3.7.80 \(\int (18+e^{e^x} (12 e^{5+3 x} x^5+e^{5+2 x} (60 x^4+24 x^5))+e^{2 e^x} (4 e^{10+5 x} x^9+e^{10+4 x} (18 x^8+8 x^9))) \, dx\)

Optimal. Leaf size=21 \[ 2 x \left (3+e^{5+e^x+2 x} x^4\right )^2 \]

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Rubi [A]  time = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.71, number of steps used = 3, number of rules used = 1, integrand size = 80, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {2288} \begin {gather*} 2 e^{4 x+2 e^x+10} x^9+12 e^{2 x+e^x+5} x^5+18 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[18 + E^E^x*(12*E^(5 + 3*x)*x^5 + E^(5 + 2*x)*(60*x^4 + 24*x^5)) + E^(2*E^x)*(4*E^(10 + 5*x)*x^9 + E^(10 +
4*x)*(18*x^8 + 8*x^9)),x]

[Out]

18*x + 12*E^(5 + E^x + 2*x)*x^5 + 2*E^(10 + 2*E^x + 4*x)*x^9

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} &=18 x+\int e^{e^x} \left (12 e^{5+3 x} x^5+e^{5+2 x} \left (60 x^4+24 x^5\right )\right ) \, dx+\int e^{2 e^x} \left (4 e^{10+5 x} x^9+e^{10+4 x} \left (18 x^8+8 x^9\right )\right ) \, dx\\ &=18 x+12 e^{5+e^x+2 x} x^5+2 e^{10+2 e^x+4 x} x^9\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 21, normalized size = 1.00 \begin {gather*} 2 x \left (3+e^{5+e^x+2 x} x^4\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[18 + E^E^x*(12*E^(5 + 3*x)*x^5 + E^(5 + 2*x)*(60*x^4 + 24*x^5)) + E^(2*E^x)*(4*E^(10 + 5*x)*x^9 + E^
(10 + 4*x)*(18*x^8 + 8*x^9)),x]

[Out]

2*x*(3 + E^(5 + E^x + 2*x)*x^4)^2

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fricas [A]  time = 0.56, size = 32, normalized size = 1.52 \begin {gather*} 2 \, x^{9} e^{\left (4 \, x + 2 \, e^{x} + 10\right )} + 12 \, x^{5} e^{\left (2 \, x + e^{x} + 5\right )} + 18 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^9*exp(5)^2*exp(x)^5+(8*x^9+18*x^8)*exp(5)^2*exp(x)^4)*exp(exp(x))^2+(12*x^5*exp(5)*exp(x)^3+(24
*x^5+60*x^4)*exp(5)*exp(x)^2)*exp(exp(x))+18,x, algorithm="fricas")

[Out]

2*x^9*e^(4*x + 2*e^x + 10) + 12*x^5*e^(2*x + e^x + 5) + 18*x

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giac [A]  time = 0.36, size = 32, normalized size = 1.52 \begin {gather*} 2 \, x^{9} e^{\left (4 \, x + 2 \, e^{x} + 10\right )} + 12 \, x^{5} e^{\left (2 \, x + e^{x} + 5\right )} + 18 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^9*exp(5)^2*exp(x)^5+(8*x^9+18*x^8)*exp(5)^2*exp(x)^4)*exp(exp(x))^2+(12*x^5*exp(5)*exp(x)^3+(24
*x^5+60*x^4)*exp(5)*exp(x)^2)*exp(exp(x))+18,x, algorithm="giac")

[Out]

2*x^9*e^(4*x + 2*e^x + 10) + 12*x^5*e^(2*x + e^x + 5) + 18*x

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maple [A]  time = 0.10, size = 33, normalized size = 1.57

method	result	size
risch	\(2 x^{9} {\mathrm e}^{10+2 \,{\mathrm e}^{x}+4 x}+12 x^{5} {\mathrm e}^{{\mathrm e}^{x}+5+2 x}+18 x\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^9*exp(5)^2*exp(x)^5+(8*x^9+18*x^8)*exp(5)^2*exp(x)^4)*exp(exp(x))^2+(12*x^5*exp(5)*exp(x)^3+(24*x^5+6
0*x^4)*exp(5)*exp(x)^2)*exp(exp(x))+18,x,method=_RETURNVERBOSE)

[Out]

2*x^9*exp(10+2*exp(x)+4*x)+12*x^5*exp(exp(x)+5+2*x)+18*x

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maxima [A]  time = 0.62, size = 32, normalized size = 1.52 \begin {gather*} 2 \, x^{9} e^{\left (4 \, x + 2 \, e^{x} + 10\right )} + 12 \, x^{5} e^{\left (2 \, x + e^{x} + 5\right )} + 18 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^9*exp(5)^2*exp(x)^5+(8*x^9+18*x^8)*exp(5)^2*exp(x)^4)*exp(exp(x))^2+(12*x^5*exp(5)*exp(x)^3+(24
*x^5+60*x^4)*exp(5)*exp(x)^2)*exp(exp(x))+18,x, algorithm="maxima")

[Out]

2*x^9*e^(4*x + 2*e^x + 10) + 12*x^5*e^(2*x + e^x + 5) + 18*x

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mupad [B]  time = 0.21, size = 34, normalized size = 1.62 \begin {gather*} 18\,x+12\,x^5\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^5+2\,x^9\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{10}\,{\mathrm {e}}^{2\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*exp(x))*(exp(4*x)*exp(10)*(18*x^8 + 8*x^9) + 4*x^9*exp(5*x)*exp(10)) + exp(exp(x))*(exp(2*x)*exp(5)*
(60*x^4 + 24*x^5) + 12*x^5*exp(3*x)*exp(5)) + 18,x)

[Out]

18*x + 12*x^5*exp(2*x)*exp(exp(x))*exp(5) + 2*x^9*exp(4*x)*exp(10)*exp(2*exp(x))

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sympy [A]  time = 0.35, size = 41, normalized size = 1.95 \begin {gather*} 2 x^{9} e^{10} e^{4 x} e^{2 e^{x}} + 12 x^{5} e^{5} e^{2 x} e^{e^{x}} + 18 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**9*exp(5)**2*exp(x)**5+(8*x**9+18*x**8)*exp(5)**2*exp(x)**4)*exp(exp(x))**2+(12*x**5*exp(5)*exp
(x)**3+(24*x**5+60*x**4)*exp(5)*exp(x)**2)*exp(exp(x))+18,x)

[Out]

2*x**9*exp(10)*exp(4*x)*exp(2*exp(x)) + 12*x**5*exp(5)*exp(2*x)*exp(exp(x)) + 18*x

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