3.58.6 \(\int \frac {2 e^{e^4+e^e}+e^{e^4} (-20-8 x)}{300 x^2+120 x^3+12 x^4+e^{2 e^e} (75+30 x+3 x^2)+e^{e^e} (-300 x-120 x^2-12 x^3)} \, dx\)

Optimal. Leaf size=29 \[ 2+\frac {2 e^{e^4}}{3 (5+x) \left (-e^{e^e}+2 x\right )} \]

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Rubi [A]  time = 0.09, antiderivative size = 33, normalized size of antiderivative = 1.14, number of steps used = 3, number of rules used = 3, integrand size = 80, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1680, 12, 261} \begin {gather*} -\frac {2 e^{e^4}}{3 \left (-2 x^2+\left (e^{e^e}-10\right ) x+5 e^{e^e}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*E^(E^4 + E^E) + E^E^4*(-20 - 8*x))/(300*x^2 + 120*x^3 + 12*x^4 + E^(2*E^E)*(75 + 30*x + 3*x^2) + E^E^E*
(-300*x - 120*x^2 - 12*x^3)),x]

[Out]

(-2*E^E^4)/(3*(5*E^E^E + (-10 + E^E^E)*x - 2*x^2))

Rule 12

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int -\frac {512 e^{e^4} x}{3 \left (100+20 e^{e^e}+e^{2 e^e}-16 x^2\right )^2} \, dx,x,\frac {1}{48} \left (120-12 e^{e^e}\right )+x\right )\\ &=-\left (\frac {1}{3} \left (512 e^{e^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (100+20 e^{e^e}+e^{2 e^e}-16 x^2\right )^2} \, dx,x,\frac {1}{48} \left (120-12 e^{e^e}\right )+x\right )\right )\\ &=-\frac {2 e^{e^4}}{3 \left (5 e^{e^e}-\left (10-e^{e^e}\right ) x-2 x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 25, normalized size = 0.86 \begin {gather*} -\frac {2 e^{e^4}}{3 \left (e^{e^e}-2 x\right ) (5+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*E^(E^4 + E^E) + E^E^4*(-20 - 8*x))/(300*x^2 + 120*x^3 + 12*x^4 + E^(2*E^E)*(75 + 30*x + 3*x^2) +
E^E^E*(-300*x - 120*x^2 - 12*x^3)),x]

[Out]

(-2*E^E^4)/(3*(E^E^E - 2*x)*(5 + x))

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fricas [A]  time = 0.81, size = 33, normalized size = 1.14 \begin {gather*} -\frac {2 \, e^{\left (2 \, e^{4}\right )}}{3 \, {\left ({\left (x + 5\right )} e^{\left (e^{4} + e^{e}\right )} - 2 \, {\left (x^{2} + 5 \, x\right )} e^{\left (e^{4}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(exp(4))*exp(exp(exp(1)))+(-8*x-20)*exp(exp(4)))/((3*x^2+30*x+75)*exp(exp(exp(1)))^2+(-12*x^3-
120*x^2-300*x)*exp(exp(exp(1)))+12*x^4+120*x^3+300*x^2),x, algorithm="fricas")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(exp(4))*exp(exp(exp(1)))+(-8*x-20)*exp(exp(4)))/((3*x^2+30*x+75)*exp(exp(exp(1)))^2+(-12*x^3-
120*x^2-300*x)*exp(exp(exp(1)))+12*x^4+120*x^3+300*x^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -2/3*((-2*exp(2*exp(exp(1)))*exp(exp(4))
-20*exp(exp(exp(1)))*exp(exp(4))+2*exp(exp(exp(1)))*exp(exp(4)+exp(exp(1)))+20*exp(exp(4)+exp(exp(1))))/(exp(2
*exp(exp(1)))^2+40*ex

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maple [A]  time = 0.16, size = 21, normalized size = 0.72

method	result	size
norman	\(-\frac {2 \,{\mathrm e}^{{\mathrm e}^{4}}}{3 \left (5+x \right ) \left (-2 x +{\mathrm e}^{{\mathrm e}^{{\mathrm e}}}\right )}\)	\(21\)
gosper	\(-\frac {2 \,{\mathrm e}^{{\mathrm e}^{4}}}{3 \left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}}} x -2 x^{2}+5 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}}}-10 x \right )}\)	\(29\)
risch	\(-\frac {2 \,{\mathrm e}^{{\mathrm e}^{4}}}{3 \left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}}} x -2 x^{2}+5 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}}}-10 x \right )}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(exp(4))*exp(exp(exp(1)))+(-8*x-20)*exp(exp(4)))/((3*x^2+30*x+75)*exp(exp(exp(1)))^2+(-12*x^3-120*x^
2-300*x)*exp(exp(exp(1)))+12*x^4+120*x^3+300*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.36, size = 28, normalized size = 0.97 \begin {gather*} \frac {2 \, e^{\left (e^{4}\right )}}{3 \, {\left (2 \, x^{2} - x {\left (e^{\left (e^{e}\right )} - 10\right )} - 5 \, e^{\left (e^{e}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(exp(4))*exp(exp(exp(1)))+(-8*x-20)*exp(exp(4)))/((3*x^2+30*x+75)*exp(exp(exp(1)))^2+(-12*x^3-
120*x^2-300*x)*exp(exp(exp(1)))+12*x^4+120*x^3+300*x^2),x, algorithm="maxima")

[Out]

2/3*e^(e^4)/(2*x^2 - x*(e^(e^e) - 10) - 5*e^(e^e))

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mupad [B]  time = 0.28, size = 86, normalized size = 2.97 \begin {gather*} \frac {4\,\left ({\mathrm {e}}^{{\mathrm {e}}^4+2\,{\mathrm {e}}^{\mathrm {e}}}+100\,{\mathrm {e}}^{{\mathrm {e}}^4}+20\,{\mathrm {e}}^{{\mathrm {e}}^4+{\mathrm {e}}^{\mathrm {e}}}\right )}{3\,{\left ({\mathrm {e}}^{{\mathrm {e}}^{\mathrm {e}}}+10\right )}^3\,\left (2\,x-{\mathrm {e}}^{{\mathrm {e}}^{\mathrm {e}}}\right )}-\frac {2\,\left ({\mathrm {e}}^{{\mathrm {e}}^4+2\,{\mathrm {e}}^{\mathrm {e}}}+100\,{\mathrm {e}}^{{\mathrm {e}}^4}+20\,{\mathrm {e}}^{{\mathrm {e}}^4+{\mathrm {e}}^{\mathrm {e}}}\right )}{3\,{\left ({\mathrm {e}}^{{\mathrm {e}}^{\mathrm {e}}}+10\right )}^3\,\left (x+5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(exp(exp(1)))*exp(exp(4)) - exp(exp(4))*(8*x + 20))/(exp(2*exp(exp(1)))*(30*x + 3*x^2 + 75) - exp(ex
p(exp(1)))*(300*x + 120*x^2 + 12*x^3) + 300*x^2 + 120*x^3 + 12*x^4),x)

[Out]

(4*(exp(exp(4) + 2*exp(exp(1))) + 100*exp(exp(4)) + 20*exp(exp(4) + exp(exp(1)))))/(3*(exp(exp(exp(1))) + 10)^
3*(2*x - exp(exp(exp(1))))) - (2*(exp(exp(4) + 2*exp(exp(1))) + 100*exp(exp(4)) + 20*exp(exp(4) + exp(exp(1)))
))/(3*(exp(exp(exp(1))) + 10)^3*(x + 5))

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sympy [A]  time = 0.55, size = 31, normalized size = 1.07 \begin {gather*} \frac {2 e^{e^{4}}}{6 x^{2} + x \left (30 - 3 e^{e^{e}}\right ) - 15 e^{e^{e}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(exp(4))*exp(exp(exp(1)))+(-8*x-20)*exp(exp(4)))/((3*x**2+30*x+75)*exp(exp(exp(1)))**2+(-12*x*
*3-120*x**2-300*x)*exp(exp(exp(1)))+12*x**4+120*x**3+300*x**2),x)

[Out]

2*exp(exp(4))/(6*x**2 + x*(30 - 3*exp(exp(E))) - 15*exp(exp(E)))

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