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

Optimal. Leaf size=21 \[ \frac {5}{e^x+16 (-2+4 (4-4 x))^4} \]

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Rubi [A]  time = 0.30, antiderivative size = 17, normalized size of antiderivative = 0.81, number of steps used = 3, number of rules used = 3, integrand size = 92, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6688, 12, 6686} \begin {gather*} \frac {5}{256 (7-8 x)^4+e^x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(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 - 76965
81394432*x^7 + 1099511627776*x^8 + E^x*(1229312 - 5619712*x + 9633792*x^2 - 7340032*x^3 + 2097152*x^4)),x]

[Out]

5/(E^x + 256*(7 - 8*x)^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6686

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

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (-e^x-8192 (-7+8 x)^3\right )}{\left (e^x+256 (7-8 x)^4\right )^2} \, dx\\ &=5 \int \frac {-e^x-8192 (-7+8 x)^3}{\left (e^x+256 (7-8 x)^4\right )^2} \, dx\\ &=\frac {5}{e^x+256 (7-8 x)^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 17, normalized size = 0.81 \begin {gather*} \frac {5}{e^x+256 (7-8 x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(14049280 - 5*E^x - 48168960*x + 55050240*x^2 - 20971520*x^3)/(377801998336 + E^(2*x) - 345418969907
2*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)),
x]

[Out]

5/(E^x + 256*(7 - 8*x)^4)

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fricas [A]  time = 0.57, size = 26, normalized size = 1.24 \begin {gather*} \frac {5}{1048576 \, x^{4} - 3670016 \, x^{3} + 4816896 \, x^{2} - 2809856 \, x + e^{x} + 614656} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*exp(x)-20971520*x^3+55050240*x^2-48168960*x+14049280)/(exp(x)^2+(2097152*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+45115
947089920*x^4-31581162962944*x^3+13816758796288*x^2-3454189699072*x+377801998336),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.24, size = 26, normalized size = 1.24 \begin {gather*} \frac {5}{1048576 \, x^{4} - 3670016 \, x^{3} + 4816896 \, x^{2} - 2809856 \, x + e^{x} + 614656} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*exp(x)-20971520*x^3+55050240*x^2-48168960*x+14049280)/(exp(x)^2+(2097152*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+45115
947089920*x^4-31581162962944*x^3+13816758796288*x^2-3454189699072*x+377801998336),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.12, size = 27, normalized size = 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	\(\frac {5}{1048576 x^{4}-3670016 x^{3}+4816896 x^{2}+{\mathrm e}^{x}-2809856 x +614656}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-5*exp(x)-20971520*x^3+55050240*x^2-48168960*x+14049280)/(exp(x)^2+(2097152*x^4-7340032*x^3+9633792*x^2-5
619712*x+1229312)*exp(x)[phone]*x^8-7696581394432*x^7+23570780520448*x^6-41248865910784*x^5+45115947089
920*x^4-31581162962944*x^3+13816758796288*x^2-3454189699072*x+377801998336),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.41, size = 26, normalized size = 1.24 \begin {gather*} \frac {5}{1048576 \, x^{4} - 3670016 \, x^{3} + 4816896 \, x^{2} - 2809856 \, x + e^{x} + 614656} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*exp(x)-20971520*x^3+55050240*x^2-48168960*x+14049280)/(exp(x)^2+(2097152*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+45115
947089920*x^4-31581162962944*x^3+13816758796288*x^2-3454189699072*x+377801998336),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(48168960*x + 5*exp(x) - 55050240*x^2 + 20971520*x^3 - 14049280)/(exp(2*x) - 3454189699072*x + exp(x)*(96
33792*x^2 - 5619712*x - 7340032*x^3 + 2097152*x^4 + 1229312) + 13816758796288*x^2 - 31581162962944*x^3 + 45115
947089920*x^4 - 41248865910784*x^5 + 23570780520448*x^6 - 7696581394432*x^7 + 1099511627776*x^8 + 377801998336
),x)

[Out]

int(-(48168960*x + 5*exp(x) - 55050240*x^2 + 20971520*x^3 - 14049280)/(exp(2*x) - 3454189699072*x + exp(x)*(96
33792*x^2 - 5619712*x - 7340032*x^3 + 2097152*x^4 + 1229312) + 13816758796288*x^2 - 31581162962944*x^3 + 45115
947089920*x^4 - 41248865910784*x^5 + 23570780520448*x^6 - 7696581394432*x^7 + 1099511627776*x^8 + 377801998336
), x)

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sympy [A]  time = 0.14, size = 24, normalized size = 1.14 \begin {gather*} \frac {5}{1048576 x^{4} - 3670016 x^{3} + 4816896 x^{2} - 2809856 x + e^{x} + 614656} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*exp(x)-20971520*x**3+55050240*x**2-48168960*x+14049280)/(exp(x)**2+(2097152*x**4-7340032*x**3+96
33792*x**2-5619712*x+1229312)*exp(x)[phone]*x**8-7696581394432*x**7+23570780520448*x**6-41248865910784*
x**5+45115947089920*x**4-31581162962944*x**3+13816758796288*x**2-3454189699072*x+377801998336),x)

[Out]

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

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