\(\int \frac {14049280-5 e^x-48168960 x+55050240 x^2-20971520 x^3}{377801998336+e^{2 x}-3454189699072 x+13816758796288 x^2-31581162962944 x^3+45115947089920 x^4-41248865910784 x^5+23570780520448 x^6-7696581394432 x^7+1099511627776 x^8+e^x (1229312-5619712 x+9633792 x^2-7340032 x^3+2097152 x^4)} \, dx\) [63]

Optimal result

Integrand size = 92, antiderivative size = 21 \[ \int \frac {14049280-5 e^x-48168960 x+55050240 x^2-20971520 x^3}{377801998336+e^{2 x}-3454189699072 x+13816758796288 x^2-31581162962944 x^3+45115947089920 x^4-41248865910784 x^5+23570780520448 x^6-7696581394432 x^7+1099511627776 x^8+e^x \left (1229312-5619712 x+9633792 x^2-7340032 x^3+2097152 x^4\right )} \, dx=\frac {5}{e^x+16 (-2+4 (4-4 x))^4} \] Output:

5/(exp(x)+4*(14-16*x)^2*(28-32*x)^2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {14049280-5 e^x-48168960 x+55050240 x^2-20971520 x^3}{377801998336+e^{2 x}-3454189699072 x+13816758796288 x^2-31581162962944 x^3+45115947089920 x^4-41248865910784 x^5+23570780520448 x^6-7696581394432 x^7+1099511627776 x^8+e^x \left (1229312-5619712 x+9633792 x^2-7340032 x^3+2097152 x^4\right )} \, dx=\frac {5}{e^x+256 (7-8 x)^4} \] Input:

Integrate[(14049280 - 5*E^x - 48168960*x + 55050240*x^2 - 20971520*x^3)/(3 
77801998336 + E^(2*x) - 3454189699072*x + 13816758796288*x^2 - 31581162962 
944*x^3 + 45115947089920*x^4 - 41248865910784*x^5 + 23570780520448*x^6 - 7 
696581394432*x^7 + 1099511627776*x^8 + E^x*(1229312 - 5619712*x + 9633792* 
x^2 - 7340032*x^3 + 2097152*x^4)),x]
 

Output:

5/(E^x + 256*(7 - 8*x)^4)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {7239, 27, 25, 7237}

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.

\(\displaystyle \int \frac {-20971520 x^3+55050240 x^2-48168960 x-5 e^x+14049280}{1099511627776 x^8-7696581394432 x^7+23570780520448 x^6-41248865910784 x^5+45115947089920 x^4-31581162962944 x^3+13816758796288 x^2+e^x \left (2097152 x^4-7340032 x^3+9633792 x^2-5619712 x+1229312\right )-3454189699072 x+e^{2 x}[phone]} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {5 \left (-8192 (8 x-7)^3-e^x\right )}{\left (256 (7-8 x)^4+e^x\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 5 \int -\frac {e^x-8192 (7-8 x)^3}{\left (256 (7-8 x)^4+e^x\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -5 \int \frac {e^x-8192 (7-8 x)^3}{\left (256 (7-8 x)^4+e^x\right )^2}dx\)

\(\Big \downarrow \) 7237

\(\displaystyle \frac {5}{256 (7-8 x)^4+e^x}\)

Input:

Int[(14049280 - 5*E^x - 48168960*x + 55050240*x^2 - 20971520*x^3)/(3778019 
98336 + E^(2*x) - 3454189699072*x + 13816758796288*x^2 - 31581162962944*x^ 
3 + 45115947089920*x^4 - 41248865910784*x^5 + 23570780520448*x^6 - 7696581 
394432*x^7 + 1099511627776*x^8 + E^x*(1229312 - 5619712*x + 9633792*x^2 - 
7340032*x^3 + 2097152*x^4)),x]
 

Output:

5/(E^x + 256*(7 - 8*x)^4)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 7237

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si 
mp[q*(y^(m + 1)/(m + 1)), x] /;  !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29

method	result	size
norman	\(\frac {5}{1048576 x^{4}-3670016 x^{3}+4816896 x^{2}+{\mathrm e}^{x}-2809856 x +614656}\)	\(27\)
risch	\(\frac {5}{1048576 x^{4}-3670016 x^{3}+4816896 x^{2}+{\mathrm e}^{x}-2809856 x +614656}\)	\(27\)
parallelrisch	\(\frac {5}{1048576 x^{4}-3670016 x^{3}+4816896 x^{2}+{\mathrm e}^{x}-2809856 x +614656}\)	\(27\)

Input:

int((-5*exp(x)-20971520*x^3+55050240*x^2-48168960*x[phone])/(exp(x)^2+(2 
097152*x^4-7340032*x^3+9633792*x^2-5619712*x+1229312)*exp(x)[phone] 
*x^8-7696581394432*x^7+23570780520448*x^6-41248865910784*x^5+4511594708992 
0*x^4-31581162962944*x^3+13816758796288*x^2-3454189699072*x[phone]), 
x,method=_RETURNVERBOSE)
 

Output:

5/(1048576*x^4-3670016*x^3+4816896*x^2+exp(x)-2809856*x+614656)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {14049280-5 e^x-48168960 x+55050240 x^2-20971520 x^3}{377801998336+e^{2 x}-3454189699072 x+13816758796288 x^2-31581162962944 x^3+45115947089920 x^4-41248865910784 x^5+23570780520448 x^6-7696581394432 x^7+1099511627776 x^8+e^x \left (1229312-5619712 x+9633792 x^2-7340032 x^3+2097152 x^4\right )} \, dx=\frac {5}{1048576 \, x^{4} - 3670016 \, x^{3} + 4816896 \, x^{2} - 2809856 \, x + e^{x} + 614656} \] Input:

integrate((-5*exp(x)-20971520*x^3+55050240*x^2-48168960*x[phone])/(exp(x 
)^2+(2097152*x^4-7340032*x^3+9633792*x^2-5619712*x+1229312)*exp(x)+1099511 
627776*x^8-7696581394432*x^7+23570780520448*x^6-41248865910784*x^5+4511594 
7089920*x^4-31581162962944*x^3+13816758796288*x^2-3454189699072*x[phone] 
8336),x, algorithm="fricas")
 

Output:

5/(1048576*x^4 - 3670016*x^3 + 4816896*x^2 - 2809856*x + e^x + 614656)
 

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {14049280-5 e^x-48168960 x+55050240 x^2-20971520 x^3}{377801998336+e^{2 x}-3454189699072 x+13816758796288 x^2-31581162962944 x^3+45115947089920 x^4-41248865910784 x^5+23570780520448 x^6-7696581394432 x^7+1099511627776 x^8+e^x \left (1229312-5619712 x+9633792 x^2-7340032 x^3+2097152 x^4\right )} \, dx=\frac {5}{1048576 x^{4} - 3670016 x^{3} + 4816896 x^{2} - 2809856 x + e^{x} + 614656} \] Input:

integrate((-5*exp(x)-20971520*x**3+55050240*x**2-48168960*x[phone])/(exp 
(x)**2+(2097152*x**4-7340032*x**3+9633792*x**2-5619712*x+1229312)*exp(x)+1 
099511627776*x**8-7696581394432*x**7+23570780520448*x**6-41248865910784*x* 
*5+45115947089920*x**4-31581162962944*x**3+13816758796288*x**2-34541896990 
72*x[phone]),x)
 

Output:

5/(1048576*x**4 - 3670016*x**3 + 4816896*x**2 - 2809856*x + exp(x) + 61465 
6)
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {14049280-5 e^x-48168960 x+55050240 x^2-20971520 x^3}{377801998336+e^{2 x}-3454189699072 x+13816758796288 x^2-31581162962944 x^3+45115947089920 x^4-41248865910784 x^5+23570780520448 x^6-7696581394432 x^7+1099511627776 x^8+e^x \left (1229312-5619712 x+9633792 x^2-7340032 x^3+2097152 x^4\right )} \, dx=\frac {5}{1048576 \, x^{4} - 3670016 \, x^{3} + 4816896 \, x^{2} - 2809856 \, x + e^{x} + 614656} \] Input:

integrate((-5*exp(x)-20971520*x^3+55050240*x^2-48168960*x[phone])/(exp(x 
)^2+(2097152*x^4-7340032*x^3+9633792*x^2-5619712*x+1229312)*exp(x)+1099511 
627776*x^8-7696581394432*x^7+23570780520448*x^6-41248865910784*x^5+4511594 
7089920*x^4-31581162962944*x^3+13816758796288*x^2-3454189699072*x[phone] 
8336),x, algorithm="maxima")
 

Output:

5/(1048576*x^4 - 3670016*x^3 + 4816896*x^2 - 2809856*x + e^x + 614656)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {14049280-5 e^x-48168960 x+55050240 x^2-20971520 x^3}{377801998336+e^{2 x}-3454189699072 x+13816758796288 x^2-31581162962944 x^3+45115947089920 x^4-41248865910784 x^5+23570780520448 x^6-7696581394432 x^7+1099511627776 x^8+e^x \left (1229312-5619712 x+9633792 x^2-7340032 x^3+2097152 x^4\right )} \, dx=\frac {5}{1048576 \, x^{4} - 3670016 \, x^{3} + 4816896 \, x^{2} - 2809856 \, x + e^{x} + 614656} \] Input:

integrate((-5*exp(x)-20971520*x^3+55050240*x^2-48168960*x[phone])/(exp(x 
)^2+(2097152*x^4-7340032*x^3+9633792*x^2-5619712*x+1229312)*exp(x)+1099511 
627776*x^8-7696581394432*x^7+23570780520448*x^6-41248865910784*x^5+4511594 
7089920*x^4-31581162962944*x^3+13816758796288*x^2-3454189699072*x[phone] 
8336),x, algorithm="giac")
 

Output:

5/(1048576*x^4 - 3670016*x^3 + 4816896*x^2 - 2809856*x + e^x + 614656)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {14049280-5 e^x-48168960 x+55050240 x^2-20971520 x^3}{377801998336+e^{2 x}-3454189699072 x+13816758796288 x^2-31581162962944 x^3+45115947089920 x^4-41248865910784 x^5+23570780520448 x^6-7696581394432 x^7+1099511627776 x^8+e^x \left (1229312-5619712 x+9633792 x^2-7340032 x^3+2097152 x^4\right )} \, dx=\int -\frac {48168960\,x+5\,{\mathrm {e}}^x-55050240\,x^2+20971520\,x^3-14049280}{{\mathrm {e}}^{2\,x}-3454189699072\,x+{\mathrm {e}}^x\,\left (2097152\,x^4-7340032\,x^3+9633792\,x^2-5619712\,x+1229312\right )[phone]\,x^2-31581162962944\,x^3+45115947089920\,x^4-41248865910784\,x^5+23570780520448\,x^6-7696581394432\,x^7+1099511627776\,x^8+377801998336} \,d x \] Input:

int(-(48168960*x + 5*exp(x) - 55050240*x^2 + 20971520*x^3 - 14049280)/(exp 
(2*x) - 3454189699072*x + exp(x)*(9633792*x^2 - 5619712*x - 7340032*x^3 + 
2097152*x^4 + 1229312) + 13816758796288*x^2 - 31581162962944*x^3 + 4511594 
7089920*x^4 - 41248865910784*x^5 + 23570780520448*x^6 - 7696581394432*x^7 
+ 1099511627776*x^8 + 377801998336),x)
 

Output:

int(-(48168960*x + 5*exp(x) - 55050240*x^2 + 20971520*x^3 - 14049280)/(exp 
(2*x) - 3454189699072*x + exp(x)*(9633792*x^2 - 5619712*x - 7340032*x^3 + 
2097152*x^4 + 1229312) + 13816758796288*x^2 - 31581162962944*x^3 + 4511594 
7089920*x^4 - 41248865910784*x^5 + 23570780520448*x^6 - 7696581394432*x^7 
+ 1099511627776*x^8 + 377801998336), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {14049280-5 e^x-48168960 x+55050240 x^2-20971520 x^3}{377801998336+e^{2 x}-3454189699072 x+13816758796288 x^2-31581162962944 x^3+45115947089920 x^4-41248865910784 x^5+23570780520448 x^6-7696581394432 x^7+1099511627776 x^8+e^x \left (1229312-5619712 x+9633792 x^2-7340032 x^3+2097152 x^4\right )} \, dx=\frac {5}{e^{x}+1048576 x^{4}-3670016 x^{3}+4816896 x^{2}-2809856 x +614656} \] Input:

int((-5*exp(x)-20971520*x^3+55050240*x^2-48168960*x[phone])/(exp(x)^2+(2 
097152*x^4-7340032*x^3+9633792*x^2-5619712*x+1229312)*exp(x)[phone] 
*x^8-7696581394432*x^7+23570780520448*x^6-41248865910784*x^5+4511594708992 
0*x^4-31581162962944*x^3+13816758796288*x^2-3454189699072*x[phone]), 
x)
 

Output:

5/(e**x + 1048576*x**4 - 3670016*x**3 + 4816896*x**2 - 2809856*x + 614656)