\(\int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx\) [114]

Optimal result

Integrand size = 25, antiderivative size = 498 \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=-\frac {720 d^2 e^{-a-b x}}{b^3}-\frac {e^{-a-b x} (a+b x)^4 (c+d x)^2}{b}-\frac {240 d e^{-a-b x} (b c+2 a d+3 b d x)}{b^3}-\frac {24 e^{-a-b x} \left (b^2 c^2+8 a b c d+6 a^2 d^2+10 b d (b c+2 a d) x+15 b^2 d^2 x^2\right )}{b^3}-\frac {24 e^{-a-b x} \left (a \left (b^2 c^2+3 a b c d+a^2 d^2\right )+b \left (b^2 c^2+8 a b c d+6 a^2 d^2\right ) x+5 b^2 d (b c+2 a d) x^2+5 b^3 d^2 x^3\right )}{b^3}-\frac {2 e^{-a-b x} \left (a^2 \left (6 b^2 c^2+8 a b c d+a^2 d^2\right )+12 a b \left (b^2 c^2+3 a b c d+a^2 d^2\right ) x+6 b^2 \left (b^2 c^2+8 a b c d+6 a^2 d^2\right ) x^2+20 b^3 d (b c+2 a d) x^3+15 b^4 d^2 x^4\right )}{b^3}-\frac {2 e^{-a-b x} \left (a^3 c (2 b c+a d)+a^2 \left (6 b^2 c^2+8 a b c d+a^2 d^2\right ) x+6 a b \left (b^2 c^2+3 a b c d+a^2 d^2\right ) x^2+2 b^2 \left (b^2 c^2+8 a b c d+6 a^2 d^2\right ) x^3+5 b^3 d (b c+2 a d) x^4+3 b^4 d^2 x^5\right )}{b^2} \] Output:

-720*d^2*exp(-b*x-a)/b^3-exp(-b*x-a)*(b*x+a)^4*(d*x+c)^2/b-240*d*exp(-b*x- 
a)*(3*b*d*x+2*a*d+b*c)/b^3-24*exp(-b*x-a)*(b^2*c^2+8*a*b*c*d+6*a^2*d^2+10* 
b*d*(2*a*d+b*c)*x+15*b^2*d^2*x^2)/b^3-24*exp(-b*x-a)*(a*(a^2*d^2+3*a*b*c*d 
+b^2*c^2)+b*(6*a^2*d^2+8*a*b*c*d+b^2*c^2)*x+5*b^2*d*(2*a*d+b*c)*x^2+5*b^3* 
d^2*x^3)/b^3-2*exp(-b*x-a)*(a^2*(a^2*d^2+8*a*b*c*d+6*b^2*c^2)+12*a*b*(a^2* 
d^2+3*a*b*c*d+b^2*c^2)*x+6*b^2*(6*a^2*d^2+8*a*b*c*d+b^2*c^2)*x^2+20*b^3*d* 
(2*a*d+b*c)*x^3+15*b^4*d^2*x^4)/b^3-2*exp(-b*x-a)*(a^3*c*(a*d+2*b*c)+a^2*( 
a^2*d^2+8*a*b*c*d+6*b^2*c^2)*x+6*a*b*(a^2*d^2+3*a*b*c*d+b^2*c^2)*x^2+2*b^2 
*(6*a^2*d^2+8*a*b*c*d+b^2*c^2)*x^3+5*b^3*d*(2*a*d+b*c)*x^4+3*b^4*d^2*x^5)/ 
b^2
 

Mathematica [A] (verified)

Time = 2.67 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.64 \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=\frac {e^{-a-b x} \left (-2 \left (360+240 a+72 a^2+12 a^3+a^4\right ) d^2-b^6 x^4 (c+d x)^2-2 b^5 x^3 (c+d x) (2 (1+a) c+(3+2 a) d x)-2 b d \left (\left (120+96 a+36 a^2+8 a^3+a^4\right ) c+\left (360+240 a+72 a^2+12 a^3+a^4\right ) d x\right )-2 b^4 x^2 \left (3 \left (2+2 a+a^2\right ) c^2+2 \left (10+8 a+3 a^2\right ) c d x+\left (15+10 a+3 a^2\right ) d^2 x^2\right )-4 b^3 x \left (\left (6+6 a+3 a^2+a^3\right ) c^2+\left (30+24 a+9 a^2+2 a^3\right ) c d x+\left (30+20 a+6 a^2+a^3\right ) d^2 x^2\right )-b^2 \left (\left (24+24 a+12 a^2+4 a^3+a^4\right ) c^2+2 \left (120+96 a+36 a^2+8 a^3+a^4\right ) c d x+\left (360+240 a+72 a^2+12 a^3+a^4\right ) d^2 x^2\right )\right )}{b^3} \] Input:

Integrate[E^(-a - b*x)*(a + b*x)^4*(c + d*x)^2,x]
 

Output:

(E^(-a - b*x)*(-2*(360 + 240*a + 72*a^2 + 12*a^3 + a^4)*d^2 - b^6*x^4*(c + 
 d*x)^2 - 2*b^5*x^3*(c + d*x)*(2*(1 + a)*c + (3 + 2*a)*d*x) - 2*b*d*((120 
+ 96*a + 36*a^2 + 8*a^3 + a^4)*c + (360 + 240*a + 72*a^2 + 12*a^3 + a^4)*d 
*x) - 2*b^4*x^2*(3*(2 + 2*a + a^2)*c^2 + 2*(10 + 8*a + 3*a^2)*c*d*x + (15 
+ 10*a + 3*a^2)*d^2*x^2) - 4*b^3*x*((6 + 6*a + 3*a^2 + a^3)*c^2 + (30 + 24 
*a + 9*a^2 + 2*a^3)*c*d*x + (30 + 20*a + 6*a^2 + a^3)*d^2*x^2) - b^2*((24 
+ 24*a + 12*a^2 + 4*a^3 + a^4)*c^2 + 2*(120 + 96*a + 36*a^2 + 8*a^3 + a^4) 
*c*d*x + (360 + 240*a + 72*a^2 + 12*a^3 + a^4)*d^2*x^2)))/b^3
 

Rubi [A] (verified)

Time = 1.59 (sec) , antiderivative size = 495, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.

\(\displaystyle \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx\)

\(\Big \downarrow \) 2626

\(\displaystyle \int \left (\frac {2 d e^{-a-b x} (a+b x)^5 (b c-a d)}{b^2}+\frac {e^{-a-b x} (a+b x)^4 (b c-a d)^2}{b^2}+\frac {d^2 e^{-a-b x} (a+b x)^6}{b^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 d e^{-a-b x} (a+b x)^5 (b c-a d)}{b^3}-\frac {e^{-a-b x} (a+b x)^4 (b c-a d)^2}{b^3}-\frac {10 d e^{-a-b x} (a+b x)^4 (b c-a d)}{b^3}-\frac {4 e^{-a-b x} (a+b x)^3 (b c-a d)^2}{b^3}-\frac {40 d e^{-a-b x} (a+b x)^3 (b c-a d)}{b^3}-\frac {12 e^{-a-b x} (a+b x)^2 (b c-a d)^2}{b^3}-\frac {120 d e^{-a-b x} (a+b x)^2 (b c-a d)}{b^3}-\frac {24 e^{-a-b x} (a+b x) (b c-a d)^2}{b^3}-\frac {240 d e^{-a-b x} (a+b x) (b c-a d)}{b^3}-\frac {24 e^{-a-b x} (b c-a d)^2}{b^3}-\frac {240 d e^{-a-b x} (b c-a d)}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {120 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {360 d^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {720 d^2 e^{-a-b x} (a+b x)}{b^3}-\frac {720 d^2 e^{-a-b x}}{b^3}\)

Input:

Int[E^(-a - b*x)*(a + b*x)^4*(c + d*x)^2,x]
 

Output:

(-720*d^2*E^(-a - b*x))/b^3 - (240*d*(b*c - a*d)*E^(-a - b*x))/b^3 - (24*( 
b*c - a*d)^2*E^(-a - b*x))/b^3 - (720*d^2*E^(-a - b*x)*(a + b*x))/b^3 - (2 
40*d*(b*c - a*d)*E^(-a - b*x)*(a + b*x))/b^3 - (24*(b*c - a*d)^2*E^(-a - b 
*x)*(a + b*x))/b^3 - (360*d^2*E^(-a - b*x)*(a + b*x)^2)/b^3 - (120*d*(b*c 
- a*d)*E^(-a - b*x)*(a + b*x)^2)/b^3 - (12*(b*c - a*d)^2*E^(-a - b*x)*(a + 
 b*x)^2)/b^3 - (120*d^2*E^(-a - b*x)*(a + b*x)^3)/b^3 - (40*d*(b*c - a*d)* 
E^(-a - b*x)*(a + b*x)^3)/b^3 - (4*(b*c - a*d)^2*E^(-a - b*x)*(a + b*x)^3) 
/b^3 - (30*d^2*E^(-a - b*x)*(a + b*x)^4)/b^3 - (10*d*(b*c - a*d)*E^(-a - b 
*x)*(a + b*x)^4)/b^3 - ((b*c - a*d)^2*E^(-a - b*x)*(a + b*x)^4)/b^3 - (6*d 
^2*E^(-a - b*x)*(a + b*x)^5)/b^3 - (2*d*(b*c - a*d)*E^(-a - b*x)*(a + b*x) 
^5)/b^3 - (d^2*E^(-a - b*x)*(a + b*x)^6)/b^3
 

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 = 0.11 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.12

method	result	size
norman	\(\left (-4 a \,b^{2} d^{2}-2 c d \,b^{3}-6 b^{2} d^{2}\right ) x^{5} {\mathrm e}^{-b x -a}+\left (-6 a^{2} b \,d^{2}-8 a \,b^{2} c d -c^{2} b^{3}-20 a b \,d^{2}-10 b^{2} c d -30 b \,d^{2}\right ) x^{4} {\mathrm e}^{-b x -a}+\left (-4 a^{3} d^{2}-12 a^{2} b c d -4 a \,b^{2} c^{2}-24 a^{2} d^{2}-32 a b c d -4 b^{2} c^{2}-80 a \,d^{2}-40 b c d -120 d^{2}\right ) x^{3} {\mathrm e}^{-b x -a}-\frac {\left (c^{2} a^{4} b^{2}+2 c d \,a^{4} b +4 c^{2} a^{3} b^{2}+2 d^{2} a^{4}+16 c d \,a^{3} b +12 c^{2} a^{2} b^{2}+24 a^{3} d^{2}+72 a^{2} b c d +24 a \,b^{2} c^{2}+144 a^{2} d^{2}+192 a b c d +24 b^{2} c^{2}+480 a \,d^{2}+240 b c d +720 d^{2}\right ) {\mathrm e}^{-b x -a}}{b^{3}}-d^{2} b^{3} x^{6} {\mathrm e}^{-b x -a}-\frac {\left (d^{2} a^{4}+8 c d \,a^{3} b +6 c^{2} a^{2} b^{2}+12 a^{3} d^{2}+36 a^{2} b c d +12 a \,b^{2} c^{2}+72 a^{2} d^{2}+96 a b c d +12 b^{2} c^{2}+240 a \,d^{2}+120 b c d +360 d^{2}\right ) x^{2} {\mathrm e}^{-b x -a}}{b}-\frac {2 \left (c d \,a^{4} b +2 c^{2} a^{3} b^{2}+d^{2} a^{4}+8 c d \,a^{3} b +6 c^{2} a^{2} b^{2}+12 a^{3} d^{2}+36 a^{2} b c d +12 a \,b^{2} c^{2}+72 a^{2} d^{2}+96 a b c d +12 b^{2} c^{2}+240 a \,d^{2}+120 b c d +360 d^{2}\right ) x \,{\mathrm e}^{-b x -a}}{b^{2}}\)	\(560\)
gosper	\(-\frac {\left (d^{2} b^{6} x^{6}+4 a \,b^{5} d^{2} x^{5}+2 b^{6} c d \,x^{5}+6 a^{2} b^{4} d^{2} x^{4}+8 a \,b^{5} c d \,x^{4}+b^{6} c^{2} x^{4}+6 b^{5} d^{2} x^{5}+4 a^{3} b^{3} d^{2} x^{3}+12 a^{2} b^{4} c d \,x^{3}+4 a \,b^{5} c^{2} x^{3}+20 a \,b^{4} d^{2} x^{4}+10 b^{5} c d \,x^{4}+a^{4} b^{2} d^{2} x^{2}+8 a^{3} b^{3} c d \,x^{2}+6 a^{2} b^{4} c^{2} x^{2}+24 a^{2} b^{3} d^{2} x^{3}+32 a \,b^{4} c d \,x^{3}+4 b^{5} c^{2} x^{3}+30 b^{4} d^{2} x^{4}+2 a^{4} b^{2} c d x +4 a^{3} b^{3} c^{2} x +12 a^{3} b^{2} d^{2} x^{2}+36 a^{2} b^{3} c d \,x^{2}+12 a \,b^{4} c^{2} x^{2}+80 a \,b^{3} d^{2} x^{3}+40 b^{4} c d \,x^{3}+c^{2} a^{4} b^{2}+2 a^{4} b \,d^{2} x +16 a^{3} b^{2} c d x +12 a^{2} b^{3} c^{2} x +72 a^{2} b^{2} d^{2} x^{2}+96 a \,b^{3} c d \,x^{2}+12 b^{4} c^{2} x^{2}+120 b^{3} d^{2} x^{3}+2 c d \,a^{4} b +4 c^{2} a^{3} b^{2}+24 a^{3} b \,d^{2} x +72 a^{2} b^{2} c d x +24 a \,b^{3} c^{2} x +240 a \,b^{2} d^{2} x^{2}+120 b^{3} c d \,x^{2}+2 d^{2} a^{4}+16 c d \,a^{3} b +12 c^{2} a^{2} b^{2}+144 a^{2} b \,d^{2} x +192 a \,b^{2} c d x +24 b^{3} c^{2} x +360 b^{2} d^{2} x^{2}+24 a^{3} d^{2}+72 a^{2} b c d +24 a \,b^{2} c^{2}+480 a b \,d^{2} x +240 x \,b^{2} d c +144 a^{2} d^{2}+192 a b c d +24 b^{2} c^{2}+720 b \,d^{2} x +480 a \,d^{2}+240 b c d +720 d^{2}\right ) {\mathrm e}^{-b x -a}}{b^{3}}\)	\(640\)
risch	\(-\frac {\left (d^{2} b^{6} x^{6}+4 a \,b^{5} d^{2} x^{5}+2 b^{6} c d \,x^{5}+6 a^{2} b^{4} d^{2} x^{4}+8 a \,b^{5} c d \,x^{4}+b^{6} c^{2} x^{4}+6 b^{5} d^{2} x^{5}+4 a^{3} b^{3} d^{2} x^{3}+12 a^{2} b^{4} c d \,x^{3}+4 a \,b^{5} c^{2} x^{3}+20 a \,b^{4} d^{2} x^{4}+10 b^{5} c d \,x^{4}+a^{4} b^{2} d^{2} x^{2}+8 a^{3} b^{3} c d \,x^{2}+6 a^{2} b^{4} c^{2} x^{2}+24 a^{2} b^{3} d^{2} x^{3}+32 a \,b^{4} c d \,x^{3}+4 b^{5} c^{2} x^{3}+30 b^{4} d^{2} x^{4}+2 a^{4} b^{2} c d x +4 a^{3} b^{3} c^{2} x +12 a^{3} b^{2} d^{2} x^{2}+36 a^{2} b^{3} c d \,x^{2}+12 a \,b^{4} c^{2} x^{2}+80 a \,b^{3} d^{2} x^{3}+40 b^{4} c d \,x^{3}+c^{2} a^{4} b^{2}+2 a^{4} b \,d^{2} x +16 a^{3} b^{2} c d x +12 a^{2} b^{3} c^{2} x +72 a^{2} b^{2} d^{2} x^{2}+96 a \,b^{3} c d \,x^{2}+12 b^{4} c^{2} x^{2}+120 b^{3} d^{2} x^{3}+2 c d \,a^{4} b +4 c^{2} a^{3} b^{2}+24 a^{3} b \,d^{2} x +72 a^{2} b^{2} c d x +24 a \,b^{3} c^{2} x +240 a \,b^{2} d^{2} x^{2}+120 b^{3} c d \,x^{2}+2 d^{2} a^{4}+16 c d \,a^{3} b +12 c^{2} a^{2} b^{2}+144 a^{2} b \,d^{2} x +192 a \,b^{2} c d x +24 b^{3} c^{2} x +360 b^{2} d^{2} x^{2}+24 a^{3} d^{2}+72 a^{2} b c d +24 a \,b^{2} c^{2}+480 a b \,d^{2} x +240 x \,b^{2} d c +144 a^{2} d^{2}+192 a b c d +24 b^{2} c^{2}+720 b \,d^{2} x +480 a \,d^{2}+240 b c d +720 d^{2}\right ) {\mathrm e}^{-b x -a}}{b^{3}}\)	\(640\)
orering	\(-\frac {\left (d^{2} b^{6} x^{6}+4 a \,b^{5} d^{2} x^{5}+2 b^{6} c d \,x^{5}+6 a^{2} b^{4} d^{2} x^{4}+8 a \,b^{5} c d \,x^{4}+b^{6} c^{2} x^{4}+6 b^{5} d^{2} x^{5}+4 a^{3} b^{3} d^{2} x^{3}+12 a^{2} b^{4} c d \,x^{3}+4 a \,b^{5} c^{2} x^{3}+20 a \,b^{4} d^{2} x^{4}+10 b^{5} c d \,x^{4}+a^{4} b^{2} d^{2} x^{2}+8 a^{3} b^{3} c d \,x^{2}+6 a^{2} b^{4} c^{2} x^{2}+24 a^{2} b^{3} d^{2} x^{3}+32 a \,b^{4} c d \,x^{3}+4 b^{5} c^{2} x^{3}+30 b^{4} d^{2} x^{4}+2 a^{4} b^{2} c d x +4 a^{3} b^{3} c^{2} x +12 a^{3} b^{2} d^{2} x^{2}+36 a^{2} b^{3} c d \,x^{2}+12 a \,b^{4} c^{2} x^{2}+80 a \,b^{3} d^{2} x^{3}+40 b^{4} c d \,x^{3}+c^{2} a^{4} b^{2}+2 a^{4} b \,d^{2} x +16 a^{3} b^{2} c d x +12 a^{2} b^{3} c^{2} x +72 a^{2} b^{2} d^{2} x^{2}+96 a \,b^{3} c d \,x^{2}+12 b^{4} c^{2} x^{2}+120 b^{3} d^{2} x^{3}+2 c d \,a^{4} b +4 c^{2} a^{3} b^{2}+24 a^{3} b \,d^{2} x +72 a^{2} b^{2} c d x +24 a \,b^{3} c^{2} x +240 a \,b^{2} d^{2} x^{2}+120 b^{3} c d \,x^{2}+2 d^{2} a^{4}+16 c d \,a^{3} b +12 c^{2} a^{2} b^{2}+144 a^{2} b \,d^{2} x +192 a \,b^{2} c d x +24 b^{3} c^{2} x +360 b^{2} d^{2} x^{2}+24 a^{3} d^{2}+72 a^{2} b c d +24 a \,b^{2} c^{2}+480 a b \,d^{2} x +240 x \,b^{2} d c +144 a^{2} d^{2}+192 a b c d +24 b^{2} c^{2}+720 b \,d^{2} x +480 a \,d^{2}+240 b c d +720 d^{2}\right ) {\mathrm e}^{-b x -a}}{b^{3}}\)	\(640\)
meijerg	\(\frac {{\mathrm e}^{-a} d^{2} \left (720-\frac {\left (7 b^{6} x^{6}+42 b^{5} x^{5}+210 b^{4} x^{4}+840 b^{3} x^{3}+2520 b^{2} x^{2}+5040 b x +5040\right ) {\mathrm e}^{-b x}}{7}\right )}{b^{3}}+\frac {2 \,{\mathrm e}^{-a} d c \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^{2}}+\frac {{\mathrm e}^{-a} c^{2} \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}+\frac {4 \,{\mathrm e}^{-a} a \,d^{2} \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 {8 \,{\mathrm e}^{-a} a d c \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^{2}}+\frac {4 \,{\mathrm e}^{-a} a \,c^{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}+\frac {6 \,{\mathrm e}^{-a} a^{2} d^{2} \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 {12 \,{\mathrm e}^{-a} a^{2} d c \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^{2}}+\frac {6 \,{\mathrm e}^{-a} a^{2} c^{2} \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b}+\frac {4 \,{\mathrm e}^{-a} a^{3} d^{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 {8 \,{\mathrm e}^{-a} a^{3} d c \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{2}}+\frac {4 \,{\mathrm e}^{-a} a^{3} c^{2} \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b}+\frac {{\mathrm e}^{-a} a^{4} d^{2} \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{3}}+\frac {2 \,{\mathrm e}^{-a} a^{4} d c \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {{\mathrm e}^{-a} a^{4} c^{2} \left (1-{\mathrm e}^{-b x}\right )}{b}\)	\(670\)
derivativedivides	\(-\frac {c^{2} \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )+\frac {d^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{6}-6 \left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}+30 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+360 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-720 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+720 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}+\frac {d^{2} a^{2} \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {2 d c \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b}+\frac {2 d^{2} a \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {2 d a c \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b}}{b}\)	\(694\)
default	\(-\frac {c^{2} \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )+\frac {d^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{6}-6 \left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}+30 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+360 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-720 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+720 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}+\frac {d^{2} a^{2} \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {2 d c \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b}+\frac {2 d^{2} a \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {2 d a c \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b}}{b}\)	\(694\)
parts	\(-d^{2} b^{3} x^{6} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} b^{2} a \,d^{2} x^{5}-2 \,{\mathrm e}^{-b x -a} b^{3} c d \,x^{5}-6 \,{\mathrm e}^{-b x -a} b \,a^{2} d^{2} x^{4}-8 \,{\mathrm e}^{-b x -a} b^{2} a c d \,x^{4}-{\mathrm e}^{-b x -a} b^{3} c^{2} x^{4}-4 \,{\mathrm e}^{-b x -a} a^{3} d^{2} x^{3}-12 \,{\mathrm e}^{-b x -a} b \,a^{2} c d \,x^{3}-4 \,{\mathrm e}^{-b x -a} b^{2} a \,c^{2} x^{3}-\frac {{\mathrm e}^{-b x -a} a^{4} d^{2} x^{2}}{b}-8 \,{\mathrm e}^{-b x -a} a^{3} c d \,x^{2}-6 \,{\mathrm e}^{-b x -a} b \,a^{2} c^{2} x^{2}-\frac {2 \,{\mathrm e}^{-b x -a} a^{4} c d x}{b}-4 \,{\mathrm e}^{-b x -a} a^{3} c^{2} x -\frac {{\mathrm e}^{-b x -a} c^{2} a^{4}}{b}-\frac {2 \left (-2 b \,c^{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 )+5 c d \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 )-\frac {3 d^{2} \left (\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}\right )}{b}-\frac {2 d^{2} 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 )}{b}+4 d a c \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 )-\frac {5 d^{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}\right )}{b^{2}}\)	\(870\)
parallelrisch	\(\text {Expression too large to display}\)	\(1171\)

Input:

int(exp(-b*x-a)*(b*x+a)^4*(d*x+c)^2,x,method=_RETURNVERBOSE)
 

Output:

(-4*a*b^2*d^2-2*b^3*c*d-6*b^2*d^2)*x^5*exp(-b*x-a)+(-6*a^2*b*d^2-8*a*b^2*c 
*d-b^3*c^2-20*a*b*d^2-10*b^2*c*d-30*b*d^2)*x^4*exp(-b*x-a)+(-4*a^3*d^2-12* 
a^2*b*c*d-4*a*b^2*c^2-24*a^2*d^2-32*a*b*c*d-4*b^2*c^2-80*a*d^2-40*b*c*d-12 
0*d^2)*x^3*exp(-b*x-a)-(a^4*b^2*c^2+2*a^4*b*c*d+4*a^3*b^2*c^2+2*a^4*d^2+16 
*a^3*b*c*d+12*a^2*b^2*c^2+24*a^3*d^2+72*a^2*b*c*d+24*a*b^2*c^2+144*a^2*d^2 
+192*a*b*c*d+24*b^2*c^2+480*a*d^2+240*b*c*d+720*d^2)/b^3*exp(-b*x-a)-d^2*b 
^3*x^6*exp(-b*x-a)-(a^4*d^2+8*a^3*b*c*d+6*a^2*b^2*c^2+12*a^3*d^2+36*a^2*b* 
c*d+12*a*b^2*c^2+72*a^2*d^2+96*a*b*c*d+12*b^2*c^2+240*a*d^2+120*b*c*d+360* 
d^2)/b*x^2*exp(-b*x-a)-2*(a^4*b*c*d+2*a^3*b^2*c^2+a^4*d^2+8*a^3*b*c*d+6*a^ 
2*b^2*c^2+12*a^3*d^2+36*a^2*b*c*d+12*a*b^2*c^2+72*a^2*d^2+96*a*b*c*d+12*b^ 
2*c^2+240*a*d^2+120*b*c*d+360*d^2)/b^2*x*exp(-b*x-a)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.71 \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=-\frac {{\left (b^{6} d^{2} x^{6} + 2 \, {\left (b^{6} c d + {\left (2 \, a + 3\right )} b^{5} d^{2}\right )} x^{5} + {\left (a^{4} + 4 \, a^{3} + 12 \, a^{2} + 24 \, a + 24\right )} b^{2} c^{2} + {\left (b^{6} c^{2} + 2 \, {\left (4 \, a + 5\right )} b^{5} c d + 2 \, {\left (3 \, a^{2} + 10 \, a + 15\right )} b^{4} d^{2}\right )} x^{4} + 2 \, {\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} b c d + 4 \, {\left ({\left (a + 1\right )} b^{5} c^{2} + {\left (3 \, a^{2} + 8 \, a + 10\right )} b^{4} c d + {\left (a^{3} + 6 \, a^{2} + 20 \, a + 30\right )} b^{3} d^{2}\right )} x^{3} + 2 \, {\left (a^{4} + 12 \, a^{3} + 72 \, a^{2} + 240 \, a + 360\right )} d^{2} + {\left (6 \, {\left (a^{2} + 2 \, a + 2\right )} b^{4} c^{2} + 4 \, {\left (2 \, a^{3} + 9 \, a^{2} + 24 \, a + 30\right )} b^{3} c d + {\left (a^{4} + 12 \, a^{3} + 72 \, a^{2} + 240 \, a + 360\right )} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (2 \, {\left (a^{3} + 3 \, a^{2} + 6 \, a + 6\right )} b^{3} c^{2} + {\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} b^{2} c d + {\left (a^{4} + 12 \, a^{3} + 72 \, a^{2} + 240 \, a + 360\right )} b d^{2}\right )} x\right )} e^{\left (-b x - a\right )}}{b^{3}} \] Input:

integrate(exp(-b*x-a)*(b*x+a)^4*(d*x+c)^2,x, algorithm="fricas")
 

Output:

-(b^6*d^2*x^6 + 2*(b^6*c*d + (2*a + 3)*b^5*d^2)*x^5 + (a^4 + 4*a^3 + 12*a^ 
2 + 24*a + 24)*b^2*c^2 + (b^6*c^2 + 2*(4*a + 5)*b^5*c*d + 2*(3*a^2 + 10*a 
+ 15)*b^4*d^2)*x^4 + 2*(a^4 + 8*a^3 + 36*a^2 + 96*a + 120)*b*c*d + 4*((a + 
 1)*b^5*c^2 + (3*a^2 + 8*a + 10)*b^4*c*d + (a^3 + 6*a^2 + 20*a + 30)*b^3*d 
^2)*x^3 + 2*(a^4 + 12*a^3 + 72*a^2 + 240*a + 360)*d^2 + (6*(a^2 + 2*a + 2) 
*b^4*c^2 + 4*(2*a^3 + 9*a^2 + 24*a + 30)*b^3*c*d + (a^4 + 12*a^3 + 72*a^2 
+ 240*a + 360)*b^2*d^2)*x^2 + 2*(2*(a^3 + 3*a^2 + 6*a + 6)*b^3*c^2 + (a^4 
+ 8*a^3 + 36*a^2 + 96*a + 120)*b^2*c*d + (a^4 + 12*a^3 + 72*a^2 + 240*a + 
360)*b*d^2)*x)*e^(-b*x - a)/b^3
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 899, normalized size of antiderivative = 1.81 \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=\begin {cases} \frac {\left (- a^{4} b^{2} c^{2} - 2 a^{4} b^{2} c d x - a^{4} b^{2} d^{2} x^{2} - 2 a^{4} b c d - 2 a^{4} b d^{2} x - 2 a^{4} d^{2} - 4 a^{3} b^{3} c^{2} x - 8 a^{3} b^{3} c d x^{2} - 4 a^{3} b^{3} d^{2} x^{3} - 4 a^{3} b^{2} c^{2} - 16 a^{3} b^{2} c d x - 12 a^{3} b^{2} d^{2} x^{2} - 16 a^{3} b c d - 24 a^{3} b d^{2} x - 24 a^{3} d^{2} - 6 a^{2} b^{4} c^{2} x^{2} - 12 a^{2} b^{4} c d x^{3} - 6 a^{2} b^{4} d^{2} x^{4} - 12 a^{2} b^{3} c^{2} x - 36 a^{2} b^{3} c d x^{2} - 24 a^{2} b^{3} d^{2} x^{3} - 12 a^{2} b^{2} c^{2} - 72 a^{2} b^{2} c d x - 72 a^{2} b^{2} d^{2} x^{2} - 72 a^{2} b c d - 144 a^{2} b d^{2} x - 144 a^{2} d^{2} - 4 a b^{5} c^{2} x^{3} - 8 a b^{5} c d x^{4} - 4 a b^{5} d^{2} x^{5} - 12 a b^{4} c^{2} x^{2} - 32 a b^{4} c d x^{3} - 20 a b^{4} d^{2} x^{4} - 24 a b^{3} c^{2} x - 96 a b^{3} c d x^{2} - 80 a b^{3} d^{2} x^{3} - 24 a b^{2} c^{2} - 192 a b^{2} c d x - 240 a b^{2} d^{2} x^{2} - 192 a b c d - 480 a b d^{2} x - 480 a d^{2} - b^{6} c^{2} x^{4} - 2 b^{6} c d x^{5} - b^{6} d^{2} x^{6} - 4 b^{5} c^{2} x^{3} - 10 b^{5} c d x^{4} - 6 b^{5} d^{2} x^{5} - 12 b^{4} c^{2} x^{2} - 40 b^{4} c d x^{3} - 30 b^{4} d^{2} x^{4} - 24 b^{3} c^{2} x - 120 b^{3} c d x^{2} - 120 b^{3} d^{2} x^{3} - 24 b^{2} c^{2} - 240 b^{2} c d x - 360 b^{2} d^{2} x^{2} - 240 b c d - 720 b d^{2} x - 720 d^{2}\right ) e^{- a - b x}}{b^{3}} & \text {for}\: b^{3} \neq 0 \\a^{4} c^{2} x + \frac {b^{4} d^{2} x^{7}}{7} + x^{6} \cdot \left (\frac {2 a b^{3} d^{2}}{3} + \frac {b^{4} c d}{3}\right ) + x^{5} \cdot \left (\frac {6 a^{2} b^{2} d^{2}}{5} + \frac {8 a b^{3} c d}{5} + \frac {b^{4} c^{2}}{5}\right ) + x^{4} \left (a^{3} b d^{2} + 3 a^{2} b^{2} c d + a b^{3} c^{2}\right ) + x^{3} \left (\frac {a^{4} d^{2}}{3} + \frac {8 a^{3} b c d}{3} + 2 a^{2} b^{2} c^{2}\right ) + x^{2} \left (a^{4} c d + 2 a^{3} b c^{2}\right ) & \text {otherwise} \end {cases} \] Input:

integrate(exp(-b*x-a)*(b*x+a)**4*(d*x+c)**2,x)
 

Output:

Piecewise(((-a**4*b**2*c**2 - 2*a**4*b**2*c*d*x - a**4*b**2*d**2*x**2 - 2* 
a**4*b*c*d - 2*a**4*b*d**2*x - 2*a**4*d**2 - 4*a**3*b**3*c**2*x - 8*a**3*b 
**3*c*d*x**2 - 4*a**3*b**3*d**2*x**3 - 4*a**3*b**2*c**2 - 16*a**3*b**2*c*d 
*x - 12*a**3*b**2*d**2*x**2 - 16*a**3*b*c*d - 24*a**3*b*d**2*x - 24*a**3*d 
**2 - 6*a**2*b**4*c**2*x**2 - 12*a**2*b**4*c*d*x**3 - 6*a**2*b**4*d**2*x** 
4 - 12*a**2*b**3*c**2*x - 36*a**2*b**3*c*d*x**2 - 24*a**2*b**3*d**2*x**3 - 
 12*a**2*b**2*c**2 - 72*a**2*b**2*c*d*x - 72*a**2*b**2*d**2*x**2 - 72*a**2 
*b*c*d - 144*a**2*b*d**2*x - 144*a**2*d**2 - 4*a*b**5*c**2*x**3 - 8*a*b**5 
*c*d*x**4 - 4*a*b**5*d**2*x**5 - 12*a*b**4*c**2*x**2 - 32*a*b**4*c*d*x**3 
- 20*a*b**4*d**2*x**4 - 24*a*b**3*c**2*x - 96*a*b**3*c*d*x**2 - 80*a*b**3* 
d**2*x**3 - 24*a*b**2*c**2 - 192*a*b**2*c*d*x - 240*a*b**2*d**2*x**2 - 192 
*a*b*c*d - 480*a*b*d**2*x - 480*a*d**2 - b**6*c**2*x**4 - 2*b**6*c*d*x**5 
- b**6*d**2*x**6 - 4*b**5*c**2*x**3 - 10*b**5*c*d*x**4 - 6*b**5*d**2*x**5 
- 12*b**4*c**2*x**2 - 40*b**4*c*d*x**3 - 30*b**4*d**2*x**4 - 24*b**3*c**2* 
x - 120*b**3*c*d*x**2 - 120*b**3*d**2*x**3 - 24*b**2*c**2 - 240*b**2*c*d*x 
 - 360*b**2*d**2*x**2 - 240*b*c*d - 720*b*d**2*x - 720*d**2)*exp(-a - b*x) 
/b**3, Ne(b**3, 0)), (a**4*c**2*x + b**4*d**2*x**7/7 + x**6*(2*a*b**3*d**2 
/3 + b**4*c*d/3) + x**5*(6*a**2*b**2*d**2/5 + 8*a*b**3*c*d/5 + b**4*c**2/5 
) + x**4*(a**3*b*d**2 + 3*a**2*b**2*c*d + a*b**3*c**2) + x**3*(a**4*d**2/3 
 + 8*a**3*b*c*d/3 + 2*a**2*b**2*c**2) + x**2*(a**4*c*d + 2*a**3*b*c**2)...
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.20 \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=-\frac {4 \, {\left (b x + 1\right )} a^{3} c^{2} e^{\left (-b x - a\right )}}{b} - \frac {a^{4} c^{2} e^{\left (-b x - a\right )}}{b} - \frac {2 \, {\left (b x + 1\right )} a^{4} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac {6 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{2} c^{2} e^{\left (-b x - a\right )}}{b} - \frac {8 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{3} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{4} d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {4 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a c^{2} e^{\left (-b x - a\right )}}{b} - \frac {12 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{2} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac {4 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{3} d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} c^{2} e^{\left (-b x - a\right )}}{b} - \frac {8 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a c d e^{\left (-b x - a\right )}}{b^{2}} - \frac {6 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a^{2} d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {2 \, {\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac {4 \, {\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} a d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {{\left (b^{6} x^{6} + 6 \, b^{5} x^{5} + 30 \, b^{4} x^{4} + 120 \, b^{3} x^{3} + 360 \, b^{2} x^{2} + 720 \, b x + 720\right )} d^{2} e^{\left (-b x - a\right )}}{b^{3}} \] Input:

integrate(exp(-b*x-a)*(b*x+a)^4*(d*x+c)^2,x, algorithm="maxima")
 

Output:

-4*(b*x + 1)*a^3*c^2*e^(-b*x - a)/b - a^4*c^2*e^(-b*x - a)/b - 2*(b*x + 1) 
*a^4*c*d*e^(-b*x - a)/b^2 - 6*(b^2*x^2 + 2*b*x + 2)*a^2*c^2*e^(-b*x - a)/b 
 - 8*(b^2*x^2 + 2*b*x + 2)*a^3*c*d*e^(-b*x - a)/b^2 - (b^2*x^2 + 2*b*x + 2 
)*a^4*d^2*e^(-b*x - a)/b^3 - 4*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*a*c^2*e^( 
-b*x - a)/b - 12*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*a^2*c*d*e^(-b*x - a)/b^ 
2 - 4*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*a^3*d^2*e^(-b*x - a)/b^3 - (b^4*x^ 
4 + 4*b^3*x^3 + 12*b^2*x^2 + 24*b*x + 24)*c^2*e^(-b*x - a)/b - 8*(b^4*x^4 
+ 4*b^3*x^3 + 12*b^2*x^2 + 24*b*x + 24)*a*c*d*e^(-b*x - a)/b^2 - 6*(b^4*x^ 
4 + 4*b^3*x^3 + 12*b^2*x^2 + 24*b*x + 24)*a^2*d^2*e^(-b*x - a)/b^3 - 2*(b^ 
5*x^5 + 5*b^4*x^4 + 20*b^3*x^3 + 60*b^2*x^2 + 120*b*x + 120)*c*d*e^(-b*x - 
 a)/b^2 - 4*(b^5*x^5 + 5*b^4*x^4 + 20*b^3*x^3 + 60*b^2*x^2 + 120*b*x + 120 
)*a*d^2*e^(-b*x - a)/b^3 - (b^6*x^6 + 6*b^5*x^5 + 30*b^4*x^4 + 120*b^3*x^3 
 + 360*b^2*x^2 + 720*b*x + 720)*d^2*e^(-b*x - a)/b^3
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.35 \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=-\frac {{\left (b^{10} d^{2} x^{6} + 2 \, b^{10} c d x^{5} + 4 \, a b^{9} d^{2} x^{5} + b^{10} c^{2} x^{4} + 8 \, a b^{9} c d x^{4} + 6 \, a^{2} b^{8} d^{2} x^{4} + 6 \, b^{9} d^{2} x^{5} + 4 \, a b^{9} c^{2} x^{3} + 12 \, a^{2} b^{8} c d x^{3} + 4 \, a^{3} b^{7} d^{2} x^{3} + 10 \, b^{9} c d x^{4} + 20 \, a b^{8} d^{2} x^{4} + 6 \, a^{2} b^{8} c^{2} x^{2} + 8 \, a^{3} b^{7} c d x^{2} + a^{4} b^{6} d^{2} x^{2} + 4 \, b^{9} c^{2} x^{3} + 32 \, a b^{8} c d x^{3} + 24 \, a^{2} b^{7} d^{2} x^{3} + 30 \, b^{8} d^{2} x^{4} + 4 \, a^{3} b^{7} c^{2} x + 2 \, a^{4} b^{6} c d x + 12 \, a b^{8} c^{2} x^{2} + 36 \, a^{2} b^{7} c d x^{2} + 12 \, a^{3} b^{6} d^{2} x^{2} + 40 \, b^{8} c d x^{3} + 80 \, a b^{7} d^{2} x^{3} + a^{4} b^{6} c^{2} + 12 \, a^{2} b^{7} c^{2} x + 16 \, a^{3} b^{6} c d x + 2 \, a^{4} b^{5} d^{2} x + 12 \, b^{8} c^{2} x^{2} + 96 \, a b^{7} c d x^{2} + 72 \, a^{2} b^{6} d^{2} x^{2} + 120 \, b^{7} d^{2} x^{3} + 4 \, a^{3} b^{6} c^{2} + 2 \, a^{4} b^{5} c d + 24 \, a b^{7} c^{2} x + 72 \, a^{2} b^{6} c d x + 24 \, a^{3} b^{5} d^{2} x + 120 \, b^{7} c d x^{2} + 240 \, a b^{6} d^{2} x^{2} + 12 \, a^{2} b^{6} c^{2} + 16 \, a^{3} b^{5} c d + 2 \, a^{4} b^{4} d^{2} + 24 \, b^{7} c^{2} x + 192 \, a b^{6} c d x + 144 \, a^{2} b^{5} d^{2} x + 360 \, b^{6} d^{2} x^{2} + 24 \, a b^{6} c^{2} + 72 \, a^{2} b^{5} c d + 24 \, a^{3} b^{4} d^{2} + 240 \, b^{6} c d x + 480 \, a b^{5} d^{2} x + 24 \, b^{6} c^{2} + 192 \, a b^{5} c d + 144 \, a^{2} b^{4} d^{2} + 720 \, b^{5} d^{2} x + 240 \, b^{5} c d + 480 \, a b^{4} d^{2} + 720 \, b^{4} d^{2}\right )} e^{\left (-b x - a\right )}}{b^{7}} \] Input:

integrate(exp(-b*x-a)*(b*x+a)^4*(d*x+c)^2,x, algorithm="giac")
 

Output:

-(b^10*d^2*x^6 + 2*b^10*c*d*x^5 + 4*a*b^9*d^2*x^5 + b^10*c^2*x^4 + 8*a*b^9 
*c*d*x^4 + 6*a^2*b^8*d^2*x^4 + 6*b^9*d^2*x^5 + 4*a*b^9*c^2*x^3 + 12*a^2*b^ 
8*c*d*x^3 + 4*a^3*b^7*d^2*x^3 + 10*b^9*c*d*x^4 + 20*a*b^8*d^2*x^4 + 6*a^2* 
b^8*c^2*x^2 + 8*a^3*b^7*c*d*x^2 + a^4*b^6*d^2*x^2 + 4*b^9*c^2*x^3 + 32*a*b 
^8*c*d*x^3 + 24*a^2*b^7*d^2*x^3 + 30*b^8*d^2*x^4 + 4*a^3*b^7*c^2*x + 2*a^4 
*b^6*c*d*x + 12*a*b^8*c^2*x^2 + 36*a^2*b^7*c*d*x^2 + 12*a^3*b^6*d^2*x^2 + 
40*b^8*c*d*x^3 + 80*a*b^7*d^2*x^3 + a^4*b^6*c^2 + 12*a^2*b^7*c^2*x + 16*a^ 
3*b^6*c*d*x + 2*a^4*b^5*d^2*x + 12*b^8*c^2*x^2 + 96*a*b^7*c*d*x^2 + 72*a^2 
*b^6*d^2*x^2 + 120*b^7*d^2*x^3 + 4*a^3*b^6*c^2 + 2*a^4*b^5*c*d + 24*a*b^7* 
c^2*x + 72*a^2*b^6*c*d*x + 24*a^3*b^5*d^2*x + 120*b^7*c*d*x^2 + 240*a*b^6* 
d^2*x^2 + 12*a^2*b^6*c^2 + 16*a^3*b^5*c*d + 2*a^4*b^4*d^2 + 24*b^7*c^2*x + 
 192*a*b^6*c*d*x + 144*a^2*b^5*d^2*x + 360*b^6*d^2*x^2 + 24*a*b^6*c^2 + 72 
*a^2*b^5*c*d + 24*a^3*b^4*d^2 + 240*b^6*c*d*x + 480*a*b^5*d^2*x + 24*b^6*c 
^2 + 192*a*b^5*c*d + 144*a^2*b^4*d^2 + 720*b^5*d^2*x + 240*b^5*c*d + 480*a 
*b^4*d^2 + 720*b^4*d^2)*e^(-b*x - a)/b^7
 

Mupad [B] (verification not implemented)

Time = 23.02 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.08 \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=-x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (120\,c\,d+b\,\left (6\,a^2\,c^2+12\,a\,c^2+12\,c^2\right )+\frac {a^4\,d^2+12\,a^3\,d^2+72\,a^2\,d^2+240\,a\,d^2+360\,d^2}{b}+96\,a\,c\,d+36\,a^2\,c\,d+8\,a^3\,c\,d\right )-x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (4\,a^3\,d^2+12\,a^2\,b\,c\,d+24\,a^2\,d^2+4\,a\,b^2\,c^2+32\,a\,b\,c\,d+80\,a\,d^2+4\,b^2\,c^2+40\,b\,c\,d+120\,d^2\right )-\frac {{\mathrm {e}}^{-a-b\,x}\,\left (a^4\,b^2\,c^2+2\,a^4\,b\,c\,d+2\,a^4\,d^2+4\,a^3\,b^2\,c^2+16\,a^3\,b\,c\,d+24\,a^3\,d^2+12\,a^2\,b^2\,c^2+72\,a^2\,b\,c\,d+144\,a^2\,d^2+24\,a\,b^2\,c^2+192\,a\,b\,c\,d+480\,a\,d^2+24\,b^2\,c^2+240\,b\,c\,d+720\,d^2\right )}{b^3}-b^3\,d^2\,x^6\,{\mathrm {e}}^{-a-b\,x}-b\,x^4\,{\mathrm {e}}^{-a-b\,x}\,\left (6\,a^2\,d^2+8\,a\,b\,c\,d+20\,a\,d^2+b^2\,c^2+10\,b\,c\,d+30\,d^2\right )-\frac {2\,x\,{\mathrm {e}}^{-a-b\,x}\,\left (a^4\,b\,c\,d+a^4\,d^2+2\,a^3\,b^2\,c^2+8\,a^3\,b\,c\,d+12\,a^3\,d^2+6\,a^2\,b^2\,c^2+36\,a^2\,b\,c\,d+72\,a^2\,d^2+12\,a\,b^2\,c^2+96\,a\,b\,c\,d+240\,a\,d^2+12\,b^2\,c^2+120\,b\,c\,d+360\,d^2\right )}{b^2}-2\,b^2\,d\,x^5\,{\mathrm {e}}^{-a-b\,x}\,\left (3\,d+2\,a\,d+b\,c\right ) \] Input:

int(exp(- a - b*x)*(a + b*x)^4*(c + d*x)^2,x)
 

Output:

- x^2*exp(- a - b*x)*(120*c*d + b*(12*a*c^2 + 12*c^2 + 6*a^2*c^2) + (240*a 
*d^2 + 360*d^2 + 72*a^2*d^2 + 12*a^3*d^2 + a^4*d^2)/b + 96*a*c*d + 36*a^2* 
c*d + 8*a^3*c*d) - x^3*exp(- a - b*x)*(80*a*d^2 + 120*d^2 + 24*a^2*d^2 + 4 
*b^2*c^2 + 4*a^3*d^2 + 4*a*b^2*c^2 + 40*b*c*d + 12*a^2*b*c*d + 32*a*b*c*d) 
 - (exp(- a - b*x)*(480*a*d^2 + 720*d^2 + 144*a^2*d^2 + 24*b^2*c^2 + 24*a^ 
3*d^2 + 2*a^4*d^2 + 24*a*b^2*c^2 + 240*b*c*d + 12*a^2*b^2*c^2 + 4*a^3*b^2* 
c^2 + a^4*b^2*c^2 + 72*a^2*b*c*d + 16*a^3*b*c*d + 2*a^4*b*c*d + 192*a*b*c* 
d))/b^3 - b^3*d^2*x^6*exp(- a - b*x) - b*x^4*exp(- a - b*x)*(20*a*d^2 + 30 
*d^2 + 6*a^2*d^2 + b^2*c^2 + 10*b*c*d + 8*a*b*c*d) - (2*x*exp(- a - b*x)*( 
240*a*d^2 + 360*d^2 + 72*a^2*d^2 + 12*b^2*c^2 + 12*a^3*d^2 + a^4*d^2 + 12* 
a*b^2*c^2 + 120*b*c*d + 6*a^2*b^2*c^2 + 2*a^3*b^2*c^2 + 36*a^2*b*c*d + 8*a 
^3*b*c*d + a^4*b*c*d + 96*a*b*c*d))/b^2 - 2*b^2*d*x^5*exp(- a - b*x)*(3*d 
+ 2*a*d + b*c)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 642, normalized size of antiderivative = 1.29 \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=\frac {-b^{6} d^{2} x^{6}-4 a \,b^{5} d^{2} x^{5}-2 b^{6} c d \,x^{5}-6 a^{2} b^{4} d^{2} x^{4}-8 a \,b^{5} c d \,x^{4}-b^{6} c^{2} x^{4}-6 b^{5} d^{2} x^{5}-4 a^{3} b^{3} d^{2} x^{3}-12 a^{2} b^{4} c d \,x^{3}-4 a \,b^{5} c^{2} x^{3}-20 a \,b^{4} d^{2} x^{4}-10 b^{5} c d \,x^{4}-a^{4} b^{2} d^{2} x^{2}-8 a^{3} b^{3} c d \,x^{2}-6 a^{2} b^{4} c^{2} x^{2}-24 a^{2} b^{3} d^{2} x^{3}-32 a \,b^{4} c d \,x^{3}-4 b^{5} c^{2} x^{3}-30 b^{4} d^{2} x^{4}-2 a^{4} b^{2} c d x -4 a^{3} b^{3} c^{2} x -12 a^{3} b^{2} d^{2} x^{2}-36 a^{2} b^{3} c d \,x^{2}-12 a \,b^{4} c^{2} x^{2}-80 a \,b^{3} d^{2} x^{3}-40 b^{4} c d \,x^{3}-a^{4} b^{2} c^{2}-2 a^{4} b \,d^{2} x -16 a^{3} b^{2} c d x -12 a^{2} b^{3} c^{2} x -72 a^{2} b^{2} d^{2} x^{2}-96 a \,b^{3} c d \,x^{2}-12 b^{4} c^{2} x^{2}-120 b^{3} d^{2} x^{3}-2 a^{4} b c d -4 a^{3} b^{2} c^{2}-24 a^{3} b \,d^{2} x -72 a^{2} b^{2} c d x -24 a \,b^{3} c^{2} x -240 a \,b^{2} d^{2} x^{2}-120 b^{3} c d \,x^{2}-2 a^{4} d^{2}-16 a^{3} b c d -12 a^{2} b^{2} c^{2}-144 a^{2} b \,d^{2} x -192 a \,b^{2} c d x -24 b^{3} c^{2} x -360 b^{2} d^{2} x^{2}-24 a^{3} d^{2}-72 a^{2} b c d -24 a \,b^{2} c^{2}-480 a b \,d^{2} x -240 b^{2} c d x -144 a^{2} d^{2}-192 a b c d -24 b^{2} c^{2}-720 b \,d^{2} x -480 a \,d^{2}-240 b c d -720 d^{2}}{e^{b x +a} b^{3}} \] Input:

int(exp(-b*x-a)*(b*x+a)^4*(d*x+c)^2,x)
 

Output:

( - a**4*b**2*c**2 - 2*a**4*b**2*c*d*x - a**4*b**2*d**2*x**2 - 2*a**4*b*c* 
d - 2*a**4*b*d**2*x - 2*a**4*d**2 - 4*a**3*b**3*c**2*x - 8*a**3*b**3*c*d*x 
**2 - 4*a**3*b**3*d**2*x**3 - 4*a**3*b**2*c**2 - 16*a**3*b**2*c*d*x - 12*a 
**3*b**2*d**2*x**2 - 16*a**3*b*c*d - 24*a**3*b*d**2*x - 24*a**3*d**2 - 6*a 
**2*b**4*c**2*x**2 - 12*a**2*b**4*c*d*x**3 - 6*a**2*b**4*d**2*x**4 - 12*a* 
*2*b**3*c**2*x - 36*a**2*b**3*c*d*x**2 - 24*a**2*b**3*d**2*x**3 - 12*a**2* 
b**2*c**2 - 72*a**2*b**2*c*d*x - 72*a**2*b**2*d**2*x**2 - 72*a**2*b*c*d - 
144*a**2*b*d**2*x - 144*a**2*d**2 - 4*a*b**5*c**2*x**3 - 8*a*b**5*c*d*x**4 
 - 4*a*b**5*d**2*x**5 - 12*a*b**4*c**2*x**2 - 32*a*b**4*c*d*x**3 - 20*a*b* 
*4*d**2*x**4 - 24*a*b**3*c**2*x - 96*a*b**3*c*d*x**2 - 80*a*b**3*d**2*x**3 
 - 24*a*b**2*c**2 - 192*a*b**2*c*d*x - 240*a*b**2*d**2*x**2 - 192*a*b*c*d 
- 480*a*b*d**2*x - 480*a*d**2 - b**6*c**2*x**4 - 2*b**6*c*d*x**5 - b**6*d* 
*2*x**6 - 4*b**5*c**2*x**3 - 10*b**5*c*d*x**4 - 6*b**5*d**2*x**5 - 12*b**4 
*c**2*x**2 - 40*b**4*c*d*x**3 - 30*b**4*d**2*x**4 - 24*b**3*c**2*x - 120*b 
**3*c*d*x**2 - 120*b**3*d**2*x**3 - 24*b**2*c**2 - 240*b**2*c*d*x - 360*b* 
*2*d**2*x**2 - 240*b*c*d - 720*b*d**2*x - 720*d**2)/(e**(a + b*x)*b**3)