3.48.9 \(\int (324 x^3+e^{9 e^{e^{e^x}}+3 x} (-750000 x^3-562500 x^4-1687500 e^{e^{e^x}+e^x+x} x^4)+e^{3 e^{e^{e^x}}+x} (-10800 x^3-2700 x^4-8100 e^{e^{e^x}+e^x+x} x^4)+e^{6 e^{e^{e^x}}+2 x} (135000 x^3+67500 x^4+202500 e^{e^{e^x}+e^x+x} x^4)+e^{12 e^{e^{e^x}}+4 x} (1562500 x^3+1562500 x^4+4687500 e^{e^{e^x}+e^x+x} x^4)) \, dx\)

Optimal. Leaf size=25 \[ \left (-x+\left (4-25 e^{3 e^{e^{e^x}}+x}\right ) x\right )^4 \]

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Rubi [B]  time = 0.66, antiderivative size = 228, normalized size of antiderivative = 9.12, number of steps used = 5, number of rules used = 1, integrand size = 180, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.006, Rules used = {2288} \begin {gather*} 81 x^4-\frac {2700 e^{x+3 e^{e^{e^x}}} \left (3 e^{x+e^{e^x}+e^x} x^4+x^4\right )}{3 e^{x+e^{e^x}+e^x}+1}+\frac {33750 e^{2 x+6 e^{e^{e^x}}} \left (3 e^{x+e^{e^x}+e^x} x^4+x^4\right )}{3 e^{x+e^{e^x}+e^x}+1}-\frac {187500 e^{3 x+9 e^{e^{e^x}}} \left (3 e^{x+e^{e^x}+e^x} x^4+x^4\right )}{3 e^{x+e^{e^x}+e^x}+1}+\frac {390625 e^{4 x+12 e^{e^{e^x}}} \left (3 e^{x+e^{e^x}+e^x} x^4+x^4\right )}{3 e^{x+e^{e^x}+e^x}+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[324*x^3 + E^(9*E^E^E^x + 3*x)*(-750000*x^3 - 562500*x^4 - 1687500*E^(E^E^x + E^x + x)*x^4) + E^(3*E^E^E^x
+ x)*(-10800*x^3 - 2700*x^4 - 8100*E^(E^E^x + E^x + x)*x^4) + E^(6*E^E^E^x + 2*x)*(135000*x^3 + 67500*x^4 + 20
2500*E^(E^E^x + E^x + x)*x^4) + E^(12*E^E^E^x + 4*x)*(1562500*x^3 + 1562500*x^4 + 4687500*E^(E^E^x + E^x + x)*
x^4),x]

[Out]

81*x^4 - (2700*E^(3*E^E^E^x + x)*(x^4 + 3*E^(E^E^x + E^x + x)*x^4))/(1 + 3*E^(E^E^x + E^x + x)) + (33750*E^(6*
E^E^E^x + 2*x)*(x^4 + 3*E^(E^E^x + E^x + x)*x^4))/(1 + 3*E^(E^E^x + E^x + x)) - (187500*E^(9*E^E^E^x + 3*x)*(x
^4 + 3*E^(E^E^x + E^x + x)*x^4))/(1 + 3*E^(E^E^x + E^x + x)) + (390625*E^(12*E^E^E^x + 4*x)*(x^4 + 3*E^(E^E^x
+ E^x + x)*x^4))/(1 + 3*E^(E^E^x + E^x + x))

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=81 x^4+\int e^{9 e^{e^{e^x}}+3 x} \left (-750000 x^3-562500 x^4-1687500 e^{e^{e^x}+e^x+x} x^4\right ) \, dx+\int e^{3 e^{e^{e^x}}+x} \left (-10800 x^3-2700 x^4-8100 e^{e^{e^x}+e^x+x} x^4\right ) \, dx+\int e^{6 e^{e^{e^x}}+2 x} \left (135000 x^3+67500 x^4+202500 e^{e^{e^x}+e^x+x} x^4\right ) \, dx+\int e^{12 e^{e^{e^x}}+4 x} \left (1562500 x^3+1562500 x^4+4687500 e^{e^{e^x}+e^x+x} x^4\right ) \, dx\\ &=81 x^4-\frac {2700 e^{3 e^{e^{e^x}}+x} \left (x^4+3 e^{e^{e^x}+e^x+x} x^4\right )}{1+3 e^{e^{e^x}+e^x+x}}+\frac {33750 e^{6 e^{e^{e^x}}+2 x} \left (x^4+3 e^{e^{e^x}+e^x+x} x^4\right )}{1+3 e^{e^{e^x}+e^x+x}}-\frac {187500 e^{9 e^{e^{e^x}}+3 x} \left (x^4+3 e^{e^{e^x}+e^x+x} x^4\right )}{1+3 e^{e^{e^x}+e^x+x}}+\frac {390625 e^{12 e^{e^{e^x}}+4 x} \left (x^4+3 e^{e^{e^x}+e^x+x} x^4\right )}{1+3 e^{e^{e^x}+e^x+x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.85, size = 23, normalized size = 0.92 \begin {gather*} \left (3-25 e^{3 e^{e^{e^x}}+x}\right )^4 x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[324*x^3 + E^(9*E^E^E^x + 3*x)*(-750000*x^3 - 562500*x^4 - 1687500*E^(E^E^x + E^x + x)*x^4) + E^(3*E^
E^E^x + x)*(-10800*x^3 - 2700*x^4 - 8100*E^(E^E^x + E^x + x)*x^4) + E^(6*E^E^E^x + 2*x)*(135000*x^3 + 67500*x^
4 + 202500*E^(E^E^x + E^x + x)*x^4) + E^(12*E^E^E^x + 4*x)*(1562500*x^3 + 1562500*x^4 + 4687500*E^(E^E^x + E^x
+ x)*x^4),x]

[Out]

(3 - 25*E^(3*E^E^E^x + x))^4*x^4

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fricas [B]  time = 0.70, size = 145, normalized size = 5.80 \begin {gather*} 390625 \, x^{4} e^{\left (4 \, {\left (x e^{\left (x + e^{x}\right )} + 3 \, e^{\left (x + e^{x} + e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-x - e^{x}\right )}\right )} - 187500 \, x^{4} e^{\left (3 \, {\left (x e^{\left (x + e^{x}\right )} + 3 \, e^{\left (x + e^{x} + e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-x - e^{x}\right )}\right )} + 33750 \, x^{4} e^{\left (2 \, {\left (x e^{\left (x + e^{x}\right )} + 3 \, e^{\left (x + e^{x} + e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-x - e^{x}\right )}\right )} - 2700 \, x^{4} e^{\left ({\left (x e^{\left (x + e^{x}\right )} + 3 \, e^{\left (x + e^{x} + e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-x - e^{x}\right )}\right )} + 81 \, x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4687500*x^4*exp(x)*exp(exp(x))*exp(exp(exp(x)))+1562500*x^4+1562500*x^3)*exp(3*exp(exp(exp(x)))+x)^
4+(-1687500*x^4*exp(x)*exp(exp(x))*exp(exp(exp(x)))-562500*x^4-750000*x^3)*exp(3*exp(exp(exp(x)))+x)^3+(202500
*x^4*exp(x)*exp(exp(x))*exp(exp(exp(x)))+67500*x^4+135000*x^3)*exp(3*exp(exp(exp(x)))+x)^2+(-8100*x^4*exp(x)*e
xp(exp(x))*exp(exp(exp(x)))-2700*x^4-10800*x^3)*exp(3*exp(exp(exp(x)))+x)+324*x^3,x, algorithm="fricas")

[Out]

390625*x^4*e^(4*(x*e^(x + e^x) + 3*e^(x + e^x + e^(e^x)))*e^(-x - e^x)) - 187500*x^4*e^(3*(x*e^(x + e^x) + 3*e
^(x + e^x + e^(e^x)))*e^(-x - e^x)) + 33750*x^4*e^(2*(x*e^(x + e^x) + 3*e^(x + e^x + e^(e^x)))*e^(-x - e^x)) -
2700*x^4*e^((x*e^(x + e^x) + 3*e^(x + e^x + e^(e^x)))*e^(-x - e^x)) + 81*x^4

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int 324 \, x^{3} + 1562500 \, {\left (3 \, x^{4} e^{\left (x + e^{x} + e^{\left (e^{x}\right )}\right )} + x^{4} + x^{3}\right )} e^{\left (4 \, x + 12 \, e^{\left (e^{\left (e^{x}\right )}\right )}\right )} - 187500 \, {\left (9 \, x^{4} e^{\left (x + e^{x} + e^{\left (e^{x}\right )}\right )} + 3 \, x^{4} + 4 \, x^{3}\right )} e^{\left (3 \, x + 9 \, e^{\left (e^{\left (e^{x}\right )}\right )}\right )} + 67500 \, {\left (3 \, x^{4} e^{\left (x + e^{x} + e^{\left (e^{x}\right )}\right )} + x^{4} + 2 \, x^{3}\right )} e^{\left (2 \, x + 6 \, e^{\left (e^{\left (e^{x}\right )}\right )}\right )} - 2700 \, {\left (3 \, x^{4} e^{\left (x + e^{x} + e^{\left (e^{x}\right )}\right )} + x^{4} + 4 \, x^{3}\right )} e^{\left (x + 3 \, e^{\left (e^{\left (e^{x}\right )}\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4687500*x^4*exp(x)*exp(exp(x))*exp(exp(exp(x)))+1562500*x^4+1562500*x^3)*exp(3*exp(exp(exp(x)))+x)^
4+(-1687500*x^4*exp(x)*exp(exp(x))*exp(exp(exp(x)))-562500*x^4-750000*x^3)*exp(3*exp(exp(exp(x)))+x)^3+(202500
*x^4*exp(x)*exp(exp(x))*exp(exp(exp(x)))+67500*x^4+135000*x^3)*exp(3*exp(exp(exp(x)))+x)^2+(-8100*x^4*exp(x)*e
xp(exp(x))*exp(exp(exp(x)))-2700*x^4-10800*x^3)*exp(3*exp(exp(exp(x)))+x)+324*x^3,x, algorithm="giac")

[Out]

integrate(324*x^3 + 1562500*(3*x^4*e^(x + e^x + e^(e^x)) + x^4 + x^3)*e^(4*x + 12*e^(e^(e^x))) - 187500*(9*x^4
*e^(x + e^x + e^(e^x)) + 3*x^4 + 4*x^3)*e^(3*x + 9*e^(e^(e^x))) + 67500*(3*x^4*e^(x + e^x + e^(e^x)) + x^4 + 2
*x^3)*e^(2*x + 6*e^(e^(e^x))) - 2700*(3*x^4*e^(x + e^x + e^(e^x)) + x^4 + 4*x^3)*e^(x + 3*e^(e^(e^x))), x)

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maple [B]  time = 0.11, size = 69, normalized size = 2.76

method	result	size
risch	\(390625 \,{\mathrm e}^{12 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}+4 x} x^{4}-187500 \,{\mathrm e}^{9 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}+3 x} x^{4}+33750 \,{\mathrm e}^{6 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}+2 x} x^{4}-2700 \,{\mathrm e}^{3 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}+x} x^{4}+81 x^{4}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4687500*x^4*exp(x)*exp(exp(x))*exp(exp(exp(x)))+1562500*x^4+1562500*x^3)*exp(3*exp(exp(exp(x)))+x)^4+(-16
87500*x^4*exp(x)*exp(exp(x))*exp(exp(exp(x)))-562500*x^4-750000*x^3)*exp(3*exp(exp(exp(x)))+x)^3+(202500*x^4*e
xp(x)*exp(exp(x))*exp(exp(exp(x)))+67500*x^4+135000*x^3)*exp(3*exp(exp(exp(x)))+x)^2+(-8100*x^4*exp(x)*exp(exp
(x))*exp(exp(exp(x)))-2700*x^4-10800*x^3)*exp(3*exp(exp(exp(x)))+x)+324*x^3,x,method=_RETURNVERBOSE)

[Out]

390625*exp(12*exp(exp(exp(x)))+4*x)*x^4-187500*exp(9*exp(exp(exp(x)))+3*x)*x^4+33750*exp(6*exp(exp(exp(x)))+2*
x)*x^4-2700*exp(3*exp(exp(exp(x)))+x)*x^4+81*x^4

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maxima [B]  time = 0.51, size = 68, normalized size = 2.72 \begin {gather*} 390625 \, x^{4} e^{\left (4 \, x + 12 \, e^{\left (e^{\left (e^{x}\right )}\right )}\right )} - 187500 \, x^{4} e^{\left (3 \, x + 9 \, e^{\left (e^{\left (e^{x}\right )}\right )}\right )} + 33750 \, x^{4} e^{\left (2 \, x + 6 \, e^{\left (e^{\left (e^{x}\right )}\right )}\right )} - 2700 \, x^{4} e^{\left (x + 3 \, e^{\left (e^{\left (e^{x}\right )}\right )}\right )} + 81 \, x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4687500*x^4*exp(x)*exp(exp(x))*exp(exp(exp(x)))+1562500*x^4+1562500*x^3)*exp(3*exp(exp(exp(x)))+x)^
4+(-1687500*x^4*exp(x)*exp(exp(x))*exp(exp(exp(x)))-562500*x^4-750000*x^3)*exp(3*exp(exp(exp(x)))+x)^3+(202500
*x^4*exp(x)*exp(exp(x))*exp(exp(exp(x)))+67500*x^4+135000*x^3)*exp(3*exp(exp(exp(x)))+x)^2+(-8100*x^4*exp(x)*e
xp(exp(x))*exp(exp(exp(x)))-2700*x^4-10800*x^3)*exp(3*exp(exp(exp(x)))+x)+324*x^3,x, algorithm="maxima")

[Out]

390625*x^4*e^(4*x + 12*e^(e^(e^x))) - 187500*x^4*e^(3*x + 9*e^(e^(e^x))) + 33750*x^4*e^(2*x + 6*e^(e^(e^x))) -
2700*x^4*e^(x + 3*e^(e^(e^x))) + 81*x^4

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mupad [B]  time = 0.32, size = 68, normalized size = 2.72 \begin {gather*} 81\,x^4-2700\,x^4\,{\mathrm {e}}^{x+3\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}}+33750\,x^4\,{\mathrm {e}}^{2\,x+6\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}}-187500\,x^4\,{\mathrm {e}}^{3\,x+9\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}}+390625\,x^4\,{\mathrm {e}}^{4\,x+12\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x + 6*exp(exp(exp(x))))*(135000*x^3 + 67500*x^4 + 202500*x^4*exp(exp(x))*exp(exp(exp(x)))*exp(x)) -
exp(x + 3*exp(exp(exp(x))))*(10800*x^3 + 2700*x^4 + 8100*x^4*exp(exp(x))*exp(exp(exp(x)))*exp(x)) - exp(3*x +
9*exp(exp(exp(x))))*(750000*x^3 + 562500*x^4 + 1687500*x^4*exp(exp(x))*exp(exp(exp(x)))*exp(x)) + exp(4*x + 12
*exp(exp(exp(x))))*(1562500*x^3 + 1562500*x^4 + 4687500*x^4*exp(exp(x))*exp(exp(exp(x)))*exp(x)) + 324*x^3,x)

[Out]

81*x^4 - 2700*x^4*exp(x + 3*exp(exp(exp(x)))) + 33750*x^4*exp(2*x + 6*exp(exp(exp(x)))) - 187500*x^4*exp(3*x +
9*exp(exp(exp(x)))) + 390625*x^4*exp(4*x + 12*exp(exp(exp(x))))

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sympy [B]  time = 7.73, size = 76, normalized size = 3.04 \begin {gather*} - 2700 x^{4} e^{x + 3 e^{e^{e^{x}}}} + 33750 x^{4} e^{2 x + 6 e^{e^{e^{x}}}} - 187500 x^{4} e^{3 x + 9 e^{e^{e^{x}}}} + 390625 x^{4} e^{4 x + 12 e^{e^{e^{x}}}} + 81 x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4687500*x**4*exp(x)*exp(exp(x))*exp(exp(exp(x)))+1562500*x**4+1562500*x**3)*exp(3*exp(exp(exp(x)))+
x)**4+(-1687500*x**4*exp(x)*exp(exp(x))*exp(exp(exp(x)))-562500*x**4-750000*x**3)*exp(3*exp(exp(exp(x)))+x)**3
+(202500*x**4*exp(x)*exp(exp(x))*exp(exp(exp(x)))+67500*x**4+135000*x**3)*exp(3*exp(exp(exp(x)))+x)**2+(-8100*
x**4*exp(x)*exp(exp(x))*exp(exp(exp(x)))-2700*x**4-10800*x**3)*exp(3*exp(exp(exp(x)))+x)+324*x**3,x)

[Out]

-2700*x**4*exp(x + 3*exp(exp(exp(x)))) + 33750*x**4*exp(2*x + 6*exp(exp(exp(x)))) - 187500*x**4*exp(3*x + 9*ex
p(exp(exp(x)))) + 390625*x**4*exp(4*x + 12*exp(exp(exp(x)))) + 81*x**4

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