\(\int e^{-40+e^{-40+8 x} (16 e^5+4 e^{45-8 x})+8 x} (e^{40-8 x}+128 e^5 x) \, dx\) [1416]

Optimal result

Integrand size = 45, antiderivative size = 21 \[ \int e^{-40+e^{-40+8 x} \left (16 e^5+4 e^{45-8 x}\right )+8 x} \left (e^{40-8 x}+128 e^5 x\right ) \, dx=e^{e^5 \left (4+16 e^{-8 (5-x)}\right )} x \]

[Out]

exp(exp(5)*(16/exp(-2*x+10)^4+4))*x

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {6873, 6820, 2326} \[ \int e^{-40+e^{-40+8 x} \left (16 e^5+4 e^{45-8 x}\right )+8 x} \left (e^{40-8 x}+128 e^5 x\right ) \, dx=e^{16 e^{8 x-35}+4 e^5} x \]

[In]

Int[E^(-40 + E^(-40 + 8*x)*(16*E^5 + 4*E^(45 - 8*x)) + 8*x)*(E^(40 - 8*x) + 128*E^5*x),x]

[Out]

E^(4*E^5 + 16*E^(-35 + 8*x))*x

Rule 2326

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]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps \begin{align*} \text {integral}& = \int \exp \left (\frac {4 \left (4 e^{8 x}-10 e^{35} \left (1-\frac {e^5}{10}\right )+2 e^{35} x\right )}{e^{35}}\right ) \left (e^{40-8 x}+128 e^5 x\right ) \, dx \\ & = \int e^{16 e^{-35+8 x}-35 \left (1-\frac {4 e^5}{35}\right )} \left (e^{35}+128 e^{8 x} x\right ) \, dx \\ & = e^{4 e^5+16 e^{-35+8 x}} x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int e^{-40+e^{-40+8 x} \left (16 e^5+4 e^{45-8 x}\right )+8 x} \left (e^{40-8 x}+128 e^5 x\right ) \, dx=e^{4 e^5+16 e^{-35+8 x}} x \]

[In]

Integrate[E^(-40 + E^(-40 + 8*x)*(16*E^5 + 4*E^(45 - 8*x)) + 8*x)*(E^(40 - 8*x) + 128*E^5*x),x]

[Out]

E^(4*E^5 + 16*E^(-35 + 8*x))*x

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10

method	result	size
risch	\(x \,{\mathrm e}^{4 \left ({\mathrm e}^{45-8 x}+4 \,{\mathrm e}^{5}\right ) {\mathrm e}^{8 x -40}}\)	\(23\)
parallelrisch	\(x \,{\mathrm e}^{4 \,{\mathrm e}^{5} \left ({\mathrm e}^{-8 x +40}+4\right ) {\mathrm e}^{8 x -40}}\)	\(26\)
norman	\(x \,{\mathrm e}^{\left (4 \,{\mathrm e}^{5} {\mathrm e}^{-8 x +40}+16 \,{\mathrm e}^{5}\right ) {\mathrm e}^{8 x -40}}\)	\(30\)

[In]

int((exp(-2*x+10)^4+128*x*exp(5))*exp((4*exp(5)*exp(-2*x+10)^4+16*exp(5))/exp(-2*x+10)^4)/exp(-2*x+10)^4,x,met
hod=_RETURNVERBOSE)

[Out]

x*exp(4*(exp(45-8*x)+4*exp(5))*exp(8*x-40))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int e^{-40+e^{-40+8 x} \left (16 e^5+4 e^{45-8 x}\right )+8 x} \left (e^{40-8 x}+128 e^5 x\right ) \, dx=x e^{\left (4 \, e^{5} + 16 \, e^{\left (8 \, x - 35\right )}\right )} \]

[In]

integrate((exp(-2*x+10)^4+128*x*exp(5))*exp((4*exp(5)*exp(-2*x+10)^4+16*exp(5))/exp(-2*x+10)^4)/exp(-2*x+10)^4
,x, algorithm="fricas")

[Out]

x*e^(4*e^5 + 16*e^(8*x - 35))

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int e^{-40+e^{-40+8 x} \left (16 e^5+4 e^{45-8 x}\right )+8 x} \left (e^{40-8 x}+128 e^5 x\right ) \, dx=x e^{\left (4 e^{5} e^{40 - 8 x} + 16 e^{5}\right ) e^{8 x - 40}} \]

[In]

integrate((exp(-2*x+10)**4+128*x*exp(5))*exp((4*exp(5)*exp(-2*x+10)**4+16*exp(5))/exp(-2*x+10)**4)/exp(-2*x+10
)**4,x)

[Out]

x*exp((4*exp(5)*exp(40 - 8*x) + 16*exp(5))*exp(8*x - 40))

Maxima [F]

\[ \int e^{-40+e^{-40+8 x} \left (16 e^5+4 e^{45-8 x}\right )+8 x} \left (e^{40-8 x}+128 e^5 x\right ) \, dx=\int { {\left (128 \, x e^{5} + e^{\left (-8 \, x + 40\right )}\right )} e^{\left (4 \, {\left (4 \, e^{5} + e^{\left (-8 \, x + 45\right )}\right )} e^{\left (8 \, x - 40\right )} + 8 \, x - 40\right )} \,d x } \]

[In]

integrate((exp(-2*x+10)^4+128*x*exp(5))*exp((4*exp(5)*exp(-2*x+10)^4+16*exp(5))/exp(-2*x+10)^4)/exp(-2*x+10)^4
,x, algorithm="maxima")

[Out]

1/8*Ei(16*e^(8*x - 35))*e^(4*e^5) + x*e^(4*e^5 + 16*e^(8*x - 35)) - integrate(e^(4*e^5 + 16*e^(8*x - 35)), x)

Giac [F]

\[ \int e^{-40+e^{-40+8 x} \left (16 e^5+4 e^{45-8 x}\right )+8 x} \left (e^{40-8 x}+128 e^5 x\right ) \, dx=\int { {\left (128 \, x e^{5} + e^{\left (-8 \, x + 40\right )}\right )} e^{\left (4 \, {\left (4 \, e^{5} + e^{\left (-8 \, x + 45\right )}\right )} e^{\left (8 \, x - 40\right )} + 8 \, x - 40\right )} \,d x } \]

[In]

integrate((exp(-2*x+10)^4+128*x*exp(5))*exp((4*exp(5)*exp(-2*x+10)^4+16*exp(5))/exp(-2*x+10)^4)/exp(-2*x+10)^4
,x, algorithm="giac")

[Out]

integrate((128*x*e^5 + e^(-8*x + 40))*e^(4*(4*e^5 + e^(-8*x + 45))*e^(8*x - 40) + 8*x - 40), x)

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int e^{-40+e^{-40+8 x} \left (16 e^5+4 e^{45-8 x}\right )+8 x} \left (e^{40-8 x}+128 e^5 x\right ) \, dx=x\,{\mathrm {e}}^{4\,{\mathrm {e}}^5}\,{\mathrm {e}}^{16\,{\mathrm {e}}^{8\,x}\,{\mathrm {e}}^{-35}} \]

[In]

int(exp(8*x - 40)*exp(exp(8*x - 40)*(16*exp(5) + 4*exp(5)*exp(40 - 8*x)))*(exp(40 - 8*x) + 128*x*exp(5)),x)

[Out]

x*exp(4*exp(5))*exp(16*exp(8*x)*exp(-35))