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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0519313, 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 \[ \frac{(c+d x) e^{\frac{e}{c+d x}}}{d}-\frac{e \text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{d} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.85324, size = 26, normalized size = 0.7 \[ - \frac{e \operatorname{Ei}{\left (\frac{e}{c + d x} \right )}}{d} + \frac{\left (c + d x\right ) e^{\frac{e}{c + d x}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(e/(d*x+c)),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0122749, size = 37, normalized size = 1. \[ \frac{(c+d x) e^{\frac{e}{c+d x}}}{d}-\frac{e \text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{d} \]

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

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 42, normalized size = 1.1 \[ -{\frac{e}{d} \left ( -{\frac{dx+c}{e}{{\rm e}^{{\frac{e}{dx+c}}}}}-{\it Ei} \left ( 1,-{\frac{e}{dx+c}} \right ) \right ) } \]

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)))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(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)
)

_______________________________________________________________________________________

Fricas [A]  time = 0.254495, size = 47, normalized size = 1.27 \[ -\frac{e{\rm Ei}\left (\frac{e}{d x + c}\right ) -{\left (d x + c\right )} e^{\left (\frac{e}{d x + c}\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(e/(d*x + c)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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

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)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int e^{\left (\frac{e}{d x + c}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(e/(d*x + c)),x, algorithm="giac")

[Out]

integrate(e^(e/(d*x + c)), x)