\(\int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 (x^3+2 x^4+x^5)+e^{2 e^5} (x^3+2 x^4+x^5)+e^2 (-50 x^3-98 x^4-46 x^5+2 x^6)+e^{e^5} (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 (2 x^3+4 x^4+2 x^5))} \, dx\) [244]

Optimal result

Integrand size = 165, antiderivative size = 24 \[ \int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx=\frac {1}{x \left (-25+e^2+e^{e^5}+x\right ) \left (x+x^2\right )} \] Output:

1/x/(exp(2)+exp(exp(5))+x-25)/(x^2+x)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx=\frac {1}{x^2 (1+x) \left (-25+e^2+e^{e^5}+x\right )} \] Input:

Integrate[(50 + E^2*(-2 - 3*x) + E^E^5*(-2 - 3*x) + 72*x - 4*x^2)/(625*x^3 
 + 1200*x^4 + 526*x^5 - 48*x^6 + x^7 + E^4*(x^3 + 2*x^4 + x^5) + E^(2*E^5) 
*(x^3 + 2*x^4 + x^5) + E^2*(-50*x^3 - 98*x^4 - 46*x^5 + 2*x^6) + E^E^5*(-5 
0*x^3 - 98*x^4 - 46*x^5 + 2*x^6 + E^2*(2*x^3 + 4*x^4 + 2*x^5))),x]
 

Output:

1/(x^2*(1 + x)*(-25 + E^2 + E^E^5 + x))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(132\) vs. \(2(24)=48\).

Time = 0.74 (sec) , antiderivative size = 132, normalized size of antiderivative = 5.50, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6, 6, 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.

\(\displaystyle \int \frac {-4 x^2+72 x+e^{e^5} (-3 x-2)+e^2 (-3 x-2)+50}{x^7-48 x^6+526 x^5+1200 x^4+625 x^3+e^{2 e^5} \left (x^5+2 x^4+x^3\right )+e^4 \left (x^5+2 x^4+x^3\right )+e^2 \left (2 x^6-46 x^5-98 x^4-50 x^3\right )+e^{e^5} \left (2 x^6-46 x^5-98 x^4-50 x^3+e^2 \left (2 x^5+4 x^4+2 x^3\right )\right )} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-4 x^2+72 x+\left (e^2+e^{e^5}\right ) (-3 x-2)+50}{x^7-48 x^6+526 x^5+1200 x^4+625 x^3+e^{2 e^5} \left (x^5+2 x^4+x^3\right )+e^4 \left (x^5+2 x^4+x^3\right )+e^2 \left (2 x^6-46 x^5-98 x^4-50 x^3\right )+e^{e^5} \left (2 x^6-46 x^5-98 x^4-50 x^3+e^2 \left (2 x^5+4 x^4+2 x^3\right )\right )}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-4 x^2+72 x+\left (e^2+e^{e^5}\right ) (-3 x-2)+50}{x^7-48 x^6+526 x^5+1200 x^4+625 x^3+\left (e^4+e^{2 e^5}\right ) \left (x^5+2 x^4+x^3\right )+e^2 \left (2 x^6-46 x^5-98 x^4-50 x^3\right )+e^{e^5} \left (2 x^6-46 x^5-98 x^4-50 x^3+e^2 \left (2 x^5+4 x^4+2 x^3\right )\right )}dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {-4 x^2+72 x+\left (e^2+e^{e^5}\right ) (-3 x-2)+50}{x^3 \left (x^4-2 \left (24-e^2-e^{e^5}\right ) x^3+\left (526-46 e^2+e^4-46 e^{e^5}+e^{2 e^5}+2 e^{2+e^5}\right ) x^2+2 \left (24-e^2-e^{e^5}\right ) \left (25-e^2-e^{e^5}\right ) x+\left (-25+e^2+e^{e^5}\right )^2\right )}dx\)

\(\Big \downarrow \) 2462

\(\displaystyle \int \left (-\frac {2}{\left (-25+e^2+e^{e^5}\right ) x^3}+\frac {-24+e^2+e^{e^5}}{\left (-25+e^2+e^{e^5}\right )^2 x^2}-\frac {1}{\left (-26+e^2+e^{e^5}\right ) (x+1)^2}+\frac {1}{\left (-26+e^2+e^{e^5}\right ) \left (-25+e^2+e^{e^5}\right )^2 \left (x+e^{e^5}+e^2-25\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{\left (25-e^2-e^{e^5}\right ) x^2}+\frac {24-e^2-e^{e^5}}{\left (25-e^2-e^{e^5}\right )^2 x}-\frac {1}{\left (26-e^2-e^{e^5}\right ) (x+1)}-\frac {1}{\left (25-e^2-e^{e^5}\right )^2 \left (26-e^2-e^{e^5}\right ) \left (-x-e^{e^5}-e^2+25\right )}\)

Input:

Int[(50 + E^2*(-2 - 3*x) + E^E^5*(-2 - 3*x) + 72*x - 4*x^2)/(625*x^3 + 120 
0*x^4 + 526*x^5 - 48*x^6 + x^7 + E^4*(x^3 + 2*x^4 + x^5) + E^(2*E^5)*(x^3 
+ 2*x^4 + x^5) + E^2*(-50*x^3 - 98*x^4 - 46*x^5 + 2*x^6) + E^E^5*(-50*x^3 
- 98*x^4 - 46*x^5 + 2*x^6 + E^2*(2*x^3 + 4*x^4 + 2*x^5))),x]
 

Output:

-(1/((25 - E^2 - E^E^5)^2*(26 - E^2 - E^E^5)*(25 - E^2 - E^E^5 - x))) - 1/ 
((25 - E^2 - E^E^5)*x^2) + (24 - E^2 - E^E^5)/((25 - E^2 - E^E^5)^2*x) - 1 
/((26 - E^2 - E^E^5)*(1 + x))
 

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 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 [A] (verified)

Time = 9.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

method	result	size
norman	\(\frac {1}{x^{2} \left (1+x \right ) \left ({\mathrm e}^{2}+{\mathrm e}^{{\mathrm e}^{5}}+x -25\right )}\)	\(20\)
gosper	\(\frac {1}{x^{2} \left ({\mathrm e}^{{\mathrm e}^{5}} x +{\mathrm e}^{2} x +x^{2}+{\mathrm e}^{{\mathrm e}^{5}}+{\mathrm e}^{2}-24 x -25\right )}\)	\(29\)
risch	\(\frac {1}{x^{2} \left ({\mathrm e}^{{\mathrm e}^{5}} x +{\mathrm e}^{2} x +x^{2}+{\mathrm e}^{{\mathrm e}^{5}}+{\mathrm e}^{2}-24 x -25\right )}\)	\(29\)
parallelrisch	\(\frac {1}{x^{2} \left ({\mathrm e}^{{\mathrm e}^{5}} x +{\mathrm e}^{2} x +x^{2}+{\mathrm e}^{{\mathrm e}^{5}}+{\mathrm e}^{2}-24 x -25\right )}\)	\(29\)

Input:

int(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x^2+72*x+50)/((x^5+2*x^4+x^3)* 
exp(exp(5))^2+((2*x^5+4*x^4+2*x^3)*exp(2)+2*x^6-46*x^5-98*x^4-50*x^3)*exp( 
exp(5))+(x^5+2*x^4+x^3)*exp(2)^2+(2*x^6-46*x^5-98*x^4-50*x^3)*exp(2)+x^7-4 
8*x^6+526*x^5+1200*x^4+625*x^3),x,method=_RETURNVERBOSE)
 

Output:

1/x^2/(1+x)/(exp(2)+exp(exp(5))+x-25)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx=\frac {1}{x^{4} - 24 \, x^{3} - 25 \, x^{2} + {\left (x^{3} + x^{2}\right )} e^{2} + {\left (x^{3} + x^{2}\right )} e^{\left (e^{5}\right )}} \] Input:

integrate(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x^2+72*x+50)/((x^5+2*x^4 
+x^3)*exp(exp(5))^2+((2*x^5+4*x^4+2*x^3)*exp(2)+2*x^6-46*x^5-98*x^4-50*x^3 
)*exp(exp(5))+(x^5+2*x^4+x^3)*exp(2)^2+(2*x^6-46*x^5-98*x^4-50*x^3)*exp(2) 
+x^7-48*x^6+526*x^5+1200*x^4+625*x^3),x, algorithm="fricas")
 

Output:

1/(x^4 - 24*x^3 - 25*x^2 + (x^3 + x^2)*e^2 + (x^3 + x^2)*e^(e^5))
 

Sympy [A] (verification not implemented)

Time = 3.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx=\frac {1}{x^{4} + x^{3} \left (-24 + e^{2} + e^{e^{5}}\right ) + x^{2} \left (-25 + e^{2} + e^{e^{5}}\right )} \] Input:

integrate(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x**2+72*x+50)/((x**5+2*x 
**4+x**3)*exp(exp(5))**2+((2*x**5+4*x**4+2*x**3)*exp(2)+2*x**6-46*x**5-98* 
x**4-50*x**3)*exp(exp(5))+(x**5+2*x**4+x**3)*exp(2)**2+(2*x**6-46*x**5-98* 
x**4-50*x**3)*exp(2)+x**7-48*x**6+526*x**5+1200*x**4+625*x**3),x)
 

Output:

1/(x**4 + x**3*(-24 + exp(2) + exp(exp(5))) + x**2*(-25 + exp(2) + exp(exp 
(5))))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx=\frac {1}{x^{4} + x^{3} {\left (e^{2} + e^{\left (e^{5}\right )} - 24\right )} + x^{2} {\left (e^{2} + e^{\left (e^{5}\right )} - 25\right )}} \] Input:

integrate(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x^2+72*x+50)/((x^5+2*x^4 
+x^3)*exp(exp(5))^2+((2*x^5+4*x^4+2*x^3)*exp(2)+2*x^6-46*x^5-98*x^4-50*x^3 
)*exp(exp(5))+(x^5+2*x^4+x^3)*exp(2)^2+(2*x^6-46*x^5-98*x^4-50*x^3)*exp(2) 
+x^7-48*x^6+526*x^5+1200*x^4+625*x^3),x, algorithm="maxima")
 

Output:

1/(x^4 + x^3*(e^2 + e^(e^5) - 24) + x^2*(e^2 + e^(e^5) - 25))
 

Giac [F]

\[ \int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx=\int { -\frac {4 \, x^{2} + {\left (3 \, x + 2\right )} e^{2} + {\left (3 \, x + 2\right )} e^{\left (e^{5}\right )} - 72 \, x - 50}{x^{7} - 48 \, x^{6} + 526 \, x^{5} + 1200 \, x^{4} + 625 \, x^{3} + {\left (x^{5} + 2 \, x^{4} + x^{3}\right )} e^{4} + 2 \, {\left (x^{6} - 23 \, x^{5} - 49 \, x^{4} - 25 \, x^{3}\right )} e^{2} + {\left (x^{5} + 2 \, x^{4} + x^{3}\right )} e^{\left (2 \, e^{5}\right )} + 2 \, {\left (x^{6} - 23 \, x^{5} - 49 \, x^{4} - 25 \, x^{3} + {\left (x^{5} + 2 \, x^{4} + x^{3}\right )} e^{2}\right )} e^{\left (e^{5}\right )}} \,d x } \] Input:

integrate(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x^2+72*x+50)/((x^5+2*x^4 
+x^3)*exp(exp(5))^2+((2*x^5+4*x^4+2*x^3)*exp(2)+2*x^6-46*x^5-98*x^4-50*x^3 
)*exp(exp(5))+(x^5+2*x^4+x^3)*exp(2)^2+(2*x^6-46*x^5-98*x^4-50*x^3)*exp(2) 
+x^7-48*x^6+526*x^5+1200*x^4+625*x^3),x, algorithm="giac")
 

Output:

undef
 

Mupad [F(-1)]

Timed out. \[ \int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx=\text {Hanged} \] Input:

int(-(exp(exp(5))*(3*x + 2) - 72*x + 4*x^2 + exp(2)*(3*x + 2) - 50)/(exp(4 
)*(x^3 + 2*x^4 + x^5) - exp(exp(5))*(50*x^3 - exp(2)*(2*x^3 + 4*x^4 + 2*x^ 
5) + 98*x^4 + 46*x^5 - 2*x^6) + exp(2*exp(5))*(x^3 + 2*x^4 + x^5) + 625*x^ 
3 + 1200*x^4 + 526*x^5 - 48*x^6 + x^7 - exp(2)*(50*x^3 + 98*x^4 + 46*x^5 - 
 2*x^6)),x)
 

Output:

\text{Hanged}
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx=\frac {1}{x^{2} \left (e^{e^{5}} x +e^{e^{5}}+e^{2} x +e^{2}+x^{2}-24 x -25\right )} \] Input:

int(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x^2+72*x+50)/((x^5+2*x^4+x^3)* 
exp(exp(5))^2+((2*x^5+4*x^4+2*x^3)*exp(2)+2*x^6-46*x^5-98*x^4-50*x^3)*exp( 
exp(5))+(x^5+2*x^4+x^3)*exp(2)^2+(2*x^6-46*x^5-98*x^4-50*x^3)*exp(2)+x^7-4 
8*x^6+526*x^5+1200*x^4+625*x^3),x)
 

Output:

1/(x**2*(e**(e**5)*x + e**(e**5) + e**2*x + e**2 + x**2 - 24*x - 25))