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

Optimal. Leaf size=102 \[ -\frac{e^{-a-b x} (a+b x)^4}{b}-\frac{4 e^{-a-b x} (a+b x)^3}{b}-\frac{12 e^{-a-b x} (a+b x)^2}{b}-\frac{24 e^{-a-b x} (a+b x)}{b}-\frac{24 e^{-a-b x}}{b} \]

[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

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Rubi [A]  time = 0.0964918, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2176, 2194} \[ -\frac{e^{-a-b x} (a+b x)^4}{b}-\frac{4 e^{-a-b x} (a+b x)^3}{b}-\frac{12 e^{-a-b x} (a+b x)^2}{b}-\frac{24 e^{-a-b x} (a+b x)}{b}-\frac{24 e^{-a-b x}}{b} \]

Antiderivative was successfully verified.

[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 2176

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] &&  !$UseGamma === True

Rule 2194

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*} \int e^{-a-b x} (a+b x)^4 \, dx &=-\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]  time = 0.061442, size = 50, normalized size = 0.49 \[ \frac{e^{-a-b x} \left (-(a+b x)^4-4 (a+b x)^3-12 (a+b x)^2-24 (a+b x)-24\right )}{b} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.004, size = 108, normalized size = 1.1 \begin{align*} -{\frac{ \left ({b}^{4}{x}^{4}+4\,{b}^{3}{x}^{3}a+6\,{a}^{2}{b}^{2}{x}^{2}+4\,{b}^{3}{x}^{3}+4\,{a}^{3}bx+12\,a{b}^{2}{x}^{2}+{a}^{4}+12\,{a}^{2}bx+12\,{b}^{2}{x}^{2}+4\,{a}^{3}+24\,abx+12\,{a}^{2}+24\,bx+24\,a+24 \right ){{\rm e}^{-bx-a}}}{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(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)*exp(-b*x-a)/b

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Maxima [A]  time = 1.12265, size = 201, normalized size = 1.97 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.42894, size = 192, normalized size = 1.88 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 0.164435, size = 158, normalized size = 1.55 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.20679, size = 178, normalized size = 1.75 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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