\(\int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx\) [341]

Optimal result

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

-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
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.98 \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx=-\frac {d e^{\frac {e}{c+d x}}}{b (b c-a d)}-\frac {e^{\frac {e}{c+d x}}}{b (a+b x)}-\frac {d e e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {b e}{b c-a d}+\frac {e}{c+d x}\right )}{(-b c+a d)^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2653, 7293, 2009}

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.

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

\(\Big \downarrow \) 2653

\(\displaystyle -\frac {d e \int \frac {e^{\frac {e}{c+d x}}}{(a+b x) (c+d x)^2}dx}{b}-\frac {e^{\frac {e}{c+d x}}}{b (a+b x)}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {d e \int \left (\frac {e^{\frac {e}{c+d x}} b^2}{(b c-a d)^2 (a+b x)}-\frac {d e^{\frac {e}{c+d x}} b}{(b c-a d)^2 (c+d x)}-\frac {d e^{\frac {e}{c+d x}}}{(b c-a d) (c+d x)^2}\right )dx}{b}-\frac {e^{\frac {e}{c+d x}}}{b (a+b x)}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d e \left (\frac {b e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^2}+\frac {e^{\frac {e}{c+d x}}}{e (b c-a d)}\right )}{b}-\frac {e^{\frac {e}{c+d x}}}{b (a+b x)}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2653

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] + S 
imp[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 7293

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.91

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

Input:

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

Output:

-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)

Time = 0.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx=-\frac {{\left (b d e x + a d e\right )} {\rm Ei}\left (-\frac {b d e x + a d e}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x}\right ) e^{\left (\frac {b e}{b c - a d}\right )} + {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x} \] Input:

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

Output:

-((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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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 = 0.14 (sec) , antiderivative size = 330, normalized size of antiderivative = 3.08 \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx=-\frac {{\left (b e^{3} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d}\right )} - \frac {b c e^{3} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d}\right )}}{d x + c} + \frac {a d e^{3} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d}\right )}}{d x + c} + b c e^{2} e^{\left (\frac {e}{d x + c}\right )} - a d e^{2} e^{\left (\frac {e}{d x + c}\right )}\right )} d}{{\left (b^{3} c^{2} e - \frac {b^{3} c^{3} e}{d x + c} - 2 \, a b^{2} c d e + \frac {3 \, a b^{2} c^{2} d e}{d x + c} + a^{2} b d^{2} e - \frac {3 \, a^{2} b c d^{2} e}{d x + c} + \frac {a^{3} d^{3} e}{d x + c}\right )} e} \] Input:

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

Output:

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

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

Output:

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

Reduce [F]

\[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx=\text {too large to display} \] Input:

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

Output:

( - 24*e**(e/(c + d*x))*a**5*c**4*d**5 - 96*e**(e/(c + d*x))*a**5*c**3*d** 
6*x + 24*e**(e/(c + d*x))*a**5*c**3*d**5*e - 144*e**(e/(c + d*x))*a**5*c** 
2*d**7*x**2 + 72*e**(e/(c + d*x))*a**5*c**2*d**6*e*x - 12*e**(e/(c + d*x)) 
*a**5*c**2*d**5*e**2 - 96*e**(e/(c + d*x))*a**5*c*d**8*x**3 + 72*e**(e/(c 
+ d*x))*a**5*c*d**7*e*x**2 - 24*e**(e/(c + d*x))*a**5*c*d**6*e**2*x + 4*e* 
*(e/(c + d*x))*a**5*c*d**5*e**3 - 24*e**(e/(c + d*x))*a**5*d**9*x**4 + 24* 
e**(e/(c + d*x))*a**5*d**8*e*x**3 - 12*e**(e/(c + d*x))*a**5*d**7*e**2*x** 
2 + 4*e**(e/(c + d*x))*a**5*d**6*e**3*x - e**(e/(c + d*x))*a**5*d**5*e**4 
+ 96*e**(e/(c + d*x))*a**4*b*c**5*d**4 + 360*e**(e/(c + d*x))*a**4*b*c**4* 
d**5*x - 108*e**(e/(c + d*x))*a**4*b*c**4*d**4*e + 480*e**(e/(c + d*x))*a* 
*4*b*c**3*d**6*x**2 - 312*e**(e/(c + d*x))*a**4*b*c**3*d**5*e*x + 60*e**(e 
/(c + d*x))*a**4*b*c**3*d**4*e**2 + 240*e**(e/(c + d*x))*a**4*b*c**2*d**7* 
x**3 - 288*e**(e/(c + d*x))*a**4*b*c**2*d**6*e*x**2 + 120*e**(e/(c + d*x)) 
*a**4*b*c**2*d**5*e**2*x - 22*e**(e/(c + d*x))*a**4*b*c**2*d**4*e**3 - 72* 
e**(e/(c + d*x))*a**4*b*c*d**7*e*x**3 + 60*e**(e/(c + d*x))*a**4*b*c*d**6* 
e**2*x**2 - 24*e**(e/(c + d*x))*a**4*b*c*d**5*e**3*x + 6*e**(e/(c + d*x))* 
a**4*b*c*d**4*e**4 - 24*e**(e/(c + d*x))*a**4*b*d**9*x**5 + 12*e**(e/(c + 
d*x))*a**4*b*d**8*e*x**4 - 2*e**(e/(c + d*x))*a**4*b*d**6*e**3*x**2 + e**( 
e/(c + d*x))*a**4*b*d**5*e**4*x - 144*e**(e/(c + d*x))*a**3*b**2*c**6*d**3 
 - 480*e**(e/(c + d*x))*a**3*b**2*c**5*d**4*x + 180*e**(e/(c + d*x))*a*...