Integrand size = 19, antiderivative size = 62 \[ \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx=-\frac {\operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{b}+\frac {e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{b} \]
Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx=\frac {-\operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )+e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (e \left (\frac {b}{-b c+a d}+\frac {1}{c+d x}\right )\right )}{b} \]
(-ExpIntegralEi[e/(c + d*x)] + E^((b*e)/(b*c - a*d))*ExpIntegralEi[e*(b/(- (b*c) + a*d) + (c + d*x)^(-1))])/b
Time = 0.52 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2652, 2639, 2658, 2609}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx\) |
\(\Big \downarrow \) 2652 |
\(\displaystyle \frac {(b c-a d) \int \frac {e^{\frac {e}{c+d x}}}{(a+b x) (c+d x)}dx}{b}+\frac {d \int \frac {e^{\frac {e}{c+d x}}}{c+d x}dx}{b}\) |
\(\Big \downarrow \) 2639 |
\(\displaystyle \frac {(b c-a d) \int \frac {e^{\frac {e}{c+d x}}}{(a+b x) (c+d x)}dx}{b}-\frac {\operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{b}\) |
\(\Big \downarrow \) 2658 |
\(\displaystyle \frac {\int \frac {\exp \left (\frac {b e}{b c-a d}-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right ) (c+d x)}{a+b x}d\frac {a+b x}{c+d x}}{b}-\frac {\operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{b}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle \frac {e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{b}-\frac {\operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{b}\) |
-(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
3.5.6.3.1 Defintions of rubi rules used
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F reeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
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] /; Free Q[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]
Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))/((e_.) + (f_.)*(x_)), x_Symbo l] :> Simp[d/f Int[F^(a + b/(c + d*x))/(c + d*x), x], x] - Simp[(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]
Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))/(((e_.) + (f_.)*(x_))*((g_.) + (h_.)*(x_))), x_Symbol] :> Simp[-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]
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 | \(-\frac {e \left (\frac {d \,{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{b e}-\frac {d \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{b e}\right )}{d}\) | \(79\) |
Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx=\frac {{\rm Ei}\left (-\frac {b d e x + a d e}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x}\right ) e^{\left (\frac {b e}{b c - a d}\right )} - {\rm Ei}\left (\frac {e}{d x + c}\right )}{b} \]
(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
\[ \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx=\int \frac {e^{\frac {e}{c + d x}}}{a + b x}\, dx \]
\[ \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx=\int { \frac {e^{\left (\frac {e}{d x + c}\right )}}{b x + a} \,d x } \]
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 \[ \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx=-\frac {{\left (\frac {2 \, b^{2} c^{2} e^{3} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d}\right )}}{{\left (d x + c\right )}^{2}} - \frac {4 \, a b c d e^{3} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d}\right )}}{{\left (d x + c\right )}^{2}} + \frac {2 \, a^{2} d^{2} e^{3} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d}\right )}}{{\left (d x + c\right )}^{2}} - \frac {2 \, b^{2} c^{2} e^{3} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + \frac {4 \, a b c d e^{3} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} - \frac {2 \, a^{2} d^{2} e^{3} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} - \frac {2 \, b^{2} c e^{4} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + \frac {2 \, a b d e^{4} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} - \frac {b^{2} e^{5} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + b^{2} e^{3} e^{\left (\frac {e}{d x + c}\right )} + \frac {2 \, b^{2} c e^{3} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} - \frac {2 \, a b d e^{3} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} + \frac {b^{2} e^{4} e^{\left (\frac {e}{d x + c}\right )}}{d x + c}\right )} {\left (d x + c\right )}^{2}}{2 \, b^{3} d e^{4}} \]
-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)
Timed out. \[ \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx=\int \frac {{\mathrm {e}}^{\frac {e}{c+d\,x}}}{a+b\,x} \,d x \]