ) - 236 \, x + 18 \, e^{2} + 225 \, \log \left (x\right ) - 450}{3 \, {\left (5 \, x^{3} - x^{2} e^{2} + 40 \, x^{2} - 3 \, x e^{2} + 75 \, x\right )}} \] Input:

integrate((((-3*x^2-18*x-27)*exp(2)^2+(30*x^3+330*x^2+1170*x+1350)*exp(2)- 
75*x^4-1200*x^3-7050*x^2-18000*x-16875)*log(x)+(12*x^2+54*x+81)*exp(2)^2+( 
-120*x^3-1106*x^2-3510*x-4050)*exp(2)+225*x^4+3560*x^3+20990*x^2+54000*x+5 
0625)/((3*x^4+18*x^3+27*x^2)*exp(2)^2+(-30*x^5-330*x^4-1170*x^3-1350*x^2)* 
exp(2)+75*x^6+1200*x^5+7050*x^4+18000*x^3+16875*x^2),x, algorithm="giac")
 

Output:

1/3*(15*x^2*log(x) - 3*x*e^2*log(x) - 30*x^2 + 9*x*e^2 + 120*x*log(x) - 9* 
e^2*log(x) - 236*x + 18*e^2 + 225*log(x) - 450)/(5*x^3 - x^2*e^2 + 40*x^2 
- 3*x*e^2 + 75*x)
 

Mupad [B] (verification not implemented)

Time = 1.85 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \frac {50625+54000 x+20990 x^2+3560 x^3+225 x^4+e^4 \left (81+54 x+12 x^2\right )+e^2 \left (-4050-3510 x-1106 x^2-120 x^3\right )+\left (-16875-18000 x-7050 x^2-1200 x^3-75 x^4+e^4 \left (-27-18 x-3 x^2\right )+e^2 \left (1350+1170 x+330 x^2+30 x^3\right )\right ) \log (x)}{16875 x^2+18000 x^3+7050 x^4+1200 x^5+75 x^6+e^4 \left (27 x^2+18 x^3+3 x^4\right )+e^2 \left (-1350 x^2-1170 x^3-330 x^4-30 x^5\right )} \, dx=\frac {\ln \left (x\right )}{x}-\frac {-30\,x^2+\left (9\,{\mathrm {e}}^2-236\right )\,x+18\,{\mathrm {e}}^2-450}{-15\,x^3+\left (3\,{\mathrm {e}}^2-120\right )\,x^2+\left (9\,{\mathrm {e}}^2-225\right )\,x} \] Input:

int((54000*x + exp(4)*(54*x + 12*x^2 + 81) - exp(2)*(3510*x + 1106*x^2 + 1 
20*x^3 + 4050) - log(x)*(18000*x + exp(4)*(18*x + 3*x^2 + 27) - exp(2)*(11 
70*x + 330*x^2 + 30*x^3 + 1350) + 7050*x^2 + 1200*x^3 + 75*x^4 + 16875) + 
20990*x^2 + 3560*x^3 + 225*x^4 + 50625)/(exp(4)*(27*x^2 + 18*x^3 + 3*x^4) 
+ 16875*x^2 + 18000*x^3 + 7050*x^4 + 1200*x^5 + 75*x^6 - exp(2)*(1350*x^2 
+ 1170*x^3 + 330*x^4 + 30*x^5)),x)
 

Output:

log(x)/x - (18*exp(2) - 30*x^2 + x*(9*exp(2) - 236) - 450)/(x^2*(3*exp(2) 
- 120) - 15*x^3 + x*(9*exp(2) - 225))
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 4.20 \[ \int \frac {50625+54000 x+20990 x^2+3560 x^3+225 x^4+e^4 \left (81+54 x+12 x^2\right )+e^2 \left (-4050-3510 x-1106 x^2-120 x^3\right )+\left (-16875-18000 x-7050 x^2-1200 x^3-75 x^4+e^4 \left (-27-18 x-3 x^2\right )+e^2 \left (1350+1170 x+330 x^2+30 x^3\right )\right ) \log (x)}{16875 x^2+18000 x^3+7050 x^4+1200 x^5+75 x^6+e^4 \left (27 x^2+18 x^3+3 x^4\right )+e^2 \left (-1350 x^2-1170 x^3-330 x^4-30 x^5\right )} \, dx=\frac {3 \,\mathrm {log}\left (x \right ) e^{4} x +9 \,\mathrm {log}\left (x \right ) e^{4}-15 \,\mathrm {log}\left (x \right ) e^{2} x^{2}-240 \,\mathrm {log}\left (x \right ) e^{2} x -585 \,\mathrm {log}\left (x \right ) e^{2}+600 \,\mathrm {log}\left (x \right ) x^{2}+4800 \,\mathrm {log}\left (x \right ) x +9000 \,\mathrm {log}\left (x \right )-9 e^{4} x -18 e^{4}-15 e^{2} x^{2}+461 e^{2} x +1170 e^{2}+225 x^{3}+600 x^{2}-6065 x -18000}{3 x \left (e^{4} x +3 e^{4}-5 e^{2} x^{2}-80 e^{2} x -195 e^{2}+200 x^{2}+1600 x +3000\right )} \] Input:

int((((-3*x^2-18*x-27)*exp(2)^2+(30*x^3+330*x^2+1170*x+1350)*exp(2)-75*x^4 
-1200*x^3-7050*x^2-18000*x-16875)*log(x)+(12*x^2+54*x+81)*exp(2)^2+(-120*x 
^3-1106*x^2-3510*x-4050)*exp(2)+225*x^4+3560*x^3+20990*x^2+54000*x+50625)/ 
((3*x^4+18*x^3+27*x^2)*exp(2)^2+(-30*x^5-330*x^4-1170*x^3-1350*x^2)*exp(2) 
+75*x^6+1200*x^5+7050*x^4+18000*x^3+16875*x^2),x)
 

Output:

(3*log(x)*e**4*x + 9*log(x)*e**4 - 15*log(x)*e**2*x**2 - 240*log(x)*e**2*x 
 - 585*log(x)*e**2 + 600*log(x)*x**2 + 4800*log(x)*x + 9000*log(x) - 9*e** 
4*x - 18*e**4 - 15*e**2*x**2 + 461*e**2*x + 1170*e**2 + 225*x**3 + 600*x** 
2 - 6065*x - 18000)/(3*x*(e**4*x + 3*e**4 - 5*e**2*x**2 - 80*e**2*x - 195* 
e**2 + 200*x**2 + 1600*x + 3000))