\(\int e^{\frac {e}{c+d x}} (a+b x) \, dx\) [404]

Optimal result

Integrand size = 17, antiderivative size = 125 \[ \int e^{\frac {e}{c+d x}} (a+b x) \, dx=-\frac {(b c-a d) e^{\frac {e}{c+d x}} (c+d x)}{d^2}+\frac {b e e^{\frac {e}{c+d x}} (c+d x)}{2 d^2}+\frac {b e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^2}+\frac {(b c-a d) e \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^2}-\frac {b e^2 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{2 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2258, 2237, 2241, 2245} \[ \int e^{\frac {e}{c+d x}} (a+b x) \, dx=\frac {e (b c-a d) \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^2}-\frac {(c+d x) (b c-a d) e^{\frac {e}{c+d x}}}{d^2}-\frac {b e^2 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{2 d^2}+\frac {b e (c+d x) e^{\frac {e}{c+d x}}}{2 d^2}+\frac {b (c+d x)^2 e^{\frac {e}{c+d x}}}{2 d^2} \]

[In]

Int[E^(e/(c + d*x))*(a + b*x),x]

[Out]

-(((b*c - a*d)*E^(e/(c + d*x))*(c + d*x))/d^2) + (b*e*E^(e/(c + d*x))*(c + d*x))/(2*d^2) + (b*E^(e/(c + d*x))*
(c + d*x)^2)/(2*d^2) + ((b*c - a*d)*e*ExpIntegralEi[e/(c + d*x)])/d^2 - (b*e^2*ExpIntegralEi[e/(c + d*x)])/(2*
d^2)

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]

Rule 2245

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))), x] - Dist[b*n*(Log[F]/(m + 1)), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2258

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d) e^{\frac {e}{c+d x}}}{d}+\frac {b e^{\frac {e}{c+d x}} (c+d x)}{d}\right ) \, dx \\ & = \frac {b \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{d}+\frac {(-b c+a d) \int e^{\frac {e}{c+d x}} \, dx}{d} \\ & = -\frac {(b c-a d) e^{\frac {e}{c+d x}} (c+d x)}{d^2}+\frac {b e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^2}+\frac {(b e) \int e^{\frac {e}{c+d x}} \, dx}{2 d}+\frac {((-b c+a d) e) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{d} \\ & = -\frac {(b c-a d) e^{\frac {e}{c+d x}} (c+d x)}{d^2}+\frac {b e e^{\frac {e}{c+d x}} (c+d x)}{2 d^2}+\frac {b e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^2}+\frac {(b c-a d) e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^2}+\frac {\left (b e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{2 d} \\ & = -\frac {(b c-a d) e^{\frac {e}{c+d x}} (c+d x)}{d^2}+\frac {b e e^{\frac {e}{c+d x}} (c+d x)}{2 d^2}+\frac {b e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^2}+\frac {(b c-a d) e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^2}-\frac {b e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.73 \[ \int e^{\frac {e}{c+d x}} (a+b x) \, dx=\frac {c (2 a d+b (-c+e)) e^{\frac {e}{c+d x}}}{2 d^2}+\frac {d e^{\frac {e}{c+d x}} x (2 a d+b (e+d x))-e (2 a d+b (-2 c+e)) \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{2 d^2} \]

[In]

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

[Out]

(c*(2*a*d + b*(-c + e))*E^(e/(c + d*x)))/(2*d^2) + (d*E^(e/(c + d*x))*x*(2*a*d + b*(e + d*x)) - e*(2*a*d + b*(
-2*c + e))*ExpIntegralEi[e/(c + d*x)])/(2*d^2)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.20

method	result	size
derivativedivides	\(-\frac {e \left (a \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )+\frac {b e \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d}-\frac {b c \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d}\right )}{d}\)	\(150\)
default	\(-\frac {e \left (a \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )+\frac {b e \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d}-\frac {b c \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d}\right )}{d}\)	\(150\)
risch	\(a \,{\mathrm e}^{\frac {e}{d x +c}} x +\frac {a \,{\mathrm e}^{\frac {e}{d x +c}} c}{d}+\frac {e a \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d}+\frac {b \,{\mathrm e}^{\frac {e}{d x +c}} x^{2}}{2}-\frac {b \,{\mathrm e}^{\frac {e}{d x +c}} c^{2}}{2 d^{2}}+\frac {e b \,{\mathrm e}^{\frac {e}{d x +c}} x}{2 d}+\frac {e b \,{\mathrm e}^{\frac {e}{d x +c}} c}{2 d^{2}}+\frac {e^{2} b \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2 d^{2}}-\frac {e b c \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d^{2}}\)	\(161\)
parts	\(b \,{\mathrm e}^{\frac {e}{d x +c}} x^{2}+a \,{\mathrm e}^{\frac {e}{d x +c}} x +\frac {b \,{\mathrm e}^{\frac {e}{d x +c}} c x}{d}+\frac {a \,{\mathrm e}^{\frac {e}{d x +c}} c}{d}+\frac {e \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b x}{d}+\frac {e a \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d}-\frac {x^{2} {\mathrm e}^{\frac {e}{d x +c}} b \,c^{2} d^{2}+2 x \,{\mathrm e}^{\frac {e}{d x +c}} b \,c^{3} d -x \,{\mathrm e}^{\frac {e}{d x +c}} b \,c^{2} d e +2 x \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b \,c^{2} d e +{\mathrm e}^{\frac {e}{d x +c}} b \,c^{4}-{\mathrm e}^{\frac {e}{d x +c}} b \,c^{3} e +2 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b \,c^{3} e -\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b \,c^{2} e^{2}}{2 c^{2} d^{2}}\)	\(260\)

[In]

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

[Out]

-1/d*e*(a*(-(d*x+c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c)))+b/d*e*(-1/2*exp(e/(d*x+c))*(d*x+c)^2/e^2-1/2*(d*x+c)/e*
exp(e/(d*x+c))-1/2*Ei(1,-e/(d*x+c)))-b/d*c*(-(d*x+c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c))))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.66 \[ \int e^{\frac {e}{c+d x}} (a+b x) \, dx=-\frac {{\left (b e^{2} - 2 \, {\left (b c - a d\right )} e\right )} {\rm Ei}\left (\frac {e}{d x + c}\right ) - {\left (b d^{2} x^{2} - b c^{2} + 2 \, a c d + b c e + {\left (2 \, a d^{2} + b d e\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{2 \, d^{2}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/2*(b*d*x^2 + (2*a*d + b*e)*x)*e^(e/(d*x + c))/d + integrate(-1/2*(b*c^2*e - (2*a*d^2*e - (2*c*d*e - d*e^2)*b
)*x)*e^(e/(d*x + c))/(d^3*x^2 + 2*c*d^2*x + c^2*d), x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.38 \[ \int e^{\frac {e}{c+d x}} (a+b x) \, dx=\frac {{\left (\frac {2 \, b c e^{4} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} - \frac {2 \, a d e^{4} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} - \frac {b e^{5} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + b e^{3} e^{\left (\frac {e}{d x + c}\right )} - \frac {2 \, b c e^{3} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} + \frac {2 \, a d e^{3} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} + \frac {b e^{4} e^{\left (\frac {e}{d x + c}\right )}}{d x + c}\right )} {\left (d x + c\right )}^{2}}{2 \, d^{2} e^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.22 \[ \int e^{\frac {e}{c+d x}} (a+b x) \, dx=\frac {\frac {{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (2\,a\,c^2\,d-b\,c^3+b\,c^2\,e\right )}{2\,d^2}+x\,{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (2\,a\,c-\frac {\frac {b\,c^2}{2}-b\,c\,e}{d}\right )+x^2\,{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (a\,d+\frac {b\,c}{2}+\frac {b\,e}{2}\right )+\frac {b\,d\,x^3\,{\mathrm {e}}^{\frac {e}{c+d\,x}}}{2}}{c+d\,x}-\frac {\mathrm {ei}\left (\frac {e}{c+d\,x}\right )\,\left (b\,e^2+2\,a\,d\,e-2\,b\,c\,e\right )}{2\,d^2} \]

[In]

int(exp(e/(c + d*x))*(a + b*x),x)

[Out]

((exp(e/(c + d*x))*(2*a*c^2*d - b*c^3 + b*c^2*e))/(2*d^2) + x*exp(e/(c + d*x))*(2*a*c - ((b*c^2)/2 - b*c*e)/d)
+ x^2*exp(e/(c + d*x))*(a*d + (b*c)/2 + (b*e)/2) + (b*d*x^3*exp(e/(c + d*x)))/2)/(c + d*x) - (ei(e/(c + d*x))
*(b*e^2 + 2*a*d*e - 2*b*c*e))/(2*d^2)