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]

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

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Rubi [A]  time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 37, normalized size = 1.00 \[ \frac {(c+d x) e^{\frac {e}{c+d x}}}{d}-\frac {e \text {Ei}\left (\frac {e}{c+d x}\right )}{d} \]

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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fricas [A]  time = 0.40, size = 35, normalized size = 0.95 \[ -\frac {e {\rm Ei}\left (\frac {e}{d x + c}\right ) - {\left (d x + c\right )} e^{\left (\frac {e}{d x + c}\right )}}{d} \]

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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giac [A]  time = 0.31, size = 49, normalized size = 1.32 \[ -\frac {{\left (d x + c\right )} {\left (\frac {{\rm Ei}\left (\frac {e}{d x + c}\right ) e^{3}}{d x + c} - e^{\left (\frac {e}{d x + c} + 2\right )}\right )} e^{\left (-2\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*x + c)*(Ei(e/(d*x + c))*e^3/(d*x + c) - e^(e/(d*x + c) + 2))*e^(-2)/d

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maple [A]  time = 0.01, size = 42, normalized size = 1.14 \[ -\frac {\left (-\Ei \left (1, -\frac {e}{d x +c}\right )-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}\right ) e}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ d e \int \frac {x e^{\left (\frac {e}{d x + c}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} + x e^{\left (\frac {e}{d x + c}\right )} \]

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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mupad [B]  time = 3.62, size = 44, normalized size = 1.19 \[ x\,{\mathrm {e}}^{\frac {e}{c+d\,x}}-\frac {e\,\mathrm {ei}\left (\frac {e}{c+d\,x}\right )-c\,{\mathrm {e}}^{\frac {e}{c+d\,x}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(e/(c + d*x)),x)

[Out]

x*exp(e/(c + d*x)) - (e*ei(e/(c + d*x)) - c*exp(e/(c + d*x)))/d

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\frac {e}{c + d x}}\, dx \]

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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