Integrand size = 23, antiderivative size = 275 \[ \int e^{-a-b x} (a+b x)^4 (c+d x) \, dx=-\frac {120 d e^{-a-b x}}{b^2}-\frac {e^{-a-b x} (a+b x)^4 (c+d x)}{b}-\frac {24 e^{-a-b x} (b c+4 a d+5 b d x)}{b^2}-\frac {12 e^{-a-b x} \left (a (2 b c+3 a d)+2 b (b c+4 a d) x+5 b^2 d x^2\right )}{b^2}-\frac {4 e^{-a-b x} \left (a^2 (3 b c+2 a d)+3 a b (2 b c+3 a d) x+3 b^2 (b c+4 a d) x^2+5 b^3 d x^3\right )}{b^2}-\frac {e^{-a-b x} \left (a^3 (4 b c+a d)+4 a^2 b (3 b c+2 a d) x+6 a b^2 (2 b c+3 a d) x^2+4 b^3 (b c+4 a d) x^3+5 b^4 d x^4\right )}{b^2} \] Output:
-120*d*exp(-b*x-a)/b^2-exp(-b*x-a)*(b*x+a)^4*(d*x+c)/b-24*exp(-b*x-a)*(5*b *d*x+4*a*d+b*c)/b^2-12*exp(-b*x-a)*(a*(3*a*d+2*b*c)+2*b*(4*a*d+b*c)*x+5*b^ 2*d*x^2)/b^2-4*exp(-b*x-a)*(a^2*(2*a*d+3*b*c)+3*a*b*(3*a*d+2*b*c)*x+3*b^2* (4*a*d+b*c)*x^2+5*b^3*d*x^3)/b^2-exp(-b*x-a)*(a^3*(a*d+4*b*c)+4*a^2*b*(2*a *d+3*b*c)*x+6*a*b^2*(3*a*d+2*b*c)*x^2+4*b^3*(4*a*d+b*c)*x^3+5*b^4*d*x^4)/b ^2
Time = 0.94 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.69 \[ \int e^{-a-b x} (a+b x)^4 (c+d x) \, dx=\frac {e^{-a-b x} \left (-\left (\left (120+96 a+36 a^2+8 a^3+a^4\right ) d\right )-b^5 x^4 (c+d x)-b^4 x^3 (4 (1+a) c+(5+4 a) d x)-2 b^3 x^2 \left (3 \left (2+2 a+a^2\right ) c+\left (10+8 a+3 a^2\right ) d x\right )-2 b^2 x \left (2 \left (6+6 a+3 a^2+a^3\right ) c+\left (30+24 a+9 a^2+2 a^3\right ) d x\right )-b \left (\left (24+24 a+12 a^2+4 a^3+a^4\right ) c+\left (120+96 a+36 a^2+8 a^3+a^4\right ) d x\right )\right )}{b^2} \] Input:
Integrate[E^(-a - b*x)*(a + b*x)^4*(c + d*x),x]
Output:
(E^(-a - b*x)*(-((120 + 96*a + 36*a^2 + 8*a^3 + a^4)*d) - b^5*x^4*(c + d*x ) - b^4*x^3*(4*(1 + a)*c + (5 + 4*a)*d*x) - 2*b^3*x^2*(3*(2 + 2*a + a^2)*c + (10 + 8*a + 3*a^2)*d*x) - 2*b^2*x*(2*(6 + 6*a + 3*a^2 + a^3)*c + (30 + 24*a + 9*a^2 + 2*a^3)*d*x) - b*((24 + 24*a + 12*a^2 + 4*a^3 + a^4)*c + (12 0 + 96*a + 36*a^2 + 8*a^3 + a^4)*d*x)))/b^2
Time = 0.97 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int e^{-a-b x} (a+b x)^4 (c+d x) \, dx\) |
| \(\Big \downarrow \) 2626 |
| \(\displaystyle \int \left (\frac {e^{-a-b x} (a+b x)^4 (b c-a d)}{b}+\frac {d e^{-a-b x} (a+b x)^5}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {e^{-a-b x} (a+b x)^4 (b c-a d)}{b^2}-\frac {4 e^{-a-b x} (a+b x)^3 (b c-a d)}{b^2}-\frac {12 e^{-a-b x} (a+b x)^2 (b c-a d)}{b^2}-\frac {24 e^{-a-b x} (a+b x) (b c-a d)}{b^2}-\frac {24 e^{-a-b x} (b c-a d)}{b^2}-\frac {d e^{-a-b x} (a+b x)^5}{b^2}-\frac {5 d e^{-a-b x} (a+b x)^4}{b^2}-\frac {20 d e^{-a-b x} (a+b x)^3}{b^2}-\frac {60 d e^{-a-b x} (a+b x)^2}{b^2}-\frac {120 d e^{-a-b x} (a+b x)}{b^2}-\frac {120 d e^{-a-b x}}{b^2}\) |
Input:
Int[E^(-a - b*x)*(a + b*x)^4*(c + d*x),x]
Output:
(-120*d*E^(-a - b*x))/b^2 - (24*(b*c - a*d)*E^(-a - b*x))/b^2 - (120*d*E^( -a - b*x)*(a + b*x))/b^2 - (24*(b*c - a*d)*E^(-a - b*x)*(a + b*x))/b^2 - ( 60*d*E^(-a - b*x)*(a + b*x)^2)/b^2 - (12*(b*c - a*d)*E^(-a - b*x)*(a + b*x )^2)/b^2 - (20*d*E^(-a - b*x)*(a + b*x)^3)/b^2 - (4*(b*c - a*d)*E^(-a - b* x)*(a + b*x)^3)/b^2 - (5*d*E^(-a - b*x)*(a + b*x)^4)/b^2 - ((b*c - a*d)*E^ (-a - b*x)*(a + b*x)^4)/b^2 - (d*E^(-a - b*x)*(a + b*x)^5)/b^2
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]
Time = 0.08 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00
| method | result | size |
| norman | \(\left (-4 a \,b^{2} d -c \,b^{3}-5 b^{2} d \right ) x^{4} {\mathrm e}^{-b x -a}+\left (-6 a^{2} b d -4 a \,b^{2} c -16 a b d -4 b^{2} c -20 b d \right ) x^{3} {\mathrm e}^{-b x -a}+\left (-4 a^{3} d -6 a^{2} b c -18 a^{2} d -12 a b c -48 d a -12 b c -60 d \right ) x^{2} {\mathrm e}^{-b x -a}-\frac {\left (c \,a^{4} b +d \,a^{4}+4 c \,a^{3} b +8 a^{3} d +12 a^{2} b c +36 a^{2} d +24 a b c +96 d a +24 b c +120 d \right ) {\mathrm e}^{-b x -a}}{b^{2}}-d \,b^{3} x^{5} {\mathrm e}^{-b x -a}-\frac {\left (d \,a^{4}+4 c \,a^{3} b +8 a^{3} d +12 a^{2} b c +36 a^{2} d +24 a b c +96 d a +24 b c +120 d \right ) x \,{\mathrm e}^{-b x -a}}{b}\) | \(276\) |
| gosper | \(-\frac {\left (d \,b^{5} x^{5}+4 a \,b^{4} d \,x^{4}+b^{5} c \,x^{4}+6 a^{2} b^{3} d \,x^{3}+4 a \,b^{4} c \,x^{3}+5 b^{4} d \,x^{4}+4 a^{3} b^{2} d \,x^{2}+6 a^{2} b^{3} c \,x^{2}+16 a \,b^{3} d \,x^{3}+4 b^{4} c \,x^{3}+a^{4} d x b +4 a^{3} b^{2} c x +18 a^{2} b^{2} d \,x^{2}+12 a \,b^{3} c \,x^{2}+20 b^{3} d \,x^{3}+c \,a^{4} b +8 a^{3} d x b +12 a^{2} b^{2} c x +48 a \,b^{2} d \,x^{2}+12 b^{3} c \,x^{2}+d \,a^{4}+4 c \,a^{3} b +36 a^{2} d x b +24 a \,b^{2} c x +60 b^{2} d \,x^{2}+8 a^{3} d +12 a^{2} b c +96 a b d x +24 b^{2} c x +36 a^{2} d +24 a b c +120 b d x +96 d a +24 b c +120 d \right ) {\mathrm e}^{-b x -a}}{b^{2}}\) | \(297\) |
| risch | \(-\frac {\left (d \,b^{5} x^{5}+4 a \,b^{4} d \,x^{4}+b^{5} c \,x^{4}+6 a^{2} b^{3} d \,x^{3}+4 a \,b^{4} c \,x^{3}+5 b^{4} d \,x^{4}+4 a^{3} b^{2} d \,x^{2}+6 a^{2} b^{3} c \,x^{2}+16 a \,b^{3} d \,x^{3}+4 b^{4} c \,x^{3}+a^{4} d x b +4 a^{3} b^{2} c x +18 a^{2} b^{2} d \,x^{2}+12 a \,b^{3} c \,x^{2}+20 b^{3} d \,x^{3}+c \,a^{4} b +8 a^{3} d x b +12 a^{2} b^{2} c x +48 a \,b^{2} d \,x^{2}+12 b^{3} c \,x^{2}+d \,a^{4}+4 c \,a^{3} b +36 a^{2} d x b +24 a \,b^{2} c x +60 b^{2} d \,x^{2}+8 a^{3} d +12 a^{2} b c +96 a b d x +24 b^{2} c x +36 a^{2} d +24 a b c +120 b d x +96 d a +24 b c +120 d \right ) {\mathrm e}^{-b x -a}}{b^{2}}\) | \(297\) |
| orering | \(-\frac {\left (d \,b^{5} x^{5}+4 a \,b^{4} d \,x^{4}+b^{5} c \,x^{4}+6 a^{2} b^{3} d \,x^{3}+4 a \,b^{4} c \,x^{3}+5 b^{4} d \,x^{4}+4 a^{3} b^{2} d \,x^{2}+6 a^{2} b^{3} c \,x^{2}+16 a \,b^{3} d \,x^{3}+4 b^{4} c \,x^{3}+a^{4} d x b +4 a^{3} b^{2} c x +18 a^{2} b^{2} d \,x^{2}+12 a \,b^{3} c \,x^{2}+20 b^{3} d \,x^{3}+c \,a^{4} b +8 a^{3} d x b +12 a^{2} b^{2} c x +48 a \,b^{2} d \,x^{2}+12 b^{3} c \,x^{2}+d \,a^{4}+4 c \,a^{3} b +36 a^{2} d x b +24 a \,b^{2} c x +60 b^{2} d \,x^{2}+8 a^{3} d +12 a^{2} b c +96 a b d x +24 b^{2} c x +36 a^{2} d +24 a b c +120 b d x +96 d a +24 b c +120 d \right ) {\mathrm e}^{-b x -a}}{b^{2}}\) | \(297\) |
| derivativedivides | \(-\frac {c \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )-\frac {d \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b}-\frac {d a \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b}}{b}\) | \(322\) |
| default | \(-\frac {c \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )-\frac {d \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b}-\frac {d a \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b}}{b}\) | \(322\) |
| meijerg | \(\frac {{\mathrm e}^{-a} d \left (120-\frac {\left (6 b^{5} x^{5}+30 b^{4} x^{4}+120 b^{3} x^{3}+360 b^{2} x^{2}+720 b x +720\right ) {\mathrm e}^{-b x}}{6}\right )}{b^{2}}+\frac {{\mathrm e}^{-a} c \left (24-\frac {\left (5 b^{4} x^{4}+20 b^{3} x^{3}+60 b^{2} x^{2}+120 b x +120\right ) {\mathrm e}^{-b x}}{5}\right )}{b}+\frac {4 \,{\mathrm e}^{-a} a d \left (24-\frac {\left (5 b^{4} x^{4}+20 b^{3} x^{3}+60 b^{2} x^{2}+120 b x +120\right ) {\mathrm e}^{-b x}}{5}\right )}{b^{2}}+\frac {4 \,{\mathrm e}^{-a} a c \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b}+\frac {6 \,{\mathrm e}^{-a} a^{2} d \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b^{2}}+\frac {6 \,{\mathrm e}^{-a} a^{2} c \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b}+\frac {4 \,{\mathrm e}^{-a} a^{3} d \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{2}}+\frac {4 \,{\mathrm e}^{-a} a^{3} c \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b}+\frac {{\mathrm e}^{-a} a^{4} d \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {{\mathrm e}^{-a} a^{4} c \left (1-{\mathrm e}^{-b x}\right )}{b}\) | \(390\) |
| parts | \(-d \,b^{3} x^{5} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} b^{2} a d \,x^{4}-{\mathrm e}^{-b x -a} b^{3} c \,x^{4}-6 \,{\mathrm e}^{-b x -a} b \,a^{2} d \,x^{3}-4 \,{\mathrm e}^{-b x -a} b^{2} a c \,x^{3}-4 \,{\mathrm e}^{-b x -a} a^{3} d \,x^{2}-6 \,{\mathrm e}^{-b x -a} b \,a^{2} c \,x^{2}-\frac {{\mathrm e}^{-b x -a} a^{4} d x}{b}-4 \,{\mathrm e}^{-b x -a} a^{3} c x -\frac {{\mathrm e}^{-b x -a} c \,a^{4}}{b}-\frac {5 d \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )-4 c b \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )+4 d a \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}\) | \(439\) |
| parallelrisch | \(-\frac {120 \,{\mathrm e}^{-b x -a} d +d \,b^{5} {\mathrm e}^{-b x -a} x^{5}+x^{4} {\mathrm e}^{-b x -a} b^{5} c +5 x^{4} {\mathrm e}^{-b x -a} b^{4} d +4 x^{3} {\mathrm e}^{-b x -a} b^{4} c +20 x^{3} {\mathrm e}^{-b x -a} b^{3} d +12 x^{2} {\mathrm e}^{-b x -a} b^{3} c +{\mathrm e}^{-b x -a} a^{4} b c +60 x^{2} {\mathrm e}^{-b x -a} b^{2} d +4 \,{\mathrm e}^{-b x -a} a^{3} b c +24 x \,{\mathrm e}^{-b x -a} b^{2} c +12 \,{\mathrm e}^{-b x -a} a^{2} b c +120 x \,{\mathrm e}^{-b x -a} b d +24 \,{\mathrm e}^{-b x -a} a b c +4 x \,{\mathrm e}^{-b x -a} a^{3} b^{2} c +48 x^{2} {\mathrm e}^{-b x -a} a \,b^{2} d +8 x \,{\mathrm e}^{-b x -a} a^{3} b d +12 x \,{\mathrm e}^{-b x -a} a^{2} b^{2} c +36 x \,{\mathrm e}^{-b x -a} a^{2} b d +24 x \,{\mathrm e}^{-b x -a} a \,b^{2} c +96 x \,{\mathrm e}^{-b x -a} a b d +4 x^{4} {\mathrm e}^{-b x -a} a \,b^{4} d +6 x^{3} {\mathrm e}^{-b x -a} a^{2} b^{3} d +4 x^{3} {\mathrm e}^{-b x -a} a \,b^{4} c +16 x^{3} {\mathrm e}^{-b x -a} a \,b^{3} d +4 x^{2} {\mathrm e}^{-b x -a} a^{3} b^{2} d +6 x^{2} {\mathrm e}^{-b x -a} a^{2} b^{3} c +18 x^{2} {\mathrm e}^{-b x -a} a^{2} b^{2} d +12 x^{2} {\mathrm e}^{-b x -a} a \,b^{3} c +x \,{\mathrm e}^{-b x -a} a^{4} b d +{\mathrm e}^{-b x -a} a^{4} d +8 \,{\mathrm e}^{-b x -a} a^{3} d +36 \,{\mathrm e}^{-b x -a} a^{2} d +96 \,{\mathrm e}^{-b x -a} a d +24 \,{\mathrm e}^{-b x -a} b c}{b^{2}}\) | \(603\) |
Input:
int(exp(-b*x-a)*(b*x+a)^4*(d*x+c),x,method=_RETURNVERBOSE)
Output:
(-4*a*b^2*d-b^3*c-5*b^2*d)*x^4*exp(-b*x-a)+(-6*a^2*b*d-4*a*b^2*c-16*a*b*d- 4*b^2*c-20*b*d)*x^3*exp(-b*x-a)+(-4*a^3*d-6*a^2*b*c-18*a^2*d-12*a*b*c-48*a *d-12*b*c-60*d)*x^2*exp(-b*x-a)-(a^4*b*c+a^4*d+4*a^3*b*c+8*a^3*d+12*a^2*b* c+36*a^2*d+24*a*b*c+96*a*d+24*b*c+120*d)/b^2*exp(-b*x-a)-d*b^3*x^5*exp(-b* x-a)-(a^4*d+4*a^3*b*c+8*a^3*d+12*a^2*b*c+36*a^2*d+24*a*b*c+96*a*d+24*b*c+1 20*d)/b*x*exp(-b*x-a)
Time = 0.08 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.72 \[ \int e^{-a-b x} (a+b x)^4 (c+d x) \, dx=-\frac {{\left (b^{5} d x^{5} + {\left (b^{5} c + {\left (4 \, a + 5\right )} b^{4} d\right )} x^{4} + 2 \, {\left (2 \, {\left (a + 1\right )} b^{4} c + {\left (3 \, a^{2} + 8 \, a + 10\right )} b^{3} d\right )} x^{3} + {\left (a^{4} + 4 \, a^{3} + 12 \, a^{2} + 24 \, a + 24\right )} b c + 2 \, {\left (3 \, {\left (a^{2} + 2 \, a + 2\right )} b^{3} c + {\left (2 \, a^{3} + 9 \, a^{2} + 24 \, a + 30\right )} b^{2} d\right )} x^{2} + {\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} d + {\left (4 \, {\left (a^{3} + 3 \, a^{2} + 6 \, a + 6\right )} b^{2} c + {\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} b d\right )} x\right )} e^{\left (-b x - a\right )}}{b^{2}} \] Input:
integrate(exp(-b*x-a)*(b*x+a)^4*(d*x+c),x, algorithm="fricas")
Output:
-(b^5*d*x^5 + (b^5*c + (4*a + 5)*b^4*d)*x^4 + 2*(2*(a + 1)*b^4*c + (3*a^2 + 8*a + 10)*b^3*d)*x^3 + (a^4 + 4*a^3 + 12*a^2 + 24*a + 24)*b*c + 2*(3*(a^ 2 + 2*a + 2)*b^3*c + (2*a^3 + 9*a^2 + 24*a + 30)*b^2*d)*x^2 + (a^4 + 8*a^3 + 36*a^2 + 96*a + 120)*d + (4*(a^3 + 3*a^2 + 6*a + 6)*b^2*c + (a^4 + 8*a^ 3 + 36*a^2 + 96*a + 120)*b*d)*x)*e^(-b*x - a)/b^2
Time = 0.13 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.63 \[ \int e^{-a-b x} (a+b x)^4 (c+d x) \, dx=\begin {cases} \frac {\left (- a^{4} b c - a^{4} b d x - a^{4} d - 4 a^{3} b^{2} c x - 4 a^{3} b^{2} d x^{2} - 4 a^{3} b c - 8 a^{3} b d x - 8 a^{3} d - 6 a^{2} b^{3} c x^{2} - 6 a^{2} b^{3} d x^{3} - 12 a^{2} b^{2} c x - 18 a^{2} b^{2} d x^{2} - 12 a^{2} b c - 36 a^{2} b d x - 36 a^{2} d - 4 a b^{4} c x^{3} - 4 a b^{4} d x^{4} - 12 a b^{3} c x^{2} - 16 a b^{3} d x^{3} - 24 a b^{2} c x - 48 a b^{2} d x^{2} - 24 a b c - 96 a b d x - 96 a d - b^{5} c x^{4} - b^{5} d x^{5} - 4 b^{4} c x^{3} - 5 b^{4} d x^{4} - 12 b^{3} c x^{2} - 20 b^{3} d x^{3} - 24 b^{2} c x - 60 b^{2} d x^{2} - 24 b c - 120 b d x - 120 d\right ) e^{- a - b x}}{b^{2}} & \text {for}\: b^{2} \neq 0 \\a^{4} c x + \frac {b^{4} d x^{6}}{6} + x^{5} \cdot \left (\frac {4 a b^{3} d}{5} + \frac {b^{4} c}{5}\right ) + x^{4} \cdot \left (\frac {3 a^{2} b^{2} d}{2} + a b^{3} c\right ) + x^{3} \cdot \left (\frac {4 a^{3} b d}{3} + 2 a^{2} b^{2} c\right ) + x^{2} \left (\frac {a^{4} d}{2} + 2 a^{3} b c\right ) & \text {otherwise} \end {cases} \] Input:
integrate(exp(-b*x-a)*(b*x+a)**4*(d*x+c),x)
Output:
Piecewise(((-a**4*b*c - a**4*b*d*x - a**4*d - 4*a**3*b**2*c*x - 4*a**3*b** 2*d*x**2 - 4*a**3*b*c - 8*a**3*b*d*x - 8*a**3*d - 6*a**2*b**3*c*x**2 - 6*a **2*b**3*d*x**3 - 12*a**2*b**2*c*x - 18*a**2*b**2*d*x**2 - 12*a**2*b*c - 3 6*a**2*b*d*x - 36*a**2*d - 4*a*b**4*c*x**3 - 4*a*b**4*d*x**4 - 12*a*b**3*c *x**2 - 16*a*b**3*d*x**3 - 24*a*b**2*c*x - 48*a*b**2*d*x**2 - 24*a*b*c - 9 6*a*b*d*x - 96*a*d - b**5*c*x**4 - b**5*d*x**5 - 4*b**4*c*x**3 - 5*b**4*d* x**4 - 12*b**3*c*x**2 - 20*b**3*d*x**3 - 24*b**2*c*x - 60*b**2*d*x**2 - 24 *b*c - 120*b*d*x - 120*d)*exp(-a - b*x)/b**2, Ne(b**2, 0)), (a**4*c*x + b* *4*d*x**6/6 + x**5*(4*a*b**3*d/5 + b**4*c/5) + x**4*(3*a**2*b**2*d/2 + a*b **3*c) + x**3*(4*a**3*b*d/3 + 2*a**2*b**2*c) + x**2*(a**4*d/2 + 2*a**3*b*c ), True))
Time = 0.06 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.25 \[ \int e^{-a-b x} (a+b x)^4 (c+d x) \, dx=-\frac {4 \, {\left (b x + 1\right )} a^{3} c e^{\left (-b x - a\right )}}{b} - \frac {a^{4} c e^{\left (-b x - a\right )}}{b} - \frac {{\left (b x + 1\right )} a^{4} d e^{\left (-b x - a\right )}}{b^{2}} - \frac {6 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{2} c e^{\left (-b x - a\right )}}{b} - \frac {4 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{3} d e^{\left (-b x - a\right )}}{b^{2}} - \frac {4 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a c e^{\left (-b x - a\right )}}{b} - \frac {6 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{2} d e^{\left (-b x - a\right )}}{b^{2}} - \frac {{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} c e^{\left (-b x - a\right )}}{b} - \frac {4 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a d e^{\left (-b x - a\right )}}{b^{2}} - \frac {{\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} d e^{\left (-b x - a\right )}}{b^{2}} \] Input:
integrate(exp(-b*x-a)*(b*x+a)^4*(d*x+c),x, algorithm="maxima")
Output:
-4*(b*x + 1)*a^3*c*e^(-b*x - a)/b - a^4*c*e^(-b*x - a)/b - (b*x + 1)*a^4*d *e^(-b*x - a)/b^2 - 6*(b^2*x^2 + 2*b*x + 2)*a^2*c*e^(-b*x - a)/b - 4*(b^2* x^2 + 2*b*x + 2)*a^3*d*e^(-b*x - a)/b^2 - 4*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*a*c*e^(-b*x - a)/b - 6*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*a^2*d*e^(-b*x - a)/b^2 - (b^4*x^4 + 4*b^3*x^3 + 12*b^2*x^2 + 24*b*x + 24)*c*e^(-b*x - a )/b - 4*(b^4*x^4 + 4*b^3*x^3 + 12*b^2*x^2 + 24*b*x + 24)*a*d*e^(-b*x - a)/ b^2 - (b^5*x^5 + 5*b^4*x^4 + 20*b^3*x^3 + 60*b^2*x^2 + 120*b*x + 120)*d*e^ (-b*x - a)/b^2
Time = 0.12 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.20 \[ \int e^{-a-b x} (a+b x)^4 (c+d x) \, dx=-\frac {{\left (b^{9} d x^{5} + b^{9} c x^{4} + 4 \, a b^{8} d x^{4} + 4 \, a b^{8} c x^{3} + 6 \, a^{2} b^{7} d x^{3} + 5 \, b^{8} d x^{4} + 6 \, a^{2} b^{7} c x^{2} + 4 \, a^{3} b^{6} d x^{2} + 4 \, b^{8} c x^{3} + 16 \, a b^{7} d x^{3} + 4 \, a^{3} b^{6} c x + a^{4} b^{5} d x + 12 \, a b^{7} c x^{2} + 18 \, a^{2} b^{6} d x^{2} + 20 \, b^{7} d x^{3} + a^{4} b^{5} c + 12 \, a^{2} b^{6} c x + 8 \, a^{3} b^{5} d x + 12 \, b^{7} c x^{2} + 48 \, a b^{6} d x^{2} + 4 \, a^{3} b^{5} c + a^{4} b^{4} d + 24 \, a b^{6} c x + 36 \, a^{2} b^{5} d x + 60 \, b^{6} d x^{2} + 12 \, a^{2} b^{5} c + 8 \, a^{3} b^{4} d + 24 \, b^{6} c x + 96 \, a b^{5} d x + 24 \, a b^{5} c + 36 \, a^{2} b^{4} d + 120 \, b^{5} d x + 24 \, b^{5} c + 96 \, a b^{4} d + 120 \, b^{4} d\right )} e^{\left (-b x - a\right )}}{b^{6}} \] Input:
integrate(exp(-b*x-a)*(b*x+a)^4*(d*x+c),x, algorithm="giac")
Output:
-(b^9*d*x^5 + b^9*c*x^4 + 4*a*b^8*d*x^4 + 4*a*b^8*c*x^3 + 6*a^2*b^7*d*x^3 + 5*b^8*d*x^4 + 6*a^2*b^7*c*x^2 + 4*a^3*b^6*d*x^2 + 4*b^8*c*x^3 + 16*a*b^7 *d*x^3 + 4*a^3*b^6*c*x + a^4*b^5*d*x + 12*a*b^7*c*x^2 + 18*a^2*b^6*d*x^2 + 20*b^7*d*x^3 + a^4*b^5*c + 12*a^2*b^6*c*x + 8*a^3*b^5*d*x + 12*b^7*c*x^2 + 48*a*b^6*d*x^2 + 4*a^3*b^5*c + a^4*b^4*d + 24*a*b^6*c*x + 36*a^2*b^5*d*x + 60*b^6*d*x^2 + 12*a^2*b^5*c + 8*a^3*b^4*d + 24*b^6*c*x + 96*a*b^5*d*x + 24*a*b^5*c + 36*a^2*b^4*d + 120*b^5*d*x + 24*b^5*c + 96*a*b^4*d + 120*b^4 *d)*e^(-b*x - a)/b^6
Time = 22.89 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.96 \[ \int e^{-a-b x} (a+b x)^4 (c+d x) \, dx=-\frac {{\mathrm {e}}^{-a-b\,x}\,\left (120\,d+96\,a\,d+24\,b\,c+36\,a^2\,d+8\,a^3\,d+a^4\,d+24\,a\,b\,c+12\,a^2\,b\,c+4\,a^3\,b\,c+a^4\,b\,c\right )}{b^2}-x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (60\,d+48\,a\,d+12\,b\,c+18\,a^2\,d+4\,a^3\,d+12\,a\,b\,c+6\,a^2\,b\,c\right )-x\,{\mathrm {e}}^{-a-b\,x}\,\left (24\,c+24\,a\,c+12\,a^2\,c+4\,a^3\,c+\frac {d\,a^4+8\,d\,a^3+36\,d\,a^2+96\,d\,a+120\,d}{b}\right )-b^3\,d\,x^5\,{\mathrm {e}}^{-a-b\,x}-b^2\,x^4\,{\mathrm {e}}^{-a-b\,x}\,\left (5\,d+4\,a\,d+b\,c\right )-2\,b\,x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (10\,d+8\,a\,d+2\,b\,c+3\,a^2\,d+2\,a\,b\,c\right ) \] Input:
int(exp(- a - b*x)*(a + b*x)^4*(c + d*x),x)
Output:
- (exp(- a - b*x)*(120*d + 96*a*d + 24*b*c + 36*a^2*d + 8*a^3*d + a^4*d + 24*a*b*c + 12*a^2*b*c + 4*a^3*b*c + a^4*b*c))/b^2 - x^2*exp(- a - b*x)*(60 *d + 48*a*d + 12*b*c + 18*a^2*d + 4*a^3*d + 12*a*b*c + 6*a^2*b*c) - x*exp( - a - b*x)*(24*c + 24*a*c + 12*a^2*c + 4*a^3*c + (120*d + 96*a*d + 36*a^2* d + 8*a^3*d + a^4*d)/b) - b^3*d*x^5*exp(- a - b*x) - b^2*x^4*exp(- a - b*x )*(5*d + 4*a*d + b*c) - 2*b*x^3*exp(- a - b*x)*(10*d + 8*a*d + 2*b*c + 3*a ^2*d + 2*a*b*c)
Time = 0.18 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.09 \[ \int e^{-a-b x} (a+b x)^4 (c+d x) \, dx=\frac {-b^{5} d \,x^{5}-4 a \,b^{4} d \,x^{4}-b^{5} c \,x^{4}-6 a^{2} b^{3} d \,x^{3}-4 a \,b^{4} c \,x^{3}-5 b^{4} d \,x^{4}-4 a^{3} b^{2} d \,x^{2}-6 a^{2} b^{3} c \,x^{2}-16 a \,b^{3} d \,x^{3}-4 b^{4} c \,x^{3}-a^{4} b d x -4 a^{3} b^{2} c x -18 a^{2} b^{2} d \,x^{2}-12 a \,b^{3} c \,x^{2}-20 b^{3} d \,x^{3}-a^{4} b c -8 a^{3} b d x -12 a^{2} b^{2} c x -48 a \,b^{2} d \,x^{2}-12 b^{3} c \,x^{2}-a^{4} d -4 a^{3} b c -36 a^{2} b d x -24 a \,b^{2} c x -60 b^{2} d \,x^{2}-8 a^{3} d -12 a^{2} b c -96 a b d x -24 b^{2} c x -36 a^{2} d -24 a b c -120 b d x -96 a d -24 b c -120 d}{e^{b x +a} b^{2}} \] Input:
int(exp(-b*x-a)*(b*x+a)^4*(d*x+c),x)
Output:
( - a**4*b*c - a**4*b*d*x - a**4*d - 4*a**3*b**2*c*x - 4*a**3*b**2*d*x**2 - 4*a**3*b*c - 8*a**3*b*d*x - 8*a**3*d - 6*a**2*b**3*c*x**2 - 6*a**2*b**3* d*x**3 - 12*a**2*b**2*c*x - 18*a**2*b**2*d*x**2 - 12*a**2*b*c - 36*a**2*b* d*x - 36*a**2*d - 4*a*b**4*c*x**3 - 4*a*b**4*d*x**4 - 12*a*b**3*c*x**2 - 1 6*a*b**3*d*x**3 - 24*a*b**2*c*x - 48*a*b**2*d*x**2 - 24*a*b*c - 96*a*b*d*x - 96*a*d - b**5*c*x**4 - b**5*d*x**5 - 4*b**4*c*x**3 - 5*b**4*d*x**4 - 12 *b**3*c*x**2 - 20*b**3*d*x**3 - 24*b**2*c*x - 60*b**2*d*x**2 - 24*b*c - 12 0*b*d*x - 120*d)/(e**(a + b*x)*b**2)