3.4.72 \(\int (4 e^3+e^{2 e^{2 x}} (e^3+4 e^{3+2 x} x)) \, dx\)

Optimal. Leaf size=16 \[ e^3 \left (4+e^{2 e^{2 x}}\right ) x \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2288} \begin {gather*} e^{2 e^{2 x}+3} x+4 e^3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[4*E^3 + E^(2*E^(2*x))*(E^3 + 4*E^(3 + 2*x)*x),x]

[Out]

4*E^3*x + E^(3 + 2*E^(2*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} &=4 e^3 x+\int e^{2 e^{2 x}} \left (e^3+4 e^{3+2 x} x\right ) \, dx\\ &=4 e^3 x+e^{3+2 e^{2 x}} x\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 19, normalized size = 1.19 \begin {gather*} e^3 \left (4 x+e^{2 e^{2 x}} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[4*E^3 + E^(2*E^(2*x))*(E^3 + 4*E^(3 + 2*x)*x),x]

[Out]

E^3*(4*x + E^(2*E^(2*x))*x)

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 17, normalized size = 1.06 \begin {gather*} 4 \, x e^{3} + x e^{\left (2 \, e^{\left (2 \, x\right )} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(3)*exp(2*x)+exp(3))*exp(exp(2*x))^2+4*exp(3),x, algorithm="fricas")

[Out]

4*x*e^3 + x*e^(2*e^(2*x) + 3)

________________________________________________________________________________________

giac [A]  time = 0.50, size = 17, normalized size = 1.06 \begin {gather*} 4 \, x e^{3} + x e^{\left (2 \, e^{\left (2 \, x\right )} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(3)*exp(2*x)+exp(3))*exp(exp(2*x))^2+4*exp(3),x, algorithm="giac")

[Out]

4*x*e^3 + x*e^(2*e^(2*x) + 3)

________________________________________________________________________________________

maple [A]  time = 0.12, size = 18, normalized size = 1.12

method	result	size
default	\(x \,{\mathrm e}^{3} {\mathrm e}^{2 \,{\mathrm e}^{2 x}}+4 x \,{\mathrm e}^{3}\)	\(18\)
norman	\(x \,{\mathrm e}^{3} {\mathrm e}^{2 \,{\mathrm e}^{2 x}}+4 x \,{\mathrm e}^{3}\)	\(18\)
risch	\(x \,{\mathrm e}^{3+2 \,{\mathrm e}^{2 x}}+4 x \,{\mathrm e}^{3}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x*exp(3)*exp(2*x)+exp(3))*exp(exp(2*x))^2+4*exp(3),x,method=_RETURNVERBOSE)

[Out]

x*exp(3)*exp(exp(2*x))^2+4*x*exp(3)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 4 \, x e^{3} + \frac {1}{2} \, {\rm Ei}\left (2 \, e^{\left (2 \, x\right )}\right ) e^{3} + x e^{\left (2 \, e^{\left (2 \, x\right )} + 3\right )} - \int e^{\left (2 \, e^{\left (2 \, x\right )} + 3\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(3)*exp(2*x)+exp(3))*exp(exp(2*x))^2+4*exp(3),x, algorithm="maxima")

[Out]

4*x*e^3 + 1/2*Ei(2*e^(2*x))*e^3 + x*e^(2*e^(2*x) + 3) - integrate(e^(2*e^(2*x) + 3), x)

________________________________________________________________________________________

mupad [B]  time = 0.40, size = 13, normalized size = 0.81 \begin {gather*} x\,{\mathrm {e}}^3\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}}+4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*exp(3) + exp(2*exp(2*x))*(exp(3) + 4*x*exp(2*x)*exp(3)),x)

[Out]

x*exp(3)*(exp(2*exp(2*x)) + 4)

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 19, normalized size = 1.19 \begin {gather*} x e^{3} e^{2 e^{2 x}} + 4 x e^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(3)*exp(2*x)+exp(3))*exp(exp(2*x))**2+4*exp(3),x)

[Out]

x*exp(3)*exp(2*exp(2*x)) + 4*x*exp(3)

________________________________________________________________________________________