\(\int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx\) [407]

Optimal result

Integrand size = 19, antiderivative size = 107 \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx=-\frac {d e^{\frac {e}{c+d x}}}{b (b c-a d)}-\frac {e^{\frac {e}{c+d x}}}{b (a+b x)}-\frac {d e e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2255, 6874, 2254, 2241, 2260, 2209, 2240} \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx=-\frac {d e e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^2}-\frac {d e^{\frac {e}{c+d x}}}{b (b c-a d)}-\frac {e^{\frac {e}{c+d x}}}{b (a+b x)} \]

[In]

Int[E^(e/(c + d*x))/(a + b*x)^2,x]

[Out]

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

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 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 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 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 2255

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))*((e_.) + (f_.)*(x_))^(m_), x_Symbol] :> Simp[(e + f*x)^(m + 1)*(
F^(a + b/(c + d*x))/(f*(m + 1))), x] + Dist[b*d*(Log[F]/(f*(m + 1))), Int[(e + f*x)^(m + 1)*(F^(a + b/(c + d*x
))/(c + d*x)^2), x], x] /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && ILtQ[m, -1]

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]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.98 \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx=-\frac {d e^{\frac {e}{c+d x}}}{b (b c-a d)}-\frac {e^{\frac {e}{c+d x}}}{b (a+b x)}-\frac {d e e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {b e}{b c-a d}+\frac {e}{c+d x}\right )}{(-b c+a d)^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(-\frac {d e \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{\frac {e}{d x +c}+\frac {b e}{a d -c b}}-{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )\right )}{\left (a d -c b \right )^{2}}\)	\(97\)
default	\(-\frac {d e \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{\frac {e}{d x +c}+\frac {b e}{a d -c b}}-{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )\right )}{\left (a d -c b \right )^{2}}\)	\(97\)
risch	\(\frac {d e \,{\mathrm e}^{\frac {e}{d x +c}}}{\left (a d -c b \right )^{2} \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )}+\frac {d e \,{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{\left (a d -c b \right )^{2}}\)	\(105\)

[In]

int(exp(e/(d*x+c))/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx=-\frac {{\left (b d e x + a d e\right )} {\rm Ei}\left (-\frac {b d e x + a d e}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x}\right ) e^{\left (\frac {b e}{b c - a d}\right )} + {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x} \]

[In]

integrate(exp(e/(d*x+c))/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx=\int \frac {e^{\frac {e}{c + d x}}}{\left (a + b x\right )^{2}}\, dx \]

[In]

integrate(exp(e/(d*x+c))/(b*x+a)**2,x)

[Out]

Integral(exp(e/(c + d*x))/(a + b*x)**2, x)

Maxima [F]

\[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx=\int { \frac {e^{\left (\frac {e}{d x + c}\right )}}{{\left (b x + a\right )}^{2}} \,d x } \]

[In]

integrate(exp(e/(d*x+c))/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (104) = 208\).

Time = 0.30 (sec) , antiderivative size = 330, normalized size of antiderivative = 3.08 \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx=-\frac {{\left (b e^{3} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d}\right )} - \frac {b c e^{3} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d}\right )}}{d x + c} + \frac {a d e^{3} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d}\right )}}{d x + c} + b c e^{2} e^{\left (\frac {e}{d x + c}\right )} - a d e^{2} e^{\left (\frac {e}{d x + c}\right )}\right )} d}{{\left (b^{3} c^{2} e - \frac {b^{3} c^{3} e}{d x + c} - 2 \, a b^{2} c d e + \frac {3 \, a b^{2} c^{2} d e}{d x + c} + a^{2} b d^{2} e - \frac {3 \, a^{2} b c d^{2} e}{d x + c} + \frac {a^{3} d^{3} e}{d x + c}\right )} e} \]

[In]

integrate(exp(e/(d*x+c))/(b*x+a)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx=\int \frac {{\mathrm {e}}^{\frac {e}{c+d\,x}}}{{\left (a+b\,x\right )}^2} \,d x \]

[In]

int(exp(e/(c + d*x))/(a + b*x)^2,x)

[Out]

int(exp(e/(c + d*x))/(a + b*x)^2, x)