3.405 \(\int e^{\frac{e}{c+d x}} \, dx\)

Optimal. Leaf size=37 \[ \frac{(c+d x) e^{\frac{e}{c+d x}}}{d}-\frac{e \text{Ei}\left (\frac{e}{c+d x}\right )}{d} \]

[Out]

(E^(e/(c + d*x))*(c + d*x))/d - (e*ExpIntegralEi[e/(c + d*x)])/d

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Rubi [A]  time = 0.0301463, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2206, 2210} \[ \frac{(c+d x) e^{\frac{e}{c+d x}}}{d}-\frac{e \text{Ei}\left (\frac{e}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[E^(e/(c + d*x)),x]

[Out]

(E^(e/(c + d*x))*(c + d*x))/d - (e*ExpIntegralEi[e/(c + d*x)])/d

Rule 2206

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[((c + d*x)*F^(a + b*(c + d*x)^n))/d, x]
- Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n]
&& ILtQ[n, 0]

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int e^{\frac{e}{c+d x}} \, dx &=\frac{e^{\frac{e}{c+d x}} (c+d x)}{d}+e \int \frac{e^{\frac{e}{c+d x}}}{c+d x} \, dx\\ &=\frac{e^{\frac{e}{c+d x}} (c+d x)}{d}-\frac{e \text{Ei}\left (\frac{e}{c+d x}\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.0129464, size = 37, normalized size = 1. \[ \frac{(c+d x) e^{\frac{e}{c+d x}}}{d}-\frac{e \text{Ei}\left (\frac{e}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(e/(c + d*x)),x]

[Out]

(E^(e/(c + d*x))*(c + d*x))/d - (e*ExpIntegralEi[e/(c + d*x)])/d

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Maple [A]  time = 0.002, size = 42, normalized size = 1.1 \begin{align*} -{\frac{e}{d} \left ( -{\frac{dx+c}{e}{{\rm e}^{{\frac{e}{dx+c}}}}}-{\it Ei} \left ( 1,-{\frac{e}{dx+c}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(e/(d*x+c)),x)

[Out]

-1/d*e*(-(d*x+c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} d e \int \frac{x e^{\left (\frac{e}{d x + c}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} + x e^{\left (\frac{e}{d x + c}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c)),x, algorithm="maxima")

[Out]

d*e*integrate(x*e^(e/(d*x + c))/(d^2*x^2 + 2*c*d*x + c^2), x) + x*e^(e/(d*x + c))

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Fricas [A]  time = 1.52595, size = 70, normalized size = 1.89 \begin{align*} -\frac{e{\rm Ei}\left (\frac{e}{d x + c}\right ) -{\left (d x + c\right )} e^{\left (\frac{e}{d x + c}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c)),x, algorithm="fricas")

[Out]

-(e*Ei(e/(d*x + c)) - (d*x + c)*e^(e/(d*x + c)))/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\frac{e}{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c)),x)

[Out]

Integral(exp(e/(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (\frac{e}{d x + c}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c)),x, algorithm="giac")

[Out]

integrate(e^(e/(d*x + c)), x)