Integrand size = 19, antiderivative size = 137 \[ \int e^{-a-b x} x (a+b x)^3 \, dx=-\frac {24 e^{-a-b x}}{b^2}-\frac {e^{-a-b x} x (a+b x)^3}{b}-\frac {6 e^{-a-b x} (3 a+4 b x)}{b^2}-\frac {6 e^{-a-b x} \left (a^2+3 a b x+2 b^2 x^2\right )}{b^2}-\frac {e^{-a-b x} \left (a^3+6 a^2 b x+9 a b^2 x^2+4 b^3 x^3\right )}{b^2} \] Output:
-24*exp(-b*x-a)/b^2-exp(-b*x-a)*x*(b*x+a)^3/b-6*exp(-b*x-a)*(4*b*x+3*a)/b^ 2-6*exp(-b*x-a)*(2*b^2*x^2+3*a*b*x+a^2)/b^2-exp(-b*x-a)*(4*b^3*x^3+9*a*b^2 *x^2+6*a^2*b*x+a^3)/b^2
Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.70 \[ \int e^{-a-b x} x (a+b x)^3 \, dx=\frac {e^{-a-b x} \left (-24-24 b x-12 b^2 x^2-4 b^3 x^3-b^4 x^4-a^3 (1+b x)-3 a^2 \left (2+2 b x+b^2 x^2\right )-3 a \left (6+6 b x+3 b^2 x^2+b^3 x^3\right )\right )}{b^2} \] Input:
Integrate[E^(-a - b*x)*x*(a + b*x)^3,x]
Output:
(E^(-a - b*x)*(-24 - 24*b*x - 12*b^2*x^2 - 4*b^3*x^3 - b^4*x^4 - a^3*(1 + b*x) - 3*a^2*(2 + 2*b*x + b^2*x^2) - 3*a*(6 + 6*b*x + 3*b^2*x^2 + b^3*x^3) ))/b^2
Time = 0.75 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, 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.
\(\displaystyle \int x e^{-a-b x} (a+b x)^3 \, dx\) |
\(\Big \downarrow \) 2626 |
\(\displaystyle \int \left (\frac {e^{-a-b x} (a+b x)^4}{b}-\frac {a e^{-a-b x} (a+b x)^3}{b}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {e^{-a-b x} (a+b x)^4}{b^2}+\frac {a e^{-a-b x} (a+b x)^3}{b^2}-\frac {4 e^{-a-b x} (a+b x)^3}{b^2}+\frac {3 a e^{-a-b x} (a+b x)^2}{b^2}-\frac {12 e^{-a-b x} (a+b x)^2}{b^2}+\frac {6 a e^{-a-b x} (a+b x)}{b^2}-\frac {24 e^{-a-b x} (a+b x)}{b^2}+\frac {6 a e^{-a-b x}}{b^2}-\frac {24 e^{-a-b x}}{b^2}\) |
Input:
Int[E^(-a - b*x)*x*(a + b*x)^3,x]
Output:
(-24*E^(-a - b*x))/b^2 + (6*a*E^(-a - b*x))/b^2 - (24*E^(-a - b*x)*(a + b* x))/b^2 + (6*a*E^(-a - b*x)*(a + b*x))/b^2 - (12*E^(-a - b*x)*(a + b*x)^2) /b^2 + (3*a*E^(-a - b*x)*(a + b*x)^2)/b^2 - (4*E^(-a - b*x)*(a + b*x)^3)/b ^2 + (a*E^(-a - b*x)*(a + b*x)^3)/b^2 - (E^(-a - b*x)*(a + b*x)^4)/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 = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.74
method | result | size |
gosper | \(-\frac {\left (b^{4} x^{4}+3 a \,b^{3} x^{3}+3 a^{2} b^{2} x^{2}+4 b^{3} x^{3}+a^{3} b x +9 a \,b^{2} x^{2}+6 a^{2} b x +12 b^{2} x^{2}+a^{3}+18 a b x +6 a^{2}+24 b x +18 a +24\right ) {\mathrm e}^{-b x -a}}{b^{2}}\) | \(102\) |
risch | \(-\frac {\left (b^{4} x^{4}+3 a \,b^{3} x^{3}+3 a^{2} b^{2} x^{2}+4 b^{3} x^{3}+a^{3} b x +9 a \,b^{2} x^{2}+6 a^{2} b x +12 b^{2} x^{2}+a^{3}+18 a b x +6 a^{2}+24 b x +18 a +24\right ) {\mathrm e}^{-b x -a}}{b^{2}}\) | \(102\) |
orering | \(-\frac {\left (b^{4} x^{4}+3 a \,b^{3} x^{3}+3 a^{2} b^{2} x^{2}+4 b^{3} x^{3}+a^{3} b x +9 a \,b^{2} x^{2}+6 a^{2} b x +12 b^{2} x^{2}+a^{3}+18 a b x +6 a^{2}+24 b x +18 a +24\right ) {\mathrm e}^{-b x -a}}{b^{2}}\) | \(102\) |
norman | \(\left (-3 a b -4 b \right ) x^{3} {\mathrm e}^{-b x -a}+\left (-3 a^{2}-9 a -12\right ) x^{2} {\mathrm e}^{-b x -a}-b^{2} x^{4} {\mathrm e}^{-b x -a}-\frac {\left (a^{3}+6 a^{2}+18 a +24\right ) {\mathrm e}^{-b x -a}}{b^{2}}-\frac {\left (a^{3}+6 a^{2}+18 a +24\right ) x \,{\mathrm e}^{-b x -a}}{b}\) | \(118\) |
meijerg | \(\frac {{\mathrm e}^{-a} \left (24-\frac {\left (5 b^{4} x^{4}+20 b^{3} x^{3}+60 b^{2} x^{2}+120 b x +120\right ) {\mathrm e}^{-b x}}{5}\right )}{b^{2}}+\frac {3 \,{\mathrm e}^{-a} a \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b^{2}}+\frac {3 \,{\mathrm e}^{-a} a^{2} \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{2}}+\frac {{\mathrm e}^{-a} a^{3} \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}\) | \(151\) |
derivativedivides | \(-\frac {\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}+a \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}\) | \(173\) |
default | \(-\frac {\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}+a \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}\) | \(173\) |
parts | \(-b^{2} x^{4} {\mathrm e}^{-b x -a}-3 \,{\mathrm e}^{-b x -a} x^{3} b a -3 \,{\mathrm e}^{-b x -a} x^{2} a^{2}-\frac {{\mathrm e}^{-b x -a} x \,a^{3}}{b}-\frac {-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}-3 a \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}\) | \(202\) |
parallelrisch | \(-\frac {b^{4} x^{4} {\mathrm e}^{-b x -a}+3 a \,b^{3} x^{3} {\mathrm e}^{-b x -a}+4 \,{\mathrm e}^{-b x -a} x^{3} b^{3}+3 a^{2} b^{2} x^{2} {\mathrm e}^{-b x -a}+9 x^{2} {\mathrm e}^{-b x -a} a \,b^{2}+a^{3} b x \,{\mathrm e}^{-b x -a}+12 b^{2} {\mathrm e}^{-b x -a} x^{2}+6 x \,{\mathrm e}^{-b x -a} a^{2} b +18 a b \,{\mathrm e}^{-b x -a} x +{\mathrm e}^{-b x -a} a^{3}+24 b \,{\mathrm e}^{-b x -a} x +6 a^{2} {\mathrm e}^{-b x -a}+18 a \,{\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}}{b^{2}}\) | \(221\) |
Input:
int(exp(-b*x-a)*x*(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
-(b^4*x^4+3*a*b^3*x^3+3*a^2*b^2*x^2+4*b^3*x^3+a^3*b*x+9*a*b^2*x^2+6*a^2*b* x+12*b^2*x^2+a^3+18*a*b*x+6*a^2+24*b*x+18*a+24)*exp(-b*x-a)/b^2
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.57 \[ \int e^{-a-b x} x (a+b x)^3 \, dx=-\frac {{\left (b^{4} x^{4} + {\left (3 \, a + 4\right )} b^{3} x^{3} + 3 \, {\left (a^{2} + 3 \, a + 4\right )} b^{2} x^{2} + a^{3} + {\left (a^{3} + 6 \, a^{2} + 18 \, a + 24\right )} b x + 6 \, a^{2} + 18 \, a + 24\right )} e^{\left (-b x - a\right )}}{b^{2}} \] Input:
integrate(exp(-b*x-a)*x*(b*x+a)^3,x, algorithm="fricas")
Output:
-(b^4*x^4 + (3*a + 4)*b^3*x^3 + 3*(a^2 + 3*a + 4)*b^2*x^2 + a^3 + (a^3 + 6 *a^2 + 18*a + 24)*b*x + 6*a^2 + 18*a + 24)*e^(-b*x - a)/b^2
Time = 0.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.08 \[ \int e^{-a-b x} x (a+b x)^3 \, dx=\begin {cases} \frac {\left (- a^{3} b x - a^{3} - 3 a^{2} b^{2} x^{2} - 6 a^{2} b x - 6 a^{2} - 3 a b^{3} x^{3} - 9 a b^{2} x^{2} - 18 a b x - 18 a - b^{4} x^{4} - 4 b^{3} x^{3} - 12 b^{2} x^{2} - 24 b x - 24\right ) e^{- a - b x}}{b^{2}} & \text {for}\: b^{2} \neq 0 \\\frac {a^{3} x^{2}}{2} + a^{2} b x^{3} + \frac {3 a b^{2} x^{4}}{4} + \frac {b^{3} x^{5}}{5} & \text {otherwise} \end {cases} \] Input:
integrate(exp(-b*x-a)*x*(b*x+a)**3,x)
Output:
Piecewise(((-a**3*b*x - a**3 - 3*a**2*b**2*x**2 - 6*a**2*b*x - 6*a**2 - 3* a*b**3*x**3 - 9*a*b**2*x**2 - 18*a*b*x - 18*a - b**4*x**4 - 4*b**3*x**3 - 12*b**2*x**2 - 24*b*x - 24)*exp(-a - b*x)/b**2, Ne(b**2, 0)), (a**3*x**2/2 + a**2*b*x**3 + 3*a*b**2*x**4/4 + b**3*x**5/5, True))
Time = 0.05 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.96 \[ \int e^{-a-b x} x (a+b x)^3 \, dx=-\frac {{\left (b x + 1\right )} a^{3} e^{\left (-b x - a\right )}}{b^{2}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{2} e^{\left (-b x - a\right )}}{b^{2}} - \frac {3 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a e^{\left (-b x - a\right )}}{b^{2}} - \frac {{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} e^{\left (-b x - a\right )}}{b^{2}} \] Input:
integrate(exp(-b*x-a)*x*(b*x+a)^3,x, algorithm="maxima")
Output:
-(b*x + 1)*a^3*e^(-b*x - a)/b^2 - 3*(b^2*x^2 + 2*b*x + 2)*a^2*e^(-b*x - a) /b^2 - 3*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*a*e^(-b*x - a)/b^2 - (b^4*x^4 + 4*b^3*x^3 + 12*b^2*x^2 + 24*b*x + 24)*e^(-b*x - a)/b^2
Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90 \[ \int e^{-a-b x} x (a+b x)^3 \, dx=-\frac {{\left (b^{7} x^{4} + 3 \, a b^{6} x^{3} + 3 \, a^{2} b^{5} x^{2} + 4 \, b^{6} x^{3} + a^{3} b^{4} x + 9 \, a b^{5} x^{2} + 6 \, a^{2} b^{4} x + 12 \, b^{5} x^{2} + a^{3} b^{3} + 18 \, a b^{4} x + 6 \, a^{2} b^{3} + 24 \, b^{4} x + 18 \, a b^{3} + 24 \, b^{3}\right )} e^{\left (-b x - a\right )}}{b^{5}} \] Input:
integrate(exp(-b*x-a)*x*(b*x+a)^3,x, algorithm="giac")
Output:
-(b^7*x^4 + 3*a*b^6*x^3 + 3*a^2*b^5*x^2 + 4*b^6*x^3 + a^3*b^4*x + 9*a*b^5* x^2 + 6*a^2*b^4*x + 12*b^5*x^2 + a^3*b^3 + 18*a*b^4*x + 6*a^2*b^3 + 24*b^4 *x + 18*a*b^3 + 24*b^3)*e^(-b*x - a)/b^5
Time = 0.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.85 \[ \int e^{-a-b x} x (a+b x)^3 \, dx=-x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (3\,a^2+9\,a+12\right )-b^2\,x^4\,{\mathrm {e}}^{-a-b\,x}-\frac {{\mathrm {e}}^{-a-b\,x}\,\left (a^3+6\,a^2+18\,a+24\right )}{b^2}-\frac {x\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+6\,a^2+18\,a+24\right )}{b}-b\,x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (3\,a+4\right ) \] Input:
int(x*exp(- a - b*x)*(a + b*x)^3,x)
Output:
- x^2*exp(- a - b*x)*(9*a + 3*a^2 + 12) - b^2*x^4*exp(- a - b*x) - (exp(- a - b*x)*(18*a + 6*a^2 + a^3 + 24))/b^2 - (x*exp(- a - b*x)*(18*a + 6*a^2 + a^3 + 24))/b - b*x^3*exp(- a - b*x)*(3*a + 4)
Time = 0.17 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.76 \[ \int e^{-a-b x} x (a+b x)^3 \, dx=\frac {-b^{4} x^{4}-3 a \,b^{3} x^{3}-3 a^{2} b^{2} x^{2}-4 b^{3} x^{3}-a^{3} b x -9 a \,b^{2} x^{2}-6 a^{2} b x -12 b^{2} x^{2}-a^{3}-18 a b x -6 a^{2}-24 b x -18 a -24}{e^{b x +a} b^{2}} \] Input:
int(exp(-b*x-a)*x*(b*x+a)^3,x)
Output:
( - a**3*b*x - a**3 - 3*a**2*b**2*x**2 - 6*a**2*b*x - 6*a**2 - 3*a*b**3*x* *3 - 9*a*b**2*x**2 - 18*a*b*x - 18*a - b**4*x**4 - 4*b**3*x**3 - 12*b**2*x **2 - 24*b*x - 24)/(e**(a + b*x)*b**2)