Integrand size = 21, antiderivative size = 318 \[ \int e^{-a-b x} x^2 (a+b x)^3 \, dx=-\frac {120 e^{-a-b x}}{b^3}-\frac {72 a e^{-a-b x}}{b^3}-\frac {18 a^2 e^{-a-b x}}{b^3}-\frac {2 a^3 e^{-a-b x}}{b^3}-\frac {120 e^{-a-b x} x}{b^2}-\frac {72 a e^{-a-b x} x}{b^2}-\frac {18 a^2 e^{-a-b x} x}{b^2}-\frac {2 a^3 e^{-a-b x} x}{b^2}-\frac {60 e^{-a-b x} x^2}{b}-\frac {36 a e^{-a-b x} x^2}{b}-\frac {9 a^2 e^{-a-b x} x^2}{b}-\frac {a^3 e^{-a-b x} x^2}{b}-20 e^{-a-b x} x^3-12 a e^{-a-b x} x^3-3 a^2 e^{-a-b x} x^3-5 b e^{-a-b x} x^4-3 a b e^{-a-b x} x^4-b^2 e^{-a-b x} x^5 \]
-120*exp(-b*x-a)/b^3-72*a*exp(-b*x-a)/b^3-18*a^2*exp(-b*x-a)/b^3-2*a^3*exp (-b*x-a)/b^3-120*exp(-b*x-a)*x/b^2-72*a*exp(-b*x-a)*x/b^2-18*a^2*exp(-b*x- a)*x/b^2-2*a^3*exp(-b*x-a)*x/b^2-60*exp(-b*x-a)*x^2/b-36*a*exp(-b*x-a)*x^2 /b-9*a^2*exp(-b*x-a)*x^2/b-a^3*exp(-b*x-a)*x^2/b-20*exp(-b*x-a)*x^3-12*a*e xp(-b*x-a)*x^3-3*a^2*exp(-b*x-a)*x^3-5*b*exp(-b*x-a)*x^4-3*a*b*exp(-b*x-a) *x^4-b^2*exp(-b*x-a)*x^5
Time = 0.22 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.41 \[ \int e^{-a-b x} x^2 (a+b x)^3 \, dx=e^{-b x} \left (-\frac {2 \left (60+36 a+9 a^2+a^3\right ) e^{-a}}{b^3}-\frac {2 \left (60+36 a+9 a^2+a^3\right ) e^{-a} x}{b^2}-\frac {\left (60+36 a+9 a^2+a^3\right ) e^{-a} x^2}{b}-\left (20+12 a+3 a^2\right ) e^{-a} x^3-(5+3 a) b e^{-a} x^4-b^2 e^{-a} x^5\right ) \]
((-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)
Time = 0.64 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.00, 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 |
\(\displaystyle -\frac {2 a^3 e^{-a-b x}}{b^3}-\frac {2 a^3 x e^{-a-b x}}{b^2}-\frac {a^3 x^2 e^{-a-b x}}{b}-\frac {18 a^2 e^{-a-b x}}{b^3}-\frac {18 a^2 x e^{-a-b x}}{b^2}-3 a^2 x^3 e^{-a-b x}-\frac {9 a^2 x^2 e^{-a-b x}}{b}-\frac {72 a e^{-a-b x}}{b^3}-\frac {120 e^{-a-b x}}{b^3}-b^2 x^5 e^{-a-b x}-\frac {72 a x e^{-a-b x}}{b^2}-\frac {120 x e^{-a-b x}}{b^2}-3 a b x^4 e^{-a-b x}-5 b x^4 e^{-a-b x}-12 a x^3 e^{-a-b x}-20 x^3 e^{-a-b x}-\frac {36 a x^2 e^{-a-b x}}{b}-\frac {60 x^2 e^{-a-b x}}{b}\) |
(-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
3.1.57.3.1 Defintions of rubi rules used
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 = 0.16 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.45
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 | \(-\frac {\left (b^{5} x^{5}+3 a \,b^{4} x^{4}+3 a^{2} b^{3} x^{3}+5 b^{4} x^{4}+a^{3} b^{2} x^{2}+12 a \,b^{3} x^{3}+9 a^{2} b^{2} x^{2}+20 b^{3} x^{3}+2 a^{3} b x +36 a \,b^{2} x^{2}+18 a^{2} b x +60 b^{2} x^{2}+2 a^{3}+72 a b x +18 a^{2}+120 b x +72 a +120\right ) {\mathrm e}^{-b x -a}}{b^{3}}\) | \(143\) |
norman | \(\left (-3 a b -5 b \right ) x^{4} {\mathrm e}^{-b x -a}+\left (-3 a^{2}-12 a -20\right ) x^{3} {\mathrm e}^{-b x -a}-b^{2} {\mathrm e}^{-b x -a} x^{5}-\frac {2 \left (a^{3}+9 a^{2}+36 a +60\right ) {\mathrm e}^{-b x -a}}{b^{3}}-\frac {2 \left (a^{3}+9 a^{2}+36 a +60\right ) x \,{\mathrm e}^{-b x -a}}{b^{2}}-\frac {\left (a^{3}+9 a^{2}+36 a +60\right ) x^{2} {\mathrm e}^{-b x -a}}{b}\) | \(148\) |
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 | \(-\frac {b^{5} {\mathrm e}^{-b x -a} x^{5}+3 x^{4} {\mathrm e}^{-b x -a} a \,b^{4}+5 \,{\mathrm e}^{-b x -a} b^{4} x^{4}+3 x^{3} {\mathrm e}^{-b x -a} a^{2} b^{3}+12 \,{\mathrm e}^{-b x -a} a \,b^{3} x^{3}+x^{2} {\mathrm e}^{-b x -a} a^{3} b^{2}+20 \,{\mathrm e}^{-b x -a} x^{3} b^{3}+9 \,{\mathrm e}^{-b x -a} a^{2} b^{2} x^{2}+36 x^{2} {\mathrm e}^{-b x -a} a \,b^{2}+2 \,{\mathrm e}^{-b x -a} a^{3} b x +60 b^{2} {\mathrm e}^{-b x -a} x^{2}+18 x \,{\mathrm e}^{-b x -a} a^{2} b +72 a b \,{\mathrm e}^{-b x -a} x +2 \,{\mathrm e}^{-b x -a} a^{3}+120 b \,{\mathrm e}^{-b x -a} x +18 a^{2} {\mathrm e}^{-b x -a}+72 a \,{\mathrm e}^{-b x -a}+120 \,{\mathrm e}^{-b x -a}}{b^{3}}\) | \(297\) |
parts | \(-b^{2} {\mathrm e}^{-b x -a} x^{5}-3 a b \,{\mathrm e}^{-b x -a} x^{4}-3 a^{2} {\mathrm e}^{-b x -a} x^{3}-\frac {a^{3} {\mathrm e}^{-b x -a} x^{2}}{b}-\frac {\frac {5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+120 \,{\mathrm e}^{-b x -a}}{b}+\frac {3 a^{2} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b}+\frac {8 a \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b}}{b^{2}}\) | \(313\) |
-(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
Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.32 \[ \int e^{-a-b x} x^2 (a+b x)^3 \, dx=-\frac {{\left (b^{5} x^{5} + {\left (3 \, a + 5\right )} b^{4} x^{4} + {\left (3 \, a^{2} + 12 \, a + 20\right )} b^{3} x^{3} + {\left (a^{3} + 9 \, a^{2} + 36 \, a + 60\right )} b^{2} x^{2} + 2 \, a^{3} + 2 \, {\left (a^{3} + 9 \, a^{2} + 36 \, a + 60\right )} b x + 18 \, a^{2} + 72 \, a + 120\right )} e^{\left (-b x - a\right )}}{b^{3}} \]
-(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
Time = 0.10 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.62 \[ \int e^{-a-b x} x^2 (a+b x)^3 \, dx=\begin {cases} \frac {\left (- a^{3} b^{2} x^{2} - 2 a^{3} b x - 2 a^{3} - 3 a^{2} b^{3} x^{3} - 9 a^{2} b^{2} x^{2} - 18 a^{2} b x - 18 a^{2} - 3 a b^{4} x^{4} - 12 a b^{3} x^{3} - 36 a b^{2} x^{2} - 72 a b x - 72 a - b^{5} x^{5} - 5 b^{4} x^{4} - 20 b^{3} x^{3} - 60 b^{2} x^{2} - 120 b x - 120\right ) e^{- a - b x}}{b^{3}} & \text {for}\: b^{3} \neq 0 \\\frac {a^{3} x^{3}}{3} + \frac {3 a^{2} b x^{4}}{4} + \frac {3 a b^{2} x^{5}}{5} + \frac {b^{3} x^{6}}{6} & \text {otherwise} \end {cases} \]
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))
Time = 0.19 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.52 \[ \int e^{-a-b x} x^2 (a+b x)^3 \, dx=-\frac {{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{3} e^{\left (-b x - a\right )}}{b^{3}} - \frac {3 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {3 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a e^{\left (-b x - a\right )}}{b^{3}} - \frac {{\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} e^{\left (-b x - a\right )}}{b^{3}} \]
-(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
Time = 0.34 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.51 \[ \int e^{-a-b x} x^2 (a+b x)^3 \, dx=-\frac {{\left (b^{8} x^{5} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{3} + 5 \, b^{7} x^{4} + a^{3} b^{5} x^{2} + 12 \, a b^{6} x^{3} + 9 \, a^{2} b^{5} x^{2} + 20 \, b^{6} x^{3} + 2 \, a^{3} b^{4} x + 36 \, a b^{5} x^{2} + 18 \, a^{2} b^{4} x + 60 \, b^{5} x^{2} + 2 \, a^{3} b^{3} + 72 \, a b^{4} x + 18 \, a^{2} b^{3} + 120 \, b^{4} x + 72 \, a b^{3} + 120 \, b^{3}\right )} e^{\left (-b x - a\right )}}{b^{6}} \]
-(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
Time = 0.20 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.40 \[ \int e^{-a-b x} x^2 (a+b x)^3 \, dx=-x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (3\,a^2+3\,a\,b\,x+12\,a+b^2\,x^2+5\,b\,x+20\right )-\frac {2\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+9\,a^2+36\,a+60\right )}{b^3}-\frac {2\,x\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+9\,a^2+36\,a+60\right )}{b^2}-\frac {x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+9\,a^2+36\,a+60\right )}{b} \]