\(\int e^{-a-b x} (a+b x)^4 \, dx\) [77]

Optimal result

Integrand size = 18, antiderivative size = 102 \[ \int e^{-a-b x} (a+b x)^4 \, dx=-\frac {24 e^{-a-b x}}{b}-\frac {24 e^{-a-b x} (a+b x)}{b}-\frac {12 e^{-a-b x} (a+b x)^2}{b}-\frac {4 e^{-a-b x} (a+b x)^3}{b}-\frac {e^{-a-b x} (a+b x)^4}{b} \]

[Out]

-24*exp(-b*x-a)/b-24*exp(-b*x-a)*(b*x+a)/b-12*exp(-b*x-a)*(b*x+a)^2/b-4*exp(-b*x-a)*(b*x+a)^3/b-exp(-b*x-a)*(b
*x+a)^4/b

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2207, 2225} \[ \int e^{-a-b x} (a+b x)^4 \, dx=-\frac {e^{-a-b x} (a+b x)^4}{b}-\frac {4 e^{-a-b x} (a+b x)^3}{b}-\frac {12 e^{-a-b x} (a+b x)^2}{b}-\frac {24 e^{-a-b x} (a+b x)}{b}-\frac {24 e^{-a-b x}}{b} \]

[In]

Int[E^(-a - b*x)*(a + b*x)^4,x]

[Out]

(-24*E^(-a - b*x))/b - (24*E^(-a - b*x)*(a + b*x))/b - (12*E^(-a - b*x)*(a + b*x)^2)/b - (4*E^(-a - b*x)*(a +
b*x)^3)/b - (E^(-a - b*x)*(a + b*x)^4)/b

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-a-b x} (a+b x)^4}{b}+4 \int e^{-a-b x} (a+b x)^3 \, dx \\ & = -\frac {4 e^{-a-b x} (a+b x)^3}{b}-\frac {e^{-a-b x} (a+b x)^4}{b}+12 \int e^{-a-b x} (a+b x)^2 \, dx \\ & = -\frac {12 e^{-a-b x} (a+b x)^2}{b}-\frac {4 e^{-a-b x} (a+b x)^3}{b}-\frac {e^{-a-b x} (a+b x)^4}{b}+24 \int e^{-a-b x} (a+b x) \, dx \\ & = -\frac {24 e^{-a-b x} (a+b x)}{b}-\frac {12 e^{-a-b x} (a+b x)^2}{b}-\frac {4 e^{-a-b x} (a+b x)^3}{b}-\frac {e^{-a-b x} (a+b x)^4}{b}+24 \int e^{-a-b x} \, dx \\ & = -\frac {24 e^{-a-b x}}{b}-\frac {24 e^{-a-b x} (a+b x)}{b}-\frac {12 e^{-a-b x} (a+b x)^2}{b}-\frac {4 e^{-a-b x} (a+b x)^3}{b}-\frac {e^{-a-b x} (a+b x)^4}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.49 \[ \int e^{-a-b x} (a+b x)^4 \, dx=\frac {e^{-a-b x} \left (-24-24 (a+b x)-12 (a+b x)^2-4 (a+b x)^3-(a+b x)^4\right )}{b} \]

[In]

Integrate[E^(-a - b*x)*(a + b*x)^4,x]

[Out]

(E^(-a - b*x)*(-24 - 24*(a + b*x) - 12*(a + b*x)^2 - 4*(a + b*x)^3 - (a + b*x)^4))/b

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.97

method	result	size
derivativedivides	\(-\frac {\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}}{b}\)	\(99\)
default	\(-\frac {\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}}{b}\)	\(99\)
gosper	\(-\frac {\left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 b^{3} x^{3}+4 a^{3} b x +12 a \,b^{2} x^{2}+a^{4}+12 a^{2} b x +12 b^{2} x^{2}+4 a^{3}+24 a b x +12 a^{2}+24 b x +24 a +24\right ) {\mathrm e}^{-b x -a}}{b}\)	\(108\)
risch	\(-\frac {\left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 b^{3} x^{3}+4 a^{3} b x +12 a \,b^{2} x^{2}+a^{4}+12 a^{2} b x +12 b^{2} x^{2}+4 a^{3}+24 a b x +12 a^{2}+24 b x +24 a +24\right ) {\mathrm e}^{-b x -a}}{b}\)	\(108\)
norman	\(\left (-4 b^{2} a -4 b^{2}\right ) x^{3} {\mathrm e}^{-b x -a}+\left (-4 a^{3}-12 a^{2}-24 a -24\right ) x \,{\mathrm e}^{-b x -a}-b^{3} x^{4} {\mathrm e}^{-b x -a}-\frac {\left (a^{4}+4 a^{3}+12 a^{2}+24 a +24\right ) {\mathrm e}^{-b x -a}}{b}-6 b \left (a^{2}+2 a +2\right ) x^{2} {\mathrm e}^{-b x -a}\)	\(125\)
parts	\(-b^{3} x^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} b^{2} a \,x^{3}-6 \,{\mathrm e}^{-b x -a} b \,a^{2} x^{2}-4 \,{\mathrm e}^{-b x -a} a^{3} x -\frac {{\mathrm e}^{-b x -a} a^{4}}{b}+\frac {4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-24 \,{\mathrm e}^{-b x -a}}{b}\)	\(164\)
meijerg	\(\frac {{\mathrm e}^{-a} \left (24-\frac {\left (5 b^{4} x^{4}+20 b^{3} x^{3}+60 b^{2} x^{2}+120 b x +120\right ) {\mathrm e}^{-b x}}{5}\right )}{b}+\frac {4 \,{\mathrm e}^{-a} a \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b}+\frac {6 \,{\mathrm e}^{-a} a^{2} \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b}+\frac {4 \,{\mathrm e}^{-a} a^{3} \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b}+\frac {{\mathrm e}^{-a} a^{4} \left (1-{\mathrm e}^{-b x}\right )}{b}\)	\(172\)
parallelrisch	\(-\frac {{\mathrm e}^{-b x -a} b^{4} x^{4}+4 \,{\mathrm e}^{-b x -a} a \,b^{3} x^{3}+4 \,{\mathrm e}^{-b x -a} x^{3} b^{3}+6 \,{\mathrm e}^{-b x -a} a^{2} b^{2} x^{2}+12 x^{2} {\mathrm e}^{-b x -a} a \,b^{2}+4 \,{\mathrm e}^{-b x -a} a^{3} b x +12 b^{2} {\mathrm e}^{-b x -a} x^{2}+12 x \,{\mathrm e}^{-b x -a} a^{2} b +{\mathrm e}^{-b x -a} a^{4}+24 a b \,{\mathrm e}^{-b x -a} x +4 \,{\mathrm e}^{-b x -a} a^{3}+24 b \,{\mathrm e}^{-b x -a} x +12 a^{2} {\mathrm e}^{-b x -a}+24 a \,{\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}}{b}\)	\(236\)

[In]

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

[Out]

-1/b*((-b*x-a)^4*exp(-b*x-a)-4*exp(-b*x-a)*(-b*x-a)^3+12*(-b*x-a)^2*exp(-b*x-a)-24*(-b*x-a)*exp(-b*x-a)+24*exp
(-b*x-a))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.81 \[ \int e^{-a-b x} (a+b x)^4 \, dx=-\frac {{\left (b^{4} x^{4} + 4 \, {\left (a + 1\right )} b^{3} x^{3} + 6 \, {\left (a^{2} + 2 \, a + 2\right )} b^{2} x^{2} + a^{4} + 4 \, a^{3} + 4 \, {\left (a^{3} + 3 \, a^{2} + 6 \, a + 6\right )} b x + 12 \, a^{2} + 24 \, a + 24\right )} e^{\left (-b x - a\right )}}{b} \]

[In]

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

[Out]

-(b^4*x^4 + 4*(a + 1)*b^3*x^3 + 6*(a^2 + 2*a + 2)*b^2*x^2 + a^4 + 4*a^3 + 4*(a^3 + 3*a^2 + 6*a + 6)*b*x + 12*a
^2 + 24*a + 24)*e^(-b*x - a)/b

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.55 \[ \int e^{-a-b x} (a+b x)^4 \, dx=\begin {cases} \frac {\left (- a^{4} - 4 a^{3} b x - 4 a^{3} - 6 a^{2} b^{2} x^{2} - 12 a^{2} b x - 12 a^{2} - 4 a b^{3} x^{3} - 12 a b^{2} x^{2} - 24 a b x - 24 a - b^{4} x^{4} - 4 b^{3} x^{3} - 12 b^{2} x^{2} - 24 b x - 24\right ) e^{- a - b x}}{b} & \text {for}\: b \neq 0 \\a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac {b^{4} x^{5}}{5} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise(((-a**4 - 4*a**3*b*x - 4*a**3 - 6*a**2*b**2*x**2 - 12*a**2*b*x - 12*a**2 - 4*a*b**3*x**3 - 12*a*b**2
*x**2 - 24*a*b*x - 24*a - b**4*x**4 - 4*b**3*x**3 - 12*b**2*x**2 - 24*b*x - 24)*exp(-a - b*x)/b, Ne(b, 0)), (a
**4*x + 2*a**3*b*x**2 + 2*a**2*b**2*x**3 + a*b**3*x**4 + b**4*x**5/5, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.46 \[ \int e^{-a-b x} (a+b x)^4 \, dx=-\frac {4 \, {\left (b x + 1\right )} a^{3} e^{\left (-b x - a\right )}}{b} - \frac {a^{4} e^{\left (-b x - a\right )}}{b} - \frac {6 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{2} e^{\left (-b x - a\right )}}{b} - \frac {4 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a e^{\left (-b x - a\right )}}{b} - \frac {{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} e^{\left (-b x - a\right )}}{b} \]

[In]

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

[Out]

-4*(b*x + 1)*a^3*e^(-b*x - a)/b - a^4*e^(-b*x - a)/b - 6*(b^2*x^2 + 2*b*x + 2)*a^2*e^(-b*x - a)/b - 4*(b^3*x^3
+ 3*b^2*x^2 + 6*b*x + 6)*a*e^(-b*x - a)/b - (b^4*x^4 + 4*b^3*x^3 + 12*b^2*x^2 + 24*b*x + 24)*e^(-b*x - a)/b

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.29 \[ \int e^{-a-b x} (a+b x)^4 \, dx=-\frac {{\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, b^{7} x^{3} + 4 \, a^{3} b^{5} x + 12 \, a b^{6} x^{2} + a^{4} b^{4} + 12 \, a^{2} b^{5} x + 12 \, b^{6} x^{2} + 4 \, a^{3} b^{4} + 24 \, a b^{5} x + 12 \, a^{2} b^{4} + 24 \, b^{5} x + 24 \, a b^{4} + 24 \, b^{4}\right )} e^{\left (-b x - a\right )}}{b^{5}} \]

[In]

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

[Out]

-(b^8*x^4 + 4*a*b^7*x^3 + 6*a^2*b^6*x^2 + 4*b^7*x^3 + 4*a^3*b^5*x + 12*a*b^6*x^2 + a^4*b^4 + 12*a^2*b^5*x + 12
*b^6*x^2 + 4*a^3*b^4 + 24*a*b^5*x + 12*a^2*b^4 + 24*b^5*x + 24*a*b^4 + 24*b^4)*e^(-b*x - a)/b^5

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.18 \[ \int e^{-a-b x} (a+b x)^4 \, dx=-b^3\,x^4\,{\mathrm {e}}^{-a-b\,x}-x\,{\mathrm {e}}^{-a-b\,x}\,\left (4\,a^3+12\,a^2+24\,a+24\right )-\frac {{\mathrm {e}}^{-a-b\,x}\,\left (a^4+4\,a^3+12\,a^2+24\,a+24\right )}{b}-6\,b\,x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (a^2+2\,a+2\right )-4\,b^2\,x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (a+1\right ) \]

[In]

int(exp(- a - b*x)*(a + b*x)^4,x)

[Out]

- b^3*x^4*exp(- a - b*x) - x*exp(- a - b*x)*(24*a + 12*a^2 + 4*a^3 + 24) - (exp(- a - b*x)*(24*a + 12*a^2 + 4*
a^3 + a^4 + 24))/b - 6*b*x^2*exp(- a - b*x)*(2*a + a^2 + 2) - 4*b^2*x^3*exp(- a - b*x)*(a + 1)