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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.20, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, 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 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

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 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 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]

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 = 0.07, size = 56, normalized size = 0.90 \[ \frac {e^{\frac {b e}{b c-a d}} \text {Ei}\left (e \left (\frac {b}{a d-b c}+\frac {1}{c+d x}\right )\right )-\text {Ei}\left (\frac {e}{c+d x}\right )}{b} \]

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

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 71, normalized size = 1.15 \[ \frac {{\rm Ei}\left (-\frac {b d e x + a d e}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x}\right ) e^{\left (\frac {b e}{b c - a d}\right )} - {\rm Ei}\left (\frac {e}{d x + c}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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

________________________________________________________________________________________

giac [B]  time = 1.95, size = 492, normalized size = 7.94 \[ \frac {{\left (\frac {2 \, b^{2} c^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{3}}{{\left (d x + c\right )}^{2}} - \frac {4 \, a b c d {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{3}}{{\left (d x + c\right )}^{2}} + \frac {2 \, a^{2} d^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{3}}{{\left (d x + c\right )}^{2}} - \frac {2 \, b^{2} c^{2} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 3\right )}}{{\left (d x + c\right )}^{2}} + \frac {4 \, a b c d {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 3\right )}}{{\left (d x + c\right )}^{2}} - \frac {2 \, a^{2} d^{2} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 3\right )}}{{\left (d x + c\right )}^{2}} + \frac {2 \, b^{2} c {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{4}}{{\left (d x + c\right )}^{2}} - \frac {2 \, a b d {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{4}}{{\left (d x + c\right )}^{2}} - b^{2} e^{\left (\frac {e}{d x + c} + 3\right )} - \frac {2 \, b^{2} c e^{\left (\frac {e}{d x + c} + 3\right )}}{d x + c} + \frac {2 \, a b d e^{\left (\frac {e}{d x + c} + 3\right )}}{d x + c} + \frac {b^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{5}}{{\left (d x + c\right )}^{2}} - \frac {b^{2} e^{\left (\frac {e}{d x + c} + 4\right )}}{d x + c}\right )} {\left (d x + c\right )}^{2} e^{\left (-4\right )}}{2 \, b^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b^2*c^2*Ei(e/(d*x + c))*e^3/(d*x + c)^2 - 4*a*b*c*d*Ei(e/(d*x + c))*e^3/(d*x + c)^2 + 2*a^2*d^2*Ei(e/(d
*x + c))*e^3/(d*x + c)^2 - 2*b^2*c^2*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c -
a*d) + 3)/(d*x + c)^2 + 4*a*b*c*d*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d
) + 3)/(d*x + c)^2 - 2*a^2*d^2*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) +
3)/(d*x + c)^2 + 2*b^2*c*Ei(e/(d*x + c))*e^4/(d*x + c)^2 - 2*a*b*d*Ei(e/(d*x + c))*e^4/(d*x + c)^2 - b^2*e^(e
/(d*x + c) + 3) - 2*b^2*c*e^(e/(d*x + c) + 3)/(d*x + c) + 2*a*b*d*e^(e/(d*x + c) + 3)/(d*x + c) + b^2*Ei(e/(d*
x + c))*e^5/(d*x + c)^2 - b^2*e^(e/(d*x + c) + 4)/(d*x + c))*(d*x + c)^2*e^(-4)/(b^3*d)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 79, normalized size = 1.27 \[ -\frac {\left (\frac {d \Ei \left (1, -\frac {b e}{a d -b c}-\frac {e}{d x +c}\right ) {\mathrm e}^{-\frac {b e}{a d -b c}}}{b e}-\frac {d \Ei \left (1, -\frac {e}{d x +c}\right )}{b e}\right ) e}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{\frac {e}{c+d\,x}}}{a+b\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(e/(c + d*x))/(a + b*x),x)

[Out]

int(exp(e/(c + d*x))/(a + b*x), x)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________