3.406 \(\int \frac{e^{\frac{e}{c+d x}}}{a+b x} \, dx\)

Optimal. Leaf size=62 \[ \frac{e^{\frac{b e}{b c-a d}} \text{ExpIntegralEi}\left (-\frac{d e (a+b x)}{(c+d x) (b c-a d)}\right )}{b}-\frac{\text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.310957, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{e^{\frac{b e}{b c-a d}} \text{ExpIntegralEi}\left (-\frac{d e (a+b x)}{(c+d x) (b c-a d)}\right )}{b}-\frac{\text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Int[E^(e/(c + d*x))/(a + b*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 25.1949, size = 44, normalized size = 0.71 \[ - \frac{\operatorname{Ei}{\left (\frac{e}{c + d x} \right )}}{b} + \frac{e^{- \frac{b e}{a d - b c}} \operatorname{Ei}{\left (\frac{d e \left (a + b x\right )}{\left (c + d x\right ) \left (a d - b c\right )} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0495554, size = 56, normalized size = 0.9 \[ \frac{e^{\frac{b e}{b c-a d}} \text{ExpIntegralEi}\left (e \left (\frac{b}{a d-b c}+\frac{1}{c+d x}\right )\right )-\text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[E^(e/(c + d*x))/(a + b*x),x]

[Out]

(-ExpIntegralEi[e/(c + d*x)] + E^((b*e)/(b*c - a*d))*ExpIntegralEi[e*(b/(-(b*c)
+ a*d) + (c + d*x)^(-1))])/b

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Maple [A]  time = 0.016, size = 79, normalized size = 1.3 \[ -{\frac{e}{d} \left ( -{\frac{d}{be}{\it Ei} \left ( 1,-{\frac{e}{dx+c}} \right ) }+{\frac{d}{be}{{\rm e}^{-{\frac{be}{ad-cb}}}}{\it Ei} \left ( 1,-{\frac{e}{dx+c}}-{\frac{be}{ad-cb}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(e/(d*x+c))/(b*x+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.255409, size = 96, normalized size = 1.55 \[ \frac{{\rm Ei}\left (-\frac{b d e x + a d e}{b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x}\right ) e^{\left (\frac{b e}{b c - a d}\right )} -{\rm Ei}\left (\frac{e}{d x + c}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(e/(d*x+c))/(b*x+a),x)

[Out]

Integral(exp(e/(c + d*x))/(a + b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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