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

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

[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 = 0.202567, 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, Rules used = {2222, 2210, 2228, 2178} \[ \frac{e^{\frac{b e}{b c-a d}} \text{Ei}\left (-\frac{d e (a+b x)}{(b c-a d) (c+d x)}\right )}{b}-\frac{\text{Ei}\left (\frac{e}{c+d x}\right )}{b} \]

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

Rule 2222

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

Rule 2228

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))/(((e_.) + (f_.)*(x_))*((g_.) + (h_.)*(x_))), x_Symbol] :> -Dist[
d/(f*(d*g - c*h)), Subst[Int[F^(a - (b*h)/(d*g - c*h) + (d*b*x)/(d*g - c*h))/x, x], x, (g + h*x)/(c + d*x)], x
] /; FreeQ[{F, a, b, c, d, e, f}, x] && EqQ[d*e - c*f, 0]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

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

Mathematica [A]  time = 0.0625747, size = 56, normalized size = 0.9 \[ \frac{e^{\frac{b e}{b c-a d}} \text{Ei}\left (e \left (\frac{b}{a d-b c}+\frac{1}{c+d x}\right )\right )-\text{Ei}\left (\frac{e}{c+d x}\right )}{b} \]

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 = 0.011, size = 79, normalized size = 1.3 \begin{align*} -{\frac{e}{d} \left ( -{\frac{d}{be}{\it Ei} \left ( 1,-{\frac{e}{dx+c}} \right ) }+{\frac{d}{be}{{\rm e}^{-{\frac{be}{ad-bc}}}}{\it Ei} \left ( 1,-{\frac{e}{dx+c}}-{\frac{be}{ad-bc}} \right ) } \right ) } \end{align*}

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. \begin{align*} \int \frac{e^{\left (\frac{e}{d x + c}\right )}}{b x + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(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 = 1.57333, size = 138, normalized size = 2.23 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(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. \begin{align*} \int \frac{e^{\frac{e}{c + d x}}}{a + b x}\, dx \end{align*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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