\(\int e^{-a-b x} x^2 (a+b x)^3 \, dx\) [96]

Optimal result

Integrand size = 21, antiderivative size = 187 \[ \int e^{-a-b x} x^2 (a+b x)^3 \, dx=-\frac {120 e^{-a-b x}}{b^3}-\frac {e^{-a-b x} x^2 (a+b x)^3}{b}-\frac {24 e^{-a-b x} (3 a+5 b x)}{b^3}-\frac {6 e^{-a-b x} \left (3 a^2+12 a b x+10 b^2 x^2\right )}{b^3}-\frac {e^{-a-b x} x \left (2 a^3+9 a^2 b x+12 a b^2 x^2+5 b^3 x^3\right )}{b^2}-\frac {2 e^{-a-b x} \left (a^3+9 a^2 b x+18 a b^2 x^2+10 b^3 x^3\right )}{b^3} \] Output:

-120*exp(-b*x-a)/b^3-exp(-b*x-a)*x^2*(b*x+a)^3/b-24*exp(-b*x-a)*(5*b*x+3*a 
)/b^3-6*exp(-b*x-a)*(10*b^2*x^2+12*a*b*x+3*a^2)/b^3-exp(-b*x-a)*x*(5*b^3*x 
^3+12*a*b^2*x^2+9*a^2*b*x+2*a^3)/b^2-2*exp(-b*x-a)*(10*b^3*x^3+18*a*b^2*x^ 
2+9*a^2*b*x+a^3)/b^3
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.70 \[ \int e^{-a-b x} x^2 (a+b x)^3 \, dx=e^{-b x} \left (-\frac {2 \left (60+36 a+9 a^2+a^3\right ) e^{-a}}{b^3}-\frac {2 \left (60+36 a+9 a^2+a^3\right ) e^{-a} x}{b^2}-\frac {\left (60+36 a+9 a^2+a^3\right ) e^{-a} x^2}{b}-\left (20+12 a+3 a^2\right ) e^{-a} x^3-(5+3 a) b e^{-a} x^4-b^2 e^{-a} x^5\right ) \] Input:

Integrate[E^(-a - b*x)*x^2*(a + b*x)^3,x]
 

Output:

((-2*(60 + 36*a + 9*a^2 + a^3))/(b^3*E^a) - (2*(60 + 36*a + 9*a^2 + a^3)*x 
)/(b^2*E^a) - ((60 + 36*a + 9*a^2 + a^3)*x^2)/(b*E^a) - ((20 + 12*a + 3*a^ 
2)*x^3)/E^a - ((5 + 3*a)*b*x^4)/E^a - (b^2*x^5)/E^a)/E^(b*x)
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.70, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2626, 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.

\(\displaystyle \int x^2 e^{-a-b x} (a+b x)^3 \, dx\)

\(\Big \downarrow \) 2626

\(\displaystyle \int \left (a^3 x^2 e^{-a-b x}+3 a^2 b x^3 e^{-a-b x}+b^3 x^5 e^{-a-b x}+3 a b^2 x^4 e^{-a-b x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 a^3 e^{-a-b x}}{b^3}-\frac {2 a^3 x e^{-a-b x}}{b^2}-\frac {a^3 x^2 e^{-a-b x}}{b}-\frac {18 a^2 e^{-a-b x}}{b^3}-\frac {18 a^2 x e^{-a-b x}}{b^2}-3 a^2 x^3 e^{-a-b x}-\frac {9 a^2 x^2 e^{-a-b x}}{b}-\frac {72 a e^{-a-b x}}{b^3}-\frac {120 e^{-a-b x}}{b^3}-b^2 x^5 e^{-a-b x}-\frac {72 a x e^{-a-b x}}{b^2}-\frac {120 x e^{-a-b x}}{b^2}-3 a b x^4 e^{-a-b x}-5 b x^4 e^{-a-b x}-12 a x^3 e^{-a-b x}-20 x^3 e^{-a-b x}-\frac {36 a x^2 e^{-a-b x}}{b}-\frac {60 x^2 e^{-a-b x}}{b}\)

Input:

Int[E^(-a - b*x)*x^2*(a + b*x)^3,x]
 

Output:

(-120*E^(-a - b*x))/b^3 - (72*a*E^(-a - b*x))/b^3 - (18*a^2*E^(-a - b*x))/ 
b^3 - (2*a^3*E^(-a - b*x))/b^3 - (120*E^(-a - b*x)*x)/b^2 - (72*a*E^(-a - 
b*x)*x)/b^2 - (18*a^2*E^(-a - b*x)*x)/b^2 - (2*a^3*E^(-a - b*x)*x)/b^2 - ( 
60*E^(-a - b*x)*x^2)/b - (36*a*E^(-a - b*x)*x^2)/b - (9*a^2*E^(-a - b*x)*x 
^2)/b - (a^3*E^(-a - b*x)*x^2)/b - 20*E^(-a - b*x)*x^3 - 12*a*E^(-a - b*x) 
*x^3 - 3*a^2*E^(-a - b*x)*x^3 - 5*b*E^(-a - b*x)*x^4 - 3*a*b*E^(-a - b*x)* 
x^4 - b^2*E^(-a - b*x)*x^5
 

Defintions of rubi rules used

rule 2009

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

rule 2626

Int[(F_)^(v_)*(Px_), x_Symbol] :> Int[ExpandIntegrand[F^v, Px, x], x] /; Fr 
eeQ[F, x] && PolynomialQ[Px, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]
 

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.76

method	result	size
gosper	\(-\frac {\left (b^{5} x^{5}+3 a \,b^{4} x^{4}+3 a^{2} b^{3} x^{3}+5 b^{4} x^{4}+a^{3} b^{2} x^{2}+12 a \,b^{3} x^{3}+9 a^{2} b^{2} x^{2}+20 b^{3} x^{3}+2 a^{3} b x +36 a \,b^{2} x^{2}+18 a^{2} b x +60 b^{2} x^{2}+2 a^{3}+72 a b x +18 a^{2}+120 b x +72 a +120\right ) {\mathrm e}^{-b x -a}}{b^{3}}\)	\(143\)
risch	\(-\frac {\left (b^{5} x^{5}+3 a \,b^{4} x^{4}+3 a^{2} b^{3} x^{3}+5 b^{4} x^{4}+a^{3} b^{2} x^{2}+12 a \,b^{3} x^{3}+9 a^{2} b^{2} x^{2}+20 b^{3} x^{3}+2 a^{3} b x +36 a \,b^{2} x^{2}+18 a^{2} b x +60 b^{2} x^{2}+2 a^{3}+72 a b x +18 a^{2}+120 b x +72 a +120\right ) {\mathrm e}^{-b x -a}}{b^{3}}\)	\(143\)
orering	\(-\frac {\left (b^{5} x^{5}+3 a \,b^{4} x^{4}+3 a^{2} b^{3} x^{3}+5 b^{4} x^{4}+a^{3} b^{2} x^{2}+12 a \,b^{3} x^{3}+9 a^{2} b^{2} x^{2}+20 b^{3} x^{3}+2 a^{3} b x +36 a \,b^{2} x^{2}+18 a^{2} b x +60 b^{2} x^{2}+2 a^{3}+72 a b x +18 a^{2}+120 b x +72 a +120\right ) {\mathrm e}^{-b x -a}}{b^{3}}\)	\(143\)
norman	\(\left (-3 a b -5 b \right ) x^{4} {\mathrm e}^{-b x -a}+\left (-3 a^{2}-12 a -20\right ) x^{3} {\mathrm e}^{-b x -a}-b^{2} x^{5} {\mathrm e}^{-b x -a}-\frac {2 \left (a^{3}+9 a^{2}+36 a +60\right ) {\mathrm e}^{-b x -a}}{b^{3}}-\frac {2 \left (a^{3}+9 a^{2}+36 a +60\right ) x \,{\mathrm e}^{-b x -a}}{b^{2}}-\frac {\left (a^{3}+9 a^{2}+36 a +60\right ) x^{2} {\mathrm e}^{-b x -a}}{b}\)	\(148\)
meijerg	\(\frac {{\mathrm e}^{-a} \left (120-\frac {\left (6 b^{5} x^{5}+30 b^{4} x^{4}+120 b^{3} x^{3}+360 b^{2} x^{2}+720 b x +720\right ) {\mathrm e}^{-b x}}{6}\right )}{b^{3}}+\frac {3 \,{\mathrm e}^{-a} a \left (24-\frac {\left (5 b^{4} x^{4}+20 b^{3} x^{3}+60 b^{2} x^{2}+120 b x +120\right ) {\mathrm e}^{-b x}}{5}\right )}{b^{3}}+\frac {3 \,{\mathrm e}^{-a} a^{2} \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b^{3}}+\frac {{\mathrm e}^{-a} a^{3} \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{3}}\)	\(183\)
derivativedivides	\(\frac {\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}+a^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )+2 a \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b^{3}}\)	\(291\)
default	\(\frac {\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}+a^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )+2 a \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b^{3}}\)	\(291\)
parallelrisch	\(-\frac {b^{5} {\mathrm e}^{-b x -a} x^{5}+3 x^{4} {\mathrm e}^{-b x -a} a \,b^{4}+5 b^{4} x^{4} {\mathrm e}^{-b x -a}+3 x^{3} {\mathrm e}^{-b x -a} a^{2} b^{3}+12 a \,b^{3} x^{3} {\mathrm e}^{-b x -a}+x^{2} {\mathrm e}^{-b x -a} a^{3} b^{2}+20 \,{\mathrm e}^{-b x -a} x^{3} b^{3}+9 a^{2} b^{2} x^{2} {\mathrm e}^{-b x -a}+36 x^{2} {\mathrm e}^{-b x -a} a \,b^{2}+2 a^{3} b x \,{\mathrm e}^{-b x -a}+60 b^{2} {\mathrm e}^{-b x -a} x^{2}+18 x \,{\mathrm e}^{-b x -a} a^{2} b +72 a b \,{\mathrm e}^{-b x -a} x +2 \,{\mathrm e}^{-b x -a} a^{3}+120 b \,{\mathrm e}^{-b x -a} x +18 a^{2} {\mathrm e}^{-b x -a}+72 a \,{\mathrm e}^{-b x -a}+120 \,{\mathrm e}^{-b x -a}}{b^{3}}\)	\(297\)
parts	\(-b^{2} x^{5} {\mathrm e}^{-b x -a}-3 \,{\mathrm e}^{-b x -a} x^{4} b a -3 \,{\mathrm e}^{-b x -a} x^{3} a^{2}-\frac {{\mathrm e}^{-b x -a} x^{2} a^{3}}{b}-\frac {\frac {5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+120 \,{\mathrm e}^{-b x -a}}{b}+\frac {3 a^{2} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b}+\frac {8 a \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b}}{b^{2}}\)	\(313\)

Input:

int(exp(-b*x-a)*x^2*(b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-(b^5*x^5+3*a*b^4*x^4+3*a^2*b^3*x^3+5*b^4*x^4+a^3*b^2*x^2+12*a*b^3*x^3+9*a 
^2*b^2*x^2+20*b^3*x^3+2*a^3*b*x+36*a*b^2*x^2+18*a^2*b*x+60*b^2*x^2+2*a^3+7 
2*a*b*x+18*a^2+120*b*x+72*a+120)*exp(-b*x-a)/b^3
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.55 \[ \int e^{-a-b x} x^2 (a+b x)^3 \, dx=-\frac {{\left (b^{5} x^{5} + {\left (3 \, a + 5\right )} b^{4} x^{4} + {\left (3 \, a^{2} + 12 \, a + 20\right )} b^{3} x^{3} + {\left (a^{3} + 9 \, a^{2} + 36 \, a + 60\right )} b^{2} x^{2} + 2 \, a^{3} + 2 \, {\left (a^{3} + 9 \, a^{2} + 36 \, a + 60\right )} b x + 18 \, a^{2} + 72 \, a + 120\right )} e^{\left (-b x - a\right )}}{b^{3}} \] Input:

integrate(exp(-b*x-a)*x^2*(b*x+a)^3,x, algorithm="fricas")
 

Output:

-(b^5*x^5 + (3*a + 5)*b^4*x^4 + (3*a^2 + 12*a + 20)*b^3*x^3 + (a^3 + 9*a^2 
 + 36*a + 60)*b^2*x^2 + 2*a^3 + 2*(a^3 + 9*a^2 + 36*a + 60)*b*x + 18*a^2 + 
 72*a + 120)*e^(-b*x - a)/b^3
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.05 \[ \int e^{-a-b x} x^2 (a+b x)^3 \, dx=\begin {cases} \frac {\left (- a^{3} b^{2} x^{2} - 2 a^{3} b x - 2 a^{3} - 3 a^{2} b^{3} x^{3} - 9 a^{2} b^{2} x^{2} - 18 a^{2} b x - 18 a^{2} - 3 a b^{4} x^{4} - 12 a b^{3} x^{3} - 36 a b^{2} x^{2} - 72 a b x - 72 a - b^{5} x^{5} - 5 b^{4} x^{4} - 20 b^{3} x^{3} - 60 b^{2} x^{2} - 120 b x - 120\right ) e^{- a - b x}}{b^{3}} & \text {for}\: b^{3} \neq 0 \\\frac {a^{3} x^{3}}{3} + \frac {3 a^{2} b x^{4}}{4} + \frac {3 a b^{2} x^{5}}{5} + \frac {b^{3} x^{6}}{6} & \text {otherwise} \end {cases} \] Input:

integrate(exp(-b*x-a)*x**2*(b*x+a)**3,x)
 

Output:

Piecewise(((-a**3*b**2*x**2 - 2*a**3*b*x - 2*a**3 - 3*a**2*b**3*x**3 - 9*a 
**2*b**2*x**2 - 18*a**2*b*x - 18*a**2 - 3*a*b**4*x**4 - 12*a*b**3*x**3 - 3 
6*a*b**2*x**2 - 72*a*b*x - 72*a - b**5*x**5 - 5*b**4*x**4 - 20*b**3*x**3 - 
 60*b**2*x**2 - 120*b*x - 120)*exp(-a - b*x)/b**3, Ne(b**3, 0)), (a**3*x** 
3/3 + 3*a**2*b*x**4/4 + 3*a*b**2*x**5/5 + b**3*x**6/6, True))
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.88 \[ \int e^{-a-b x} x^2 (a+b x)^3 \, dx=-\frac {{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{3} e^{\left (-b x - a\right )}}{b^{3}} - \frac {3 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {3 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a e^{\left (-b x - a\right )}}{b^{3}} - \frac {{\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} e^{\left (-b x - a\right )}}{b^{3}} \] Input:

integrate(exp(-b*x-a)*x^2*(b*x+a)^3,x, algorithm="maxima")
 

Output:

-(b^2*x^2 + 2*b*x + 2)*a^3*e^(-b*x - a)/b^3 - 3*(b^3*x^3 + 3*b^2*x^2 + 6*b 
*x + 6)*a^2*e^(-b*x - a)/b^3 - 3*(b^4*x^4 + 4*b^3*x^3 + 12*b^2*x^2 + 24*b* 
x + 24)*a*e^(-b*x - a)/b^3 - (b^5*x^5 + 5*b^4*x^4 + 20*b^3*x^3 + 60*b^2*x^ 
2 + 120*b*x + 120)*e^(-b*x - a)/b^3
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.87 \[ \int e^{-a-b x} x^2 (a+b x)^3 \, dx=-\frac {{\left (b^{8} x^{5} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{3} + 5 \, b^{7} x^{4} + a^{3} b^{5} x^{2} + 12 \, a b^{6} x^{3} + 9 \, a^{2} b^{5} x^{2} + 20 \, b^{6} x^{3} + 2 \, a^{3} b^{4} x + 36 \, a b^{5} x^{2} + 18 \, a^{2} b^{4} x + 60 \, b^{5} x^{2} + 2 \, a^{3} b^{3} + 72 \, a b^{4} x + 18 \, a^{2} b^{3} + 120 \, b^{4} x + 72 \, a b^{3} + 120 \, b^{3}\right )} e^{\left (-b x - a\right )}}{b^{6}} \] Input:

integrate(exp(-b*x-a)*x^2*(b*x+a)^3,x, algorithm="giac")
 

Output:

-(b^8*x^5 + 3*a*b^7*x^4 + 3*a^2*b^6*x^3 + 5*b^7*x^4 + a^3*b^5*x^2 + 12*a*b 
^6*x^3 + 9*a^2*b^5*x^2 + 20*b^6*x^3 + 2*a^3*b^4*x + 36*a*b^5*x^2 + 18*a^2* 
b^4*x + 60*b^5*x^2 + 2*a^3*b^3 + 72*a*b^4*x + 18*a^2*b^3 + 120*b^4*x + 72* 
a*b^3 + 120*b^3)*e^(-b*x - a)/b^6
 

Mupad [B] (verification not implemented)

Time = 22.56 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.67 \[ \int e^{-a-b x} x^2 (a+b x)^3 \, dx=-x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (3\,a^2+3\,a\,b\,x+12\,a+b^2\,x^2+5\,b\,x+20\right )-\frac {2\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+9\,a^2+36\,a+60\right )}{b^3}-\frac {2\,x\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+9\,a^2+36\,a+60\right )}{b^2}-\frac {x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+9\,a^2+36\,a+60\right )}{b} \] Input:

int(x^2*exp(- a - b*x)*(a + b*x)^3,x)
 

Output:

- x^3*exp(- a - b*x)*(12*a + 5*b*x + 3*a^2 + b^2*x^2 + 3*a*b*x + 20) - (2* 
exp(- a - b*x)*(36*a + 9*a^2 + a^3 + 60))/b^3 - (2*x*exp(- a - b*x)*(36*a 
+ 9*a^2 + a^3 + 60))/b^2 - (x^2*exp(- a - b*x)*(36*a + 9*a^2 + a^3 + 60))/ 
b
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.76 \[ \int e^{-a-b x} x^2 (a+b x)^3 \, dx=\frac {-b^{5} x^{5}-3 a \,b^{4} x^{4}-3 a^{2} b^{3} x^{3}-5 b^{4} x^{4}-a^{3} b^{2} x^{2}-12 a \,b^{3} x^{3}-9 a^{2} b^{2} x^{2}-20 b^{3} x^{3}-2 a^{3} b x -36 a \,b^{2} x^{2}-18 a^{2} b x -60 b^{2} x^{2}-2 a^{3}-72 a b x -18 a^{2}-120 b x -72 a -120}{e^{b x +a} b^{3}} \] Input:

int(exp(-b*x-a)*x^2*(b*x+a)^3,x)
 

Output:

( - a**3*b**2*x**2 - 2*a**3*b*x - 2*a**3 - 3*a**2*b**3*x**3 - 9*a**2*b**2* 
x**2 - 18*a**2*b*x - 18*a**2 - 3*a*b**4*x**4 - 12*a*b**3*x**3 - 36*a*b**2* 
x**2 - 72*a*b*x - 72*a - b**5*x**5 - 5*b**4*x**4 - 20*b**3*x**3 - 60*b**2* 
x**2 - 120*b*x - 120)/(e**(a + b*x)*b**3)