\(\int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x))+e^{-\frac {25 x}{-21+25 e^4-25 x}} (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x))}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 (-9450 x^3-11250 x^4)} \, dx\) [866]

Optimal result

Integrand size = 150, antiderivative size = 27 \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=\frac {\left (3+e^{\frac {x}{\frac {21}{25}-e^4+x}}\right )^2}{9 x^2} \] Output:

1/9*(exp(x/(21/25+x-exp(4)))+3)^2/x^2
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=\frac {\left (3+e^{\frac {25 x}{21-25 e^4+25 x}}\right )^2}{9 x^2} \] Input:

Integrate[(-7938 - 11250*E^8 - 18900*x - 11250*x^2 + E^4*(18900 + 22500*x) 
 + (-882 - 1250*E^8 - 1050*x - 1250*x^2 + E^4*(2100 + 1250*x))/E^((50*x)/( 
-21 + 25*E^4 - 25*x)) + (-5292 - 7500*E^8 - 9450*x - 7500*x^2 + E^4*(12600 
 + 11250*x))/E^((25*x)/(-21 + 25*E^4 - 25*x)))/(3969*x^3 + 5625*E^8*x^3 + 
9450*x^4 + 5625*x^5 + E^4*(-9450*x^3 - 11250*x^4)),x]
 

Output:

(3 + E^((25*x)/(21 - 25*E^4 + 25*x)))^2/(9*x^2)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(426\) vs. \(2(27)=54\).

Time = 8.11 (sec) , antiderivative size = 426, normalized size of antiderivative = 15.78, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {6, 2026, 7277, 27, 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.

\(\displaystyle \int \frac {-11250 x^2+e^{-\frac {50 x}{-25 x+25 e^4-21}} \left (-1250 x^2-1050 x+e^4 (1250 x+2100)-1250 e^8-882\right )+e^{-\frac {25 x}{-25 x+25 e^4-21}} \left (-7500 x^2-9450 x+e^4 (11250 x+12600)-7500 e^8-5292\right )-18900 x+e^4 (22500 x+18900)-11250 e^8-7938}{5625 x^5+9450 x^4+5625 e^8 x^3+3969 x^3+e^4 \left (-11250 x^4-9450 x^3\right )} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-11250 x^2+e^{-\frac {50 x}{-25 x+25 e^4-21}} \left (-1250 x^2-1050 x+e^4 (1250 x+2100)-1250 e^8-882\right )+e^{-\frac {25 x}{-25 x+25 e^4-21}} \left (-7500 x^2-9450 x+e^4 (11250 x+12600)-7500 e^8-5292\right )-18900 x+e^4 (22500 x+18900)-11250 e^8-7938}{5625 x^5+9450 x^4+\left (3969+5625 e^8\right ) x^3+e^4 \left (-11250 x^4-9450 x^3\right )}dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {-11250 x^2+e^{-\frac {50 x}{-25 x+25 e^4-21}} \left (-1250 x^2-1050 x+e^4 (1250 x+2100)-1250 e^8-882\right )+e^{-\frac {25 x}{-25 x+25 e^4-21}} \left (-7500 x^2-9450 x+e^4 (11250 x+12600)-7500 e^8-5292\right )-18900 x+e^4 (22500 x+18900)-11250 e^8-7938}{x^3 \left (5625 x^2+450 \left (21-25 e^4\right ) x+9 \left (21-25 e^4\right )^2\right )}dx\)

\(\Big \downarrow \) 7277

\(\displaystyle 22500 \int -\frac {5625 x^2+9450 x-450 e^4 (25 x+21)+3 e^{\frac {25 x}{25 x-25 e^4+21}} \left (1250 x^2+1575 x-75 e^4 (25 x+28)+2 \left (441+625 e^8\right )\right )+e^{\frac {50 x}{25 x-25 e^4+21}} \left (625 x^2+525 x-25 e^4 (25 x+42)+625 e^8+441\right )+9 \left (441+625 e^8\right )}{101250 x^3 \left (25 x-25 e^4+21\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{9} \int \frac {5625 x^2+9450 x-450 e^4 (25 x+21)+3 e^{\frac {25 x}{25 x-25 e^4+21}} \left (1250 x^2+1575 x-75 e^4 (25 x+28)+2 \left (441+625 e^8\right )\right )+e^{\frac {50 x}{25 x-25 e^4+21}} \left (625 x^2+525 x-25 e^4 (25 x+42)+625 e^8+441\right )+9 \left (441+625 e^8\right )}{x^3 \left (25 x-25 e^4+21\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {2}{9} \int \left (-\frac {450 e^4 (25 x+21)}{\left (-25 x+25 e^4-21\right )^2 x^3}+\frac {e^{\frac {50 x}{25 x-25 e^4+21}} \left (625 x^2+25 \left (21-25 e^4\right ) x+\left (21-25 e^4\right )^2\right )}{x^3 \left (25 x-25 e^4+21\right )^2}+\frac {3 e^{\frac {25 x}{25 x-25 e^4+21}} \left (1250 x^2+75 \left (21-25 e^4\right ) x+2 \left (21-25 e^4\right )^2\right )}{x^3 \left (25 x-25 e^4+21\right )^2}+\frac {5625}{x \left (25 x-25 e^4+21\right )^2}+\frac {9450}{x^2 \left (25 x-25 e^4+21\right )^2}+\frac {9 \left (441+625 e^8\right )}{\left (-25 x+25 e^4-21\right )^2 x^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2}{9} \left (-\frac {3 e^{\frac {25 x}{25 x-25 e^4+21}}}{x^2}-\frac {e^{\frac {50 x}{25 x-25 e^4+21}}}{2 x^2}-\frac {9 \left (441+625 e^8\right )}{2 \left (21-25 e^4\right )^2 x^2}+\frac {4725 e^4}{\left (21-25 e^4\right )^2 x^2}+\frac {450 \left (441+625 e^8\right )}{\left (21-25 e^4\right )^3 x}-\frac {11250 e^4 \left (21+25 e^4\right )}{\left (21-25 e^4\right )^3 x}-\frac {9450}{\left (21-25 e^4\right )^2 x}+\frac {5625 \left (441+625 e^8\right )}{\left (21-25 e^4\right )^3 \left (25 x-25 e^4+21\right )}+\frac {5625}{\left (21-25 e^4\right ) \left (25 x-25 e^4+21\right )}-\frac {236250}{\left (21-25 e^4\right )^2 \left (25 x-25 e^4+21\right )}-\frac {7031250 e^8}{\left (21-25 e^4\right )^3 \left (25 x-25 e^4+21\right )}+\frac {16875 \left (441+625 e^8\right ) \log (x)}{\left (21-25 e^4\right )^4}-\frac {281250 e^4 \left (21+50 e^4\right ) \log (x)}{\left (21-25 e^4\right )^4}+\frac {5625 \log (x)}{\left (21-25 e^4\right )^2}-\frac {472500 \log (x)}{\left (21-25 e^4\right )^3}-\frac {16875 \left (441+625 e^8\right ) \log \left (25 x-25 e^4+21\right )}{\left (21-25 e^4\right )^4}+\frac {281250 e^4 \left (21+50 e^4\right ) \log \left (25 x-25 e^4+21\right )}{\left (21-25 e^4\right )^4}-\frac {5625 \log \left (25 x-25 e^4+21\right )}{\left (21-25 e^4\right )^2}+\frac {472500 \log \left (25 x-25 e^4+21\right )}{\left (21-25 e^4\right )^3}\right )\)

Input:

Int[(-7938 - 11250*E^8 - 18900*x - 11250*x^2 + E^4*(18900 + 22500*x) + (-8 
82 - 1250*E^8 - 1050*x - 1250*x^2 + E^4*(2100 + 1250*x))/E^((50*x)/(-21 + 
25*E^4 - 25*x)) + (-5292 - 7500*E^8 - 9450*x - 7500*x^2 + E^4*(12600 + 112 
50*x))/E^((25*x)/(-21 + 25*E^4 - 25*x)))/(3969*x^3 + 5625*E^8*x^3 + 9450*x 
^4 + 5625*x^5 + E^4*(-9450*x^3 - 11250*x^4)),x]
 

Output:

(-2*((-3*E^((25*x)/(21 - 25*E^4 + 25*x)))/x^2 - E^((50*x)/(21 - 25*E^4 + 2 
5*x))/(2*x^2) + (4725*E^4)/((21 - 25*E^4)^2*x^2) - (9*(441 + 625*E^8))/(2* 
(21 - 25*E^4)^2*x^2) - 9450/((21 - 25*E^4)^2*x) - (11250*E^4*(21 + 25*E^4) 
)/((21 - 25*E^4)^3*x) + (450*(441 + 625*E^8))/((21 - 25*E^4)^3*x) - (70312 
50*E^8)/((21 - 25*E^4)^3*(21 - 25*E^4 + 25*x)) - 236250/((21 - 25*E^4)^2*( 
21 - 25*E^4 + 25*x)) + 5625/((21 - 25*E^4)*(21 - 25*E^4 + 25*x)) + (5625*( 
441 + 625*E^8))/((21 - 25*E^4)^3*(21 - 25*E^4 + 25*x)) - (472500*Log[x])/( 
21 - 25*E^4)^3 + (5625*Log[x])/(21 - 25*E^4)^2 - (281250*E^4*(21 + 50*E^4) 
*Log[x])/(21 - 25*E^4)^4 + (16875*(441 + 625*E^8)*Log[x])/(21 - 25*E^4)^4 
+ (472500*Log[21 - 25*E^4 + 25*x])/(21 - 25*E^4)^3 - (5625*Log[21 - 25*E^4 
 + 25*x])/(21 - 25*E^4)^2 + (281250*E^4*(21 + 50*E^4)*Log[21 - 25*E^4 + 25 
*x])/(21 - 25*E^4)^4 - (16875*(441 + 625*E^8)*Log[21 - 25*E^4 + 25*x])/(21 
 - 25*E^4)^4))/9
 

Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, 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 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 7277

Int[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> 
 Simp[1/(4^p*c^p)   Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} 
, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] &&  !AlgebraicFu 
nctionQ[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. \(264\) vs. \(2(21)=42\).

Time = 0.03 (sec) , antiderivative size = 265, normalized size of antiderivative = 9.81

\[-\frac {\left (25 \,{\mathrm e}^{4}-21\right ) \left (-\frac {13781250 \left (-\frac {1}{1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}}+\frac {1}{2 \left (1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}\right )^{2}}\right )}{9765625 \,{\mathrm e}^{20}-41015625 \,{\mathrm e}^{16}[phone] \,{\mathrm e}^{12}-57881250 \,{\mathrm e}^{8}[phone] \,{\mathrm e}^{4}-4084101}+\frac {32812500 \,{\mathrm e}^{4} \left (-\frac {1}{1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}}+\frac {1}{2 \left (1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}\right )^{2}}\right )}{9765625 \,{\mathrm e}^{20}-41015625 \,{\mathrm e}^{16}[phone] \,{\mathrm e}^{12}-57881250 \,{\mathrm e}^{8}[phone] \,{\mathrm e}^{4}-4084101}-\frac {19531250 \,{\mathrm e}^{8} \left (-\frac {1}{1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}}+\frac {1}{2 \left (1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}\right )^{2}}\right )}{9765625 \,{\mathrm e}^{20}-41015625 \,{\mathrm e}^{16}[phone] \,{\mathrm e}^{12}-57881250 \,{\mathrm e}^{8}[phone] \,{\mathrm e}^{4}-4084101}\right )}{25}\]

Input:

int(((-1250*exp(4)^2+(1250*x+2100)*exp(4)-1250*x^2-1050*x-882)*exp(-25*x/( 
25*exp(4)-25*x-21))^2+(-7500*exp(4)^2+(11250*x+12600)*exp(4)-7500*x^2-9450 
*x-5292)*exp(-25*x/(25*exp(4)-25*x-21))-11250*exp(4)^2+(22500*x+18900)*exp 
(4)-11250*x^2-18900*x-7938)/(5625*x^3*exp(4)^2+(-11250*x^4-9450*x^3)*exp(4 
)+5625*x^5+9450*x^4+3969*x^3),x)
 

Output:

-1/25*(25*exp(4)-21)*(-13781250/(9765625*exp(4)^5-41015625*exp(4)^4+689062 
50*exp(4)^3-57881250*exp(4)^2+24310125*exp(4)-4084101)*(-1/(1+(25*exp(4)-2 
1)/(-25*exp(4)+25*x+21))+1/2/(1+(25*exp(4)-21)/(-25*exp(4)+25*x+21))^2)+32 
812500*exp(4)/(9765625*exp(4)^5-41015625*exp(4)^4+68906250*exp(4)^3-578812 
50*exp(4)^2+24310125*exp(4)-4084101)*(-1/(1+(25*exp(4)-21)/(-25*exp(4)+25* 
x+21))+1/2/(1+(25*exp(4)-21)/(-25*exp(4)+25*x+21))^2)-19531250*exp(4)^2/(9 
765625*exp(4)^5-41015625*exp(4)^4+68906250*exp(4)^3-57881250*exp(4)^2+2431 
0125*exp(4)-4084101)*(-1/(1+(25*exp(4)-21)/(-25*exp(4)+25*x+21))+1/2/(1+(2 
5*exp(4)-21)/(-25*exp(4)+25*x+21))^2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=\frac {e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21}\right )} + 6 \, e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21}\right )} + 9}{9 \, x^{2}} \] Input:

integrate(((-1250*exp(4)^2+(1250*x+2100)*exp(4)-1250*x^2-1050*x-882)*exp(- 
25*x/(25*exp(4)-25*x-21))^2+(-7500*exp(4)^2+(11250*x+12600)*exp(4)-7500*x^ 
2-9450*x-5292)*exp(-25*x/(25*exp(4)-25*x-21))-11250*exp(4)^2+(22500*x+1890 
0)*exp(4)-11250*x^2-18900*x-7938)/(5625*x^3*exp(4)^2+(-11250*x^4-9450*x^3) 
*exp(4)+5625*x^5+9450*x^4+3969*x^3),x, algorithm="fricas")
 

Output:

1/9*(e^(50*x/(25*x - 25*e^4 + 21)) + 6*e^(25*x/(25*x - 25*e^4 + 21)) + 9)/ 
x^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (19) = 38\).

Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=\frac {1}{x^{2}} + \frac {18 x^{2} e^{- \frac {25 x}{- 25 x - 21 + 25 e^{4}}} + 3 x^{2} e^{- \frac {50 x}{- 25 x - 21 + 25 e^{4}}}}{27 x^{4}} \] Input:

integrate(((-1250*exp(4)**2+(1250*x+2100)*exp(4)-1250*x**2-1050*x-882)*exp 
(-25*x/(25*exp(4)-25*x-21))**2+(-7500*exp(4)**2+(11250*x+12600)*exp(4)-750 
0*x**2-9450*x-5292)*exp(-25*x/(25*exp(4)-25*x-21))-11250*exp(4)**2+(22500* 
x+18900)*exp(4)-11250*x**2-18900*x-7938)/(5625*x**3*exp(4)**2+(-11250*x**4 
-9450*x**3)*exp(4)+5625*x**5+9450*x**4+3969*x**3),x)
 

Output:

x**(-2) + (18*x**2*exp(-25*x/(-25*x - 21 + 25*exp(4))) + 3*x**2*exp(-50*x/ 
(-25*x - 21 + 25*exp(4))))/(27*x**4)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 716 vs. \(2 (24) = 48\).

Time = 0.22 (sec) , antiderivative size = 716, normalized size of antiderivative = 26.52 \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=\text {Too large to display} \] Input:

integrate(((-1250*exp(4)^2+(1250*x+2100)*exp(4)-1250*x^2-1050*x-882)*exp(- 
25*x/(25*exp(4)-25*x-21))^2+(-7500*exp(4)^2+(11250*x+12600)*exp(4)-7500*x^ 
2-9450*x-5292)*exp(-25*x/(25*exp(4)-25*x-21))-11250*exp(4)^2+(22500*x+1890 
0)*exp(4)-11250*x^2-18900*x-7938)/(5625*x^3*exp(4)^2+(-11250*x^4-9450*x^3) 
*exp(4)+5625*x^5+9450*x^4+3969*x^3),x, algorithm="maxima")
 

Output:

625*((3750*x^2 - 75*x*(25*e^4 - 21) - 625*e^8 + 1050*e^4 - 441)/(25*x^3*(1 
5625*e^12 - 39375*e^8 + 33075*e^4 - 9261) - x^2*(390625*e^16 - 1312500*e^1 
2 + 1653750*e^8 - 926100*e^4 + 194481)) + 3750*log(25*x - 25*e^4 + 21)/(39 
0625*e^16 - 1312500*e^12 + 1653750*e^8 - 926100*e^4 + 194481) - 3750*log(x 
)/(390625*e^16 - 1312500*e^12 + 1653750*e^8 - 926100*e^4 + 194481))*e^8 - 
1050*((3750*x^2 - 75*x*(25*e^4 - 21) - 625*e^8 + 1050*e^4 - 441)/(25*x^3*( 
15625*e^12 - 39375*e^8 + 33075*e^4 - 9261) - x^2*(390625*e^16 - 1312500*e^ 
12 + 1653750*e^8 - 926100*e^4 + 194481)) + 3750*log(25*x - 25*e^4 + 21)/(3 
90625*e^16 - 1312500*e^12 + 1653750*e^8 - 926100*e^4 + 194481) - 3750*log( 
x)/(390625*e^16 - 1312500*e^12 + 1653750*e^8 - 926100*e^4 + 194481))*e^4 - 
 2500*((50*x - 25*e^4 + 21)/(25*x^2*(625*e^8 - 1050*e^4 + 441) - x*(15625* 
e^12 - 39375*e^8 + 33075*e^4 - 9261)) + 50*log(25*x - 25*e^4 + 21)/(15625* 
e^12 - 39375*e^8 + 33075*e^4 - 9261) - 50*log(x)/(15625*e^12 - 39375*e^8 + 
 33075*e^4 - 9261))*e^4 + 441*(3750*x^2 - 75*x*(25*e^4 - 21) - 625*e^8 + 1 
050*e^4 - 441)/(25*x^3*(15625*e^12 - 39375*e^8 + 33075*e^4 - 9261) - x^2*( 
390625*e^16 - 1312500*e^12 + 1653750*e^8 - 926100*e^4 + 194481)) + 2100*(5 
0*x - 25*e^4 + 21)/(25*x^2*(625*e^8 - 1050*e^4 + 441) - x*(15625*e^12 - 39 
375*e^8 + 33075*e^4 - 9261)) + 1/9*(e^(50*e^4/(25*x - 25*e^4 + 21) + 2) + 
6*e^(25*e^4/(25*x - 25*e^4 + 21) + 21/(25*x - 25*e^4 + 21) + 1))*e^(-42/(2 
5*x - 25*e^4 + 21))/x^2 + 1653750*log(25*x - 25*e^4 + 21)/(390625*e^16 ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (24) = 48\).

Time = 2.65 (sec) , antiderivative size = 217, normalized size of antiderivative = 8.04 \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=-\frac {\frac {50 \, x e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21}\right )}}{25 \, x - 25 \, e^{4} + 21} - \frac {625 \, x^{2} e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21}\right )}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} + \frac {300 \, x e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21}\right )}}{25 \, x - 25 \, e^{4} + 21} - \frac {3750 \, x^{2} e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21}\right )}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} + \frac {450 \, x}{25 \, x - 25 \, e^{4} + 21} - e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21}\right )} - 6 \, e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21}\right )} - 9}{9 \, {\left (\frac {25 \, x^{2} e^{4}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} - \frac {21 \, x^{2}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}}\right )} {\left (25 \, e^{4} - 21\right )}} \] Input:

integrate(((-1250*exp(4)^2+(1250*x+2100)*exp(4)-1250*x^2-1050*x-882)*exp(- 
25*x/(25*exp(4)-25*x-21))^2+(-7500*exp(4)^2+(11250*x+12600)*exp(4)-7500*x^ 
2-9450*x-5292)*exp(-25*x/(25*exp(4)-25*x-21))-11250*exp(4)^2+(22500*x+1890 
0)*exp(4)-11250*x^2-18900*x-7938)/(5625*x^3*exp(4)^2+(-11250*x^4-9450*x^3) 
*exp(4)+5625*x^5+9450*x^4+3969*x^3),x, algorithm="giac")
 

Output:

-1/9*(50*x*e^(50*x/(25*x - 25*e^4 + 21))/(25*x - 25*e^4 + 21) - 625*x^2*e^ 
(50*x/(25*x - 25*e^4 + 21))/(25*x - 25*e^4 + 21)^2 + 300*x*e^(25*x/(25*x - 
 25*e^4 + 21))/(25*x - 25*e^4 + 21) - 3750*x^2*e^(25*x/(25*x - 25*e^4 + 21 
))/(25*x - 25*e^4 + 21)^2 + 450*x/(25*x - 25*e^4 + 21) - e^(50*x/(25*x - 2 
5*e^4 + 21)) - 6*e^(25*x/(25*x - 25*e^4 + 21)) - 9)/((25*x^2*e^4/(25*x - 2 
5*e^4 + 21)^2 - 21*x^2/(25*x - 25*e^4 + 21)^2)*(25*e^4 - 21))
 

Mupad [B] (verification not implemented)

Time = 3.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=\frac {{\left ({\mathrm {e}}^{\frac {25\,x}{25\,x-25\,{\mathrm {e}}^4+21}}+3\right )}^2}{9\,x^2} \] Input:

int(-(18900*x + 11250*exp(8) + exp((50*x)/(25*x - 25*exp(4) + 21))*(1050*x 
 + 1250*exp(8) + 1250*x^2 - exp(4)*(1250*x + 2100) + 882) + exp((25*x)/(25 
*x - 25*exp(4) + 21))*(9450*x + 7500*exp(8) + 7500*x^2 - exp(4)*(11250*x + 
 12600) + 5292) + 11250*x^2 - exp(4)*(22500*x + 18900) + 7938)/(5625*x^3*e 
xp(8) - exp(4)*(9450*x^3 + 11250*x^4) + 3969*x^3 + 9450*x^4 + 5625*x^5),x)
 

Output:

(exp((25*x)/(25*x - 25*exp(4) + 21)) + 3)^2/(9*x^2)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.37 \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=\frac {9 e^{\frac {50 x}{25 e^{4}-25 x -21}}+6 e^{\frac {25 x}{25 e^{4}-25 x -21}}+1}{9 e^{\frac {50 x}{25 e^{4}-25 x -21}} x^{2}} \] Input:

int(((-1250*exp(4)^2+(1250*x+2100)*exp(4)-1250*x^2-1050*x-882)*exp(-25*x/( 
25*exp(4)-25*x-21))^2+(-7500*exp(4)^2+(11250*x+12600)*exp(4)-7500*x^2-9450 
*x-5292)*exp(-25*x/(25*exp(4)-25*x-21))-11250*exp(4)^2+(22500*x+18900)*exp 
(4)-11250*x^2-18900*x-7938)/(5625*x^3*exp(4)^2+(-11250*x^4-9450*x^3)*exp(4 
)+5625*x^5+9450*x^4+3969*x^3),x)
 

Output:

(9*e**((50*x)/(25*e**4 - 25*x - 21)) + 6*e**((25*x)/(25*e**4 - 25*x - 21)) 
 + 1)/(9*e**((50*x)/(25*e**4 - 25*x - 21))*x**2)