\(\int e^{\frac {e}{c+d x}} \, dx\) [405]

Optimal result

Integrand size = 11, antiderivative size = 37 \[ \int e^{\frac {e}{c+d x}} \, dx=\frac {e^{\frac {e}{c+d x}} (c+d x)}{d}-\frac {e \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2237, 2241} \[ \int e^{\frac {e}{c+d x}} \, dx=\frac {(c+d x) e^{\frac {e}{c+d x}}}{d}-\frac {e \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d} \]

[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

Rule 2237

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(c + d*x)*(F^(a + b*(c + d*x)^n)/d), x]
- Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n]
&& ILtQ[n, 0]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {e^{\frac {e}{c+d x}} (c+d x)}{d}+e \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx \\ & = \frac {e^{\frac {e}{c+d x}} (c+d x)}{d}-\frac {e \text {Ei}\left (\frac {e}{c+d x}\right )}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int e^{\frac {e}{c+d x}} \, dx=\frac {e^{\frac {e}{c+d x}} (c+d x)}{d}-\frac {e \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d} \]

[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] (verified)

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14

method	result	size
derivativedivides	\(-\frac {e \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d}\)	\(42\)
default	\(-\frac {e \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d}\)	\(42\)
risch	\({\mathrm e}^{\frac {e}{d x +c}} x +\frac {{\mathrm e}^{\frac {e}{d x +c}} c}{d}+\frac {e \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d}\)	\(46\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int e^{\frac {e}{c+d x}} \, dx=-\frac {e {\rm Ei}\left (\frac {e}{d x + c}\right ) - {\left (d x + c\right )} e^{\left (\frac {e}{d x + c}\right )}}{d} \]

[In]

integrate(exp(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]

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

[In]

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

[Out]

Integral(exp(e/(c + d*x)), x)

Maxima [F]

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

[In]

integrate(exp(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))

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.35 \[ \int e^{\frac {e}{c+d x}} \, dx=-\frac {{\left (\frac {e^{3} {\rm Ei}\left (\frac {e}{d x + c}\right )}{d x + c} - e^{2} e^{\left (\frac {e}{d x + c}\right )}\right )} {\left (d x + c\right )}}{d e^{2}} \]

[In]

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

[Out]

-(e^3*Ei(e/(d*x + c))/(d*x + c) - e^2*e^(e/(d*x + c)))*(d*x + c)/(d*e^2)

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.19 \[ \int e^{\frac {e}{c+d x}} \, dx=x\,{\mathrm {e}}^{\frac {e}{c+d\,x}}-\frac {e\,\mathrm {ei}\left (\frac {e}{c+d\,x}\right )-c\,{\mathrm {e}}^{\frac {e}{c+d\,x}}}{d} \]

[In]

int(exp(e/(c + d*x)),x)

[Out]

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