Integrand size = 21, antiderivative size = 227 \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=-\frac {720 e^{-a-b x}}{b^4}-\frac {e^{-a-b x} x^3 (a+b x)^3}{b}-\frac {360 e^{-a-b x} (a+2 b x)}{b^4}-\frac {72 e^{-a-b x} \left (a^2+5 a b x+5 b^2 x^2\right )}{b^4}-\frac {3 e^{-a-b x} x^2 \left (a^3+4 a^2 b x+5 a b^2 x^2+2 b^3 x^3\right )}{b^2}-\frac {6 e^{-a-b x} x \left (a^3+6 a^2 b x+10 a b^2 x^2+5 b^3 x^3\right )}{b^3}-\frac {6 e^{-a-b x} \left (a^3+12 a^2 b x+30 a b^2 x^2+20 b^3 x^3\right )}{b^4} \] Output:
-720*exp(-b*x-a)/b^4-exp(-b*x-a)*x^3*(b*x+a)^3/b-360*exp(-b*x-a)*(2*b*x+a) /b^4-72*exp(-b*x-a)*(5*b^2*x^2+5*a*b*x+a^2)/b^4-3*exp(-b*x-a)*x^2*(2*b^3*x ^3+5*a*b^2*x^2+4*a^2*b*x+a^3)/b^2-6*exp(-b*x-a)*x*(5*b^3*x^3+10*a*b^2*x^2+ 6*a^2*b*x+a^3)/b^3-6*exp(-b*x-a)*(20*b^3*x^3+30*a*b^2*x^2+12*a^2*b*x+a^3)/ b^4
Time = 0.40 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.53 \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=e^{-a-b x} \left (-\frac {6 \left (120+60 a+12 a^2+a^3\right )}{b^4}-\frac {6 \left (120+60 a+12 a^2+a^3\right ) x}{b^3}-\frac {3 \left (120+60 a+12 a^2+a^3\right ) x^2}{b^2}-\frac {\left (120+60 a+12 a^2+a^3\right ) x^3}{b}-3 \left (10+5 a+a^2\right ) x^4-3 (2+a) b x^5-b^2 x^6\right ) \] Input:
Integrate[E^(-a - b*x)*x^3*(a + b*x)^3,x]
Output:
E^(-a - b*x)*((-6*(120 + 60*a + 12*a^2 + a^3))/b^4 - (6*(120 + 60*a + 12*a ^2 + a^3)*x)/b^3 - (3*(120 + 60*a + 12*a^2 + a^3)*x^2)/b^2 - ((120 + 60*a + 12*a^2 + a^3)*x^3)/b - 3*(10 + 5*a + a^2)*x^4 - 3*(2 + a)*b*x^5 - b^2*x^ 6)
Time = 1.32 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.75, 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.
| \(\displaystyle \int x^3 e^{-a-b x} (a+b x)^3 \, dx\) |
| \(\Big \downarrow \) 2626 |
| \(\displaystyle \int \left (a^3 x^3 e^{-a-b x}+3 a^2 b x^4 e^{-a-b x}+b^3 x^6 e^{-a-b x}+3 a b^2 x^5 e^{-a-b x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {6 a^3 e^{-a-b x}}{b^4}-\frac {6 a^3 x e^{-a-b x}}{b^3}-\frac {3 a^3 x^2 e^{-a-b x}}{b^2}-\frac {a^3 x^3 e^{-a-b x}}{b}-\frac {72 a^2 e^{-a-b x}}{b^4}-\frac {72 a^2 x e^{-a-b x}}{b^3}-\frac {36 a^2 x^2 e^{-a-b x}}{b^2}-3 a^2 x^4 e^{-a-b x}-\frac {12 a^2 x^3 e^{-a-b x}}{b}-\frac {360 a e^{-a-b x}}{b^4}-\frac {720 e^{-a-b x}}{b^4}-\frac {360 a x e^{-a-b x}}{b^3}-\frac {720 x e^{-a-b x}}{b^3}-b^2 x^6 e^{-a-b x}-\frac {180 a x^2 e^{-a-b x}}{b^2}-\frac {360 x^2 e^{-a-b x}}{b^2}-3 a b x^5 e^{-a-b x}-6 b x^5 e^{-a-b x}-15 a x^4 e^{-a-b x}-30 x^4 e^{-a-b x}-\frac {60 a x^3 e^{-a-b x}}{b}-\frac {120 x^3 e^{-a-b x}}{b}\) |
Input:
Int[E^(-a - b*x)*x^3*(a + b*x)^3,x]
Output:
(-720*E^(-a - b*x))/b^4 - (360*a*E^(-a - b*x))/b^4 - (72*a^2*E^(-a - b*x)) /b^4 - (6*a^3*E^(-a - b*x))/b^4 - (720*E^(-a - b*x)*x)/b^3 - (360*a*E^(-a - b*x)*x)/b^3 - (72*a^2*E^(-a - b*x)*x)/b^3 - (6*a^3*E^(-a - b*x)*x)/b^3 - (360*E^(-a - b*x)*x^2)/b^2 - (180*a*E^(-a - b*x)*x^2)/b^2 - (36*a^2*E^(-a - b*x)*x^2)/b^2 - (3*a^3*E^(-a - b*x)*x^2)/b^2 - (120*E^(-a - b*x)*x^3)/b - (60*a*E^(-a - b*x)*x^3)/b - (12*a^2*E^(-a - b*x)*x^3)/b - (a^3*E^(-a - b*x)*x^3)/b - 30*E^(-a - b*x)*x^4 - 15*a*E^(-a - b*x)*x^4 - 3*a^2*E^(-a - b*x)*x^4 - 6*b*E^(-a - b*x)*x^5 - 3*a*b*E^(-a - b*x)*x^5 - b^2*E^(-a - b*x )*x^6
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 = 178, normalized size of antiderivative = 0.78
| method | result | size |
| norman | \(\left (-3 a b -6 b \right ) x^{5} {\mathrm e}^{-b x -a}+\left (-3 a^{2}-15 a -30\right ) x^{4} {\mathrm e}^{-b x -a}-b^{2} x^{6} {\mathrm e}^{-b x -a}-\frac {6 \left (a^{3}+12 a^{2}+60 a +120\right ) {\mathrm e}^{-b x -a}}{b^{4}}-\frac {6 \left (a^{3}+12 a^{2}+60 a +120\right ) x \,{\mathrm e}^{-b x -a}}{b^{3}}-\frac {3 \left (a^{3}+12 a^{2}+60 a +120\right ) x^{2} {\mathrm e}^{-b x -a}}{b^{2}}-\frac {\left (a^{3}+12 a^{2}+60 a +120\right ) x^{3} {\mathrm e}^{-b x -a}}{b}\) | \(178\) |
| gosper | \(-\frac {\left (b^{6} x^{6}+3 b^{5} x^{5} a +3 a^{2} b^{4} x^{4}+6 b^{5} x^{5}+a^{3} b^{3} x^{3}+15 a \,b^{4} x^{4}+12 a^{2} b^{3} x^{3}+30 b^{4} x^{4}+3 a^{3} b^{2} x^{2}+60 a \,b^{3} x^{3}+36 a^{2} b^{2} x^{2}+120 b^{3} x^{3}+6 a^{3} b x +180 a \,b^{2} x^{2}+72 a^{2} b x +360 b^{2} x^{2}+6 a^{3}+360 a b x +72 a^{2}+720 b x +360 a +720\right ) {\mathrm e}^{-b x -a}}{b^{4}}\) | \(182\) |
| risch | \(-\frac {\left (b^{6} x^{6}+3 b^{5} x^{5} a +3 a^{2} b^{4} x^{4}+6 b^{5} x^{5}+a^{3} b^{3} x^{3}+15 a \,b^{4} x^{4}+12 a^{2} b^{3} x^{3}+30 b^{4} x^{4}+3 a^{3} b^{2} x^{2}+60 a \,b^{3} x^{3}+36 a^{2} b^{2} x^{2}+120 b^{3} x^{3}+6 a^{3} b x +180 a \,b^{2} x^{2}+72 a^{2} b x +360 b^{2} x^{2}+6 a^{3}+360 a b x +72 a^{2}+720 b x +360 a +720\right ) {\mathrm e}^{-b x -a}}{b^{4}}\) | \(182\) |
| orering | \(-\frac {\left (b^{6} x^{6}+3 b^{5} x^{5} a +3 a^{2} b^{4} x^{4}+6 b^{5} x^{5}+a^{3} b^{3} x^{3}+15 a \,b^{4} x^{4}+12 a^{2} b^{3} x^{3}+30 b^{4} x^{4}+3 a^{3} b^{2} x^{2}+60 a \,b^{3} x^{3}+36 a^{2} b^{2} x^{2}+120 b^{3} x^{3}+6 a^{3} b x +180 a \,b^{2} x^{2}+72 a^{2} b x +360 b^{2} x^{2}+6 a^{3}+360 a b x +72 a^{2}+720 b x +360 a +720\right ) {\mathrm e}^{-b x -a}}{b^{4}}\) | \(182\) |
| meijerg | \(\frac {{\mathrm e}^{-a} \left (720-\frac {\left (7 b^{6} x^{6}+42 b^{5} x^{5}+210 b^{4} x^{4}+840 b^{3} x^{3}+2520 b^{2} x^{2}+5040 b x +5040\right ) {\mathrm e}^{-b x}}{7}\right )}{b^{4}}+\frac {3 \,{\mathrm e}^{-a} a \left (120-\frac {\left (6 b^{5} x^{5}+30 b^{4} x^{4}+120 b^{3} x^{3}+360 b^{2} x^{2}+720 b x +720\right ) {\mathrm e}^{-b x}}{6}\right )}{b^{4}}+\frac {3 \,{\mathrm e}^{-a} a^{2} \left (24-\frac {\left (5 b^{4} x^{4}+20 b^{3} x^{3}+60 b^{2} x^{2}+120 b x +120\right ) {\mathrm e}^{-b x}}{5}\right )}{b^{4}}+\frac {{\mathrm e}^{-a} a^{3} \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b^{4}}\) | \(215\) |
| parallelrisch | \(-\frac {b^{6} {\mathrm e}^{-b x -a} x^{6}+3 b^{5} {\mathrm e}^{-b x -a} x^{5} a +6 b^{5} {\mathrm e}^{-b x -a} x^{5}+3 x^{4} {\mathrm e}^{-b x -a} a^{2} b^{4}+15 x^{4} {\mathrm e}^{-b x -a} a \,b^{4}+x^{3} {\mathrm e}^{-b x -a} a^{3} b^{3}+30 b^{4} x^{4} {\mathrm e}^{-b x -a}+12 x^{3} {\mathrm e}^{-b x -a} a^{2} b^{3}+60 a \,b^{3} x^{3} {\mathrm e}^{-b x -a}+3 x^{2} {\mathrm e}^{-b x -a} a^{3} b^{2}+120 \,{\mathrm e}^{-b x -a} x^{3} b^{3}+36 a^{2} b^{2} x^{2} {\mathrm e}^{-b x -a}+180 x^{2} {\mathrm e}^{-b x -a} a \,b^{2}+6 a^{3} b x \,{\mathrm e}^{-b x -a}+360 b^{2} {\mathrm e}^{-b x -a} x^{2}+72 x \,{\mathrm e}^{-b x -a} a^{2} b +360 a b \,{\mathrm e}^{-b x -a} x +6 \,{\mathrm e}^{-b x -a} a^{3}+720 b \,{\mathrm e}^{-b x -a} x +72 a^{2} {\mathrm e}^{-b x -a}+360 a \,{\mathrm e}^{-b x -a}+720 \,{\mathrm e}^{-b x -a}}{b^{4}}\) | \(372\) |
| derivativedivides | \(-\frac {{\mathrm e}^{-b x -a} \left (-b x -a \right )^{6}-6 \left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}+30 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+360 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-720 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+720 \,{\mathrm e}^{-b x -a}+a^{3} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )+3 a^{2} \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )+3 a \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b^{4}}\) | \(432\) |
| default | \(-\frac {{\mathrm e}^{-b x -a} \left (-b x -a \right )^{6}-6 \left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}+30 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+360 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-720 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+720 \,{\mathrm e}^{-b x -a}+a^{3} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )+3 a^{2} \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )+3 a \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b^{4}}\) | \(432\) |
| parts | \(-b^{2} x^{6} {\mathrm e}^{-b x -a}-3 \,{\mathrm e}^{-b x -a} x^{5} b a -3 \,{\mathrm e}^{-b x -a} x^{4} a^{2}-\frac {{\mathrm e}^{-b x -a} x^{3} a^{3}}{b}-\frac {3 \left (-\frac {2 \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {a^{3} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {4 a^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {5 a \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}\right )}{b^{2}}\) | \(435\) |
Input:
int(exp(-b*x-a)*x^3*(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
(-3*a*b-6*b)*x^5*exp(-b*x-a)+(-3*a^2-15*a-30)*x^4*exp(-b*x-a)-b^2*x^6*exp( -b*x-a)-6*(a^3+12*a^2+60*a+120)/b^4*exp(-b*x-a)-6*(a^3+12*a^2+60*a+120)/b^ 3*x*exp(-b*x-a)-3*(a^3+12*a^2+60*a+120)/b^2*x^2*exp(-b*x-a)-(a^3+12*a^2+60 *a+120)/b*x^3*exp(-b*x-a)
Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.53 \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=-\frac {{\left (b^{6} x^{6} + 3 \, {\left (a + 2\right )} b^{5} x^{5} + 3 \, {\left (a^{2} + 5 \, a + 10\right )} b^{4} x^{4} + {\left (a^{3} + 12 \, a^{2} + 60 \, a + 120\right )} b^{3} x^{3} + 3 \, {\left (a^{3} + 12 \, a^{2} + 60 \, a + 120\right )} b^{2} x^{2} + 6 \, a^{3} + 6 \, {\left (a^{3} + 12 \, a^{2} + 60 \, a + 120\right )} b x + 72 \, a^{2} + 360 \, a + 720\right )} e^{\left (-b x - a\right )}}{b^{4}} \] Input:
integrate(exp(-b*x-a)*x^3*(b*x+a)^3,x, algorithm="fricas")
Output:
-(b^6*x^6 + 3*(a + 2)*b^5*x^5 + 3*(a^2 + 5*a + 10)*b^4*x^4 + (a^3 + 12*a^2 + 60*a + 120)*b^3*x^3 + 3*(a^3 + 12*a^2 + 60*a + 120)*b^2*x^2 + 6*a^3 + 6 *(a^3 + 12*a^2 + 60*a + 120)*b*x + 72*a^2 + 360*a + 720)*e^(-b*x - a)/b^4
Time = 0.11 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.04 \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=\begin {cases} \frac {\left (- a^{3} b^{3} x^{3} - 3 a^{3} b^{2} x^{2} - 6 a^{3} b x - 6 a^{3} - 3 a^{2} b^{4} x^{4} - 12 a^{2} b^{3} x^{3} - 36 a^{2} b^{2} x^{2} - 72 a^{2} b x - 72 a^{2} - 3 a b^{5} x^{5} - 15 a b^{4} x^{4} - 60 a b^{3} x^{3} - 180 a b^{2} x^{2} - 360 a b x - 360 a - b^{6} x^{6} - 6 b^{5} x^{5} - 30 b^{4} x^{4} - 120 b^{3} x^{3} - 360 b^{2} x^{2} - 720 b x - 720\right ) e^{- a - b x}}{b^{4}} & \text {for}\: b^{4} \neq 0 \\\frac {a^{3} x^{4}}{4} + \frac {3 a^{2} b x^{5}}{5} + \frac {a b^{2} x^{6}}{2} + \frac {b^{3} x^{7}}{7} & \text {otherwise} \end {cases} \] Input:
integrate(exp(-b*x-a)*x**3*(b*x+a)**3,x)
Output:
Piecewise(((-a**3*b**3*x**3 - 3*a**3*b**2*x**2 - 6*a**3*b*x - 6*a**3 - 3*a **2*b**4*x**4 - 12*a**2*b**3*x**3 - 36*a**2*b**2*x**2 - 72*a**2*b*x - 72*a **2 - 3*a*b**5*x**5 - 15*a*b**4*x**4 - 60*a*b**3*x**3 - 180*a*b**2*x**2 - 360*a*b*x - 360*a - 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)*exp(-a - b*x)/b**4, Ne(b**4, 0)), (a**3* x**4/4 + 3*a**2*b*x**5/5 + a*b**2*x**6/2 + b**3*x**7/7, True))
Time = 0.06 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.86 \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=-\frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{3} e^{\left (-b x - a\right )}}{b^{4}} - \frac {3 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a^{2} e^{\left (-b x - a\right )}}{b^{4}} - \frac {3 \, {\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} a e^{\left (-b x - a\right )}}{b^{4}} - \frac {{\left (b^{6} x^{6} + 6 \, b^{5} x^{5} + 30 \, b^{4} x^{4} + 120 \, b^{3} x^{3} + 360 \, b^{2} x^{2} + 720 \, b x + 720\right )} e^{\left (-b x - a\right )}}{b^{4}} \] Input:
integrate(exp(-b*x-a)*x^3*(b*x+a)^3,x, algorithm="maxima")
Output:
-(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*a^3*e^(-b*x - a)/b^4 - 3*(b^4*x^4 + 4*b ^3*x^3 + 12*b^2*x^2 + 24*b*x + 24)*a^2*e^(-b*x - a)/b^4 - 3*(b^5*x^5 + 5*b ^4*x^4 + 20*b^3*x^3 + 60*b^2*x^2 + 120*b*x + 120)*a*e^(-b*x - a)/b^4 - (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 )*e^(-b*x - a)/b^4
Time = 0.12 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.89 \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=-\frac {{\left (b^{9} x^{6} + 3 \, a b^{8} x^{5} + 3 \, a^{2} b^{7} x^{4} + 6 \, b^{8} x^{5} + a^{3} b^{6} x^{3} + 15 \, a b^{7} x^{4} + 12 \, a^{2} b^{6} x^{3} + 30 \, b^{7} x^{4} + 3 \, a^{3} b^{5} x^{2} + 60 \, a b^{6} x^{3} + 36 \, a^{2} b^{5} x^{2} + 120 \, b^{6} x^{3} + 6 \, a^{3} b^{4} x + 180 \, a b^{5} x^{2} + 72 \, a^{2} b^{4} x + 360 \, b^{5} x^{2} + 6 \, a^{3} b^{3} + 360 \, a b^{4} x + 72 \, a^{2} b^{3} + 720 \, b^{4} x + 360 \, a b^{3} + 720 \, b^{3}\right )} e^{\left (-b x - a\right )}}{b^{7}} \] Input:
integrate(exp(-b*x-a)*x^3*(b*x+a)^3,x, algorithm="giac")
Output:
-(b^9*x^6 + 3*a*b^8*x^5 + 3*a^2*b^7*x^4 + 6*b^8*x^5 + a^3*b^6*x^3 + 15*a*b ^7*x^4 + 12*a^2*b^6*x^3 + 30*b^7*x^4 + 3*a^3*b^5*x^2 + 60*a*b^6*x^3 + 36*a ^2*b^5*x^2 + 120*b^6*x^3 + 6*a^3*b^4*x + 180*a*b^5*x^2 + 72*a^2*b^4*x + 36 0*b^5*x^2 + 6*a^3*b^3 + 360*a*b^4*x + 72*a^2*b^3 + 720*b^4*x + 360*a*b^3 + 720*b^3)*e^(-b*x - a)/b^7
Time = 22.45 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.77 \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=-x^4\,{\mathrm {e}}^{-a-b\,x}\,\left (3\,a^2+15\,a+30\right )-b^2\,x^6\,{\mathrm {e}}^{-a-b\,x}-\frac {6\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+12\,a^2+60\,a+120\right )}{b^4}-3\,b\,x^5\,{\mathrm {e}}^{-a-b\,x}\,\left (a+2\right )-\frac {6\,x\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+12\,a^2+60\,a+120\right )}{b^3}-\frac {x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+12\,a^2+60\,a+120\right )}{b}-\frac {3\,x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+12\,a^2+60\,a+120\right )}{b^2} \] Input:
int(x^3*exp(- a - b*x)*(a + b*x)^3,x)
Output:
- x^4*exp(- a - b*x)*(15*a + 3*a^2 + 30) - b^2*x^6*exp(- a - b*x) - (6*exp (- a - b*x)*(60*a + 12*a^2 + a^3 + 120))/b^4 - 3*b*x^5*exp(- a - b*x)*(a + 2) - (6*x*exp(- a - b*x)*(60*a + 12*a^2 + a^3 + 120))/b^3 - (x^3*exp(- a - b*x)*(60*a + 12*a^2 + a^3 + 120))/b - (3*x^2*exp(- a - b*x)*(60*a + 12*a ^2 + a^3 + 120))/b^2
Time = 0.17 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.80 \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=\frac {-b^{6} x^{6}-3 a \,b^{5} x^{5}-3 a^{2} b^{4} x^{4}-6 b^{5} x^{5}-a^{3} b^{3} x^{3}-15 a \,b^{4} x^{4}-12 a^{2} b^{3} x^{3}-30 b^{4} x^{4}-3 a^{3} b^{2} x^{2}-60 a \,b^{3} x^{3}-36 a^{2} b^{2} x^{2}-120 b^{3} x^{3}-6 a^{3} b x -180 a \,b^{2} x^{2}-72 a^{2} b x -360 b^{2} x^{2}-6 a^{3}-360 a b x -72 a^{2}-720 b x -360 a -720}{e^{b x +a} b^{4}} \] Input:
int(exp(-b*x-a)*x^3*(b*x+a)^3,x)
Output:
( - a**3*b**3*x**3 - 3*a**3*b**2*x**2 - 6*a**3*b*x - 6*a**3 - 3*a**2*b**4* x**4 - 12*a**2*b**3*x**3 - 36*a**2*b**2*x**2 - 72*a**2*b*x - 72*a**2 - 3*a *b**5*x**5 - 15*a*b**4*x**4 - 60*a*b**3*x**3 - 180*a*b**2*x**2 - 360*a*b*x - 360*a - 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)/(e**(a + b*x)*b**4)