3.100.27 \(\int \frac {e^{e^x} (28 x+14 e^x x^2)}{18+e} \, dx\)

Optimal. Leaf size=15 \[ \frac {14 e^{e^x} x^2}{18+e} \]

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {12, 2288} \begin {gather*} \frac {14 e^{e^x} x^2}{18+e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^E^x*(28*x + 14*E^x*x^2))/(18 + E),x]

[Out]

(14*E^E^x*x^2)/(18 + E)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {\int e^{e^x} \left (28 x+14 e^x x^2\right ) \, dx}{18+e}\\ &=\frac {14 e^{e^x} x^2}{18+e}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \frac {14 e^{e^x} x^2}{18+e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^E^x*(28*x + 14*E^x*x^2))/(18 + E),x]

[Out]

(14*E^E^x*x^2)/(18 + E)

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fricas [A]  time = 2.35, size = 14, normalized size = 0.93 \begin {gather*} \frac {14 \, x^{2} e^{\left (e^{x}\right )}}{e + 18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((14*exp(x)*x^2+28*x)*exp(exp(x))/(exp(1)+18),x, algorithm="fricas")

[Out]

14*x^2*e^(e^x)/(e + 18)

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giac [A]  time = 1.01, size = 14, normalized size = 0.93 \begin {gather*} \frac {14 \, x^{2} e^{\left (e^{x}\right )}}{e + 18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((14*exp(x)*x^2+28*x)*exp(exp(x))/(exp(1)+18),x, algorithm="giac")

[Out]

14*x^2*e^(e^x)/(e + 18)

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maple [A]  time = 0.05, size = 15, normalized size = 1.00

method	result	size
norman	\(\frac {14 x^{2} {\mathrm e}^{{\mathrm e}^{x}}}{{\mathrm e}+18}\)	\(15\)
risch	\(\frac {14 x^{2} {\mathrm e}^{{\mathrm e}^{x}}}{{\mathrm e}+18}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((14*exp(x)*x^2+28*x)*exp(exp(x))/(exp(1)+18),x,method=_RETURNVERBOSE)

[Out]

14*x^2/(exp(1)+18)*exp(exp(x))

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maxima [A]  time = 0.39, size = 14, normalized size = 0.93 \begin {gather*} \frac {14 \, x^{2} e^{\left (e^{x}\right )}}{e + 18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((14*exp(x)*x^2+28*x)*exp(exp(x))/(exp(1)+18),x, algorithm="maxima")

[Out]

14*x^2*e^(e^x)/(e + 18)

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mupad [B]  time = 7.32, size = 14, normalized size = 0.93 \begin {gather*} \frac {14\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^x}}{\mathrm {e}+18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x))*(28*x + 14*x^2*exp(x)))/(exp(1) + 18),x)

[Out]

(14*x^2*exp(exp(x)))/(exp(1) + 18)

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sympy [A]  time = 0.17, size = 14, normalized size = 0.93 \begin {gather*} \frac {14 x^{2} e^{e^{x}}}{e + 18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((14*exp(x)*x**2+28*x)*exp(exp(x))/(exp(1)+18),x)

[Out]

14*x**2*exp(exp(x))/(E + 18)

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