3.85.81 \(\int \frac {6250+e^{2 e^{8 x/5}} (-5-16 e^{8 x/5} x)}{3125} \, dx\)

Optimal. Leaf size=20 \[ 2 x-\frac {1}{625} e^{2 e^{8 x/5}} x \]

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {12, 2288} \begin {gather*} 2 x-\frac {1}{625} e^{2 e^{8 x/5}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6250 + E^(2*E^((8*x)/5))*(-5 - 16*E^((8*x)/5)*x))/3125,x]

[Out]

2*x - (E^(2*E^((8*x)/5))*x)/625

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 \left (6250+e^{2 e^{8 x/5}} \left (-5-16 e^{8 x/5} x\right )\right ) \, dx}{3125}\\ &=2 x+\frac {\int e^{2 e^{8 x/5}} \left (-5-16 e^{8 x/5} x\right ) \, dx}{3125}\\ &=2 x-\frac {1}{625} e^{2 e^{8 x/5}} x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} 2 x-\frac {1}{625} e^{2 e^{8 x/5}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6250 + E^(2*E^((8*x)/5))*(-5 - 16*E^((8*x)/5)*x))/3125,x]

[Out]

2*x - (E^(2*E^((8*x)/5))*x)/625

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fricas [A]  time = 0.46, size = 14, normalized size = 0.70 \begin {gather*} -\frac {1}{625} \, x e^{\left (2 \, e^{\left (\frac {8}{5} \, x\right )}\right )} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3125*(-16*x*exp(8/5*x)-5)*exp(exp(8/5*x))^2+2,x, algorithm="fricas")

[Out]

-1/625*x*e^(2*e^(8/5*x)) + 2*x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {1}{3125} \, {\left (16 \, x e^{\left (\frac {8}{5} \, x\right )} + 5\right )} e^{\left (2 \, e^{\left (\frac {8}{5} \, x\right )}\right )} + 2\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3125*(-16*x*exp(8/5*x)-5)*exp(exp(8/5*x))^2+2,x, algorithm="giac")

[Out]

integrate(-1/3125*(16*x*e^(8/5*x) + 5)*e^(2*e^(8/5*x)) + 2, x)

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maple [A]  time = 0.05, size = 15, normalized size = 0.75

method	result	size
default	\(2 x -\frac {{\mathrm e}^{2 \,{\mathrm e}^{\frac {8 x}{5}}} x}{625}\)	\(15\)
norman	\(2 x -\frac {{\mathrm e}^{2 \,{\mathrm e}^{\frac {8 x}{5}}} x}{625}\)	\(15\)
risch	\(2 x -\frac {{\mathrm e}^{2 \,{\mathrm e}^{\frac {8 x}{5}}} x}{625}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3125*(-16*x*exp(8/5*x)-5)*exp(exp(8/5*x))^2+2,x,method=_RETURNVERBOSE)

[Out]

2*x-1/625*exp(exp(8/5*x))^2*x

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {1}{625} \, x e^{\left (2 \, e^{\left (\frac {8}{5} \, x\right )}\right )} + 2 \, x - \frac {1}{1000} \, {\rm Ei}\left (2 \, e^{\left (\frac {8}{5} \, x\right )}\right ) + \frac {1}{625} \, \int e^{\left (2 \, e^{\left (\frac {8}{5} \, x\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3125*(-16*x*exp(8/5*x)-5)*exp(exp(8/5*x))^2+2,x, algorithm="maxima")

[Out]

-1/625*x*e^(2*e^(8/5*x)) + 2*x - 1/1000*Ei(2*e^(8/5*x)) + 1/625*integrate(e^(2*e^(8/5*x)), x)

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mupad [B]  time = 0.06, size = 12, normalized size = 0.60 \begin {gather*} -\frac {x\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^{\frac {8\,x}{5}}}-1250\right )}{625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2 - (exp(2*exp((8*x)/5))*(16*x*exp((8*x)/5) + 5))/3125,x)

[Out]

-(x*(exp(2*exp((8*x)/5)) - 1250))/625

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sympy [A]  time = 0.14, size = 15, normalized size = 0.75 \begin {gather*} - \frac {x e^{2 e^{\frac {8 x}{5}}}}{625} + 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3125*(-16*x*exp(8/5*x)-5)*exp(exp(8/5*x))**2+2,x)

[Out]

-x*exp(2*exp(8*x/5))/625 + 2*x

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