\(\int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx\) [406]

Optimal result

Integrand size = 19, antiderivative size = 62 \[ \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx=-\frac {\operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{b}+\frac {e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{b} \]

[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] (verified)

Time = 0.13 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2254, 2241, 2260, 2209} \[ \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx=\frac {e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{b}-\frac {\operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{b} \]

[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 2209

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

Rule 2241

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 2254

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 2260

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*} \text {integral}& = \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 {\text {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] (verified)

Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx=\frac {-\operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )+e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (e \left (\frac {b}{-b c+a d}+\frac {1}{c+d x}\right )\right )}{b} \]

[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

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.05

method	result	size
risch	\(-\frac {{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{b}+\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{b}\)	\(65\)
derivativedivides	\(-\frac {e \left (\frac {d \,{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{b e}-\frac {d \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{b e}\right )}{d}\)	\(79\)
default	\(-\frac {e \left (\frac {d \,{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{b e}-\frac {d \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{b e}\right )}{d}\)	\(79\)

[In]

int(exp(e/(d*x+c))/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx=\frac {{\rm Ei}\left (-\frac {b d e x + a d e}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x}\right ) e^{\left (\frac {b e}{b c - a d}\right )} - {\rm Ei}\left (\frac {e}{d x + c}\right )}{b} \]

[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

Sympy [F]

\[ \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx=\int \frac {e^{\frac {e}{c + d x}}}{a + b x}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx=\int { \frac {e^{\left (\frac {e}{d x + c}\right )}}{b x + a} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (61) = 122\).

Time = 0.57 (sec) , antiderivative size = 483, normalized size of antiderivative = 7.79 \[ \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx=-\frac {{\left (\frac {2 \, b^{2} c^{2} e^{3} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d}\right )}}{{\left (d x + c\right )}^{2}} - \frac {4 \, a b c d e^{3} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d}\right )}}{{\left (d x + c\right )}^{2}} + \frac {2 \, a^{2} d^{2} e^{3} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d}\right )}}{{\left (d x + c\right )}^{2}} - \frac {2 \, b^{2} c^{2} e^{3} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + \frac {4 \, a b c d e^{3} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} - \frac {2 \, a^{2} d^{2} e^{3} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} - \frac {2 \, b^{2} c e^{4} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + \frac {2 \, a b d e^{4} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} - \frac {b^{2} e^{5} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + b^{2} e^{3} e^{\left (\frac {e}{d x + c}\right )} + \frac {2 \, b^{2} c e^{3} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} - \frac {2 \, a b d e^{3} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} + \frac {b^{2} e^{4} e^{\left (\frac {e}{d x + c}\right )}}{d x + c}\right )} {\left (d x + c\right )}^{2}}{2 \, b^{3} d e^{4}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx=\int \frac {{\mathrm {e}}^{\frac {e}{c+d\,x}}}{a+b\,x} \,d x \]

[In]

int(exp(e/(c + d*x))/(a + b*x),x)

[Out]

int(exp(e/(c + d*x))/(a + b*x), x)