3.59 \(\int e^{-a-b x} (a+b x)^3 \, dx\)

Optimal. Leaf size=80 \[ -\frac{e^{-a-b x} (a+b x)^3}{b}-\frac{3 e^{-a-b x} (a+b x)^2}{b}-\frac{6 e^{-a-b x} (a+b x)}{b}-\frac{6 e^{-a-b x}}{b} \]

[Out]

(-6*E^(-a - b*x))/b - (6*E^(-a - b*x)*(a + b*x))/b - (3*E^(-a - b*x)*(a + b*x)^2)/b - (E^(-a - b*x)*(a + b*x)^
3)/b

________________________________________________________________________________________

Rubi [A]  time = 0.0683236, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2176, 2194} \[ -\frac{e^{-a-b x} (a+b x)^3}{b}-\frac{3 e^{-a-b x} (a+b x)^2}{b}-\frac{6 e^{-a-b x} (a+b x)}{b}-\frac{6 e^{-a-b x}}{b} \]

Antiderivative was successfully verified.

[In]

Int[E^(-a - b*x)*(a + b*x)^3,x]

[Out]

(-6*E^(-a - b*x))/b - (6*E^(-a - b*x)*(a + b*x))/b - (3*E^(-a - b*x)*(a + b*x)^2)/b - (E^(-a - b*x)*(a + b*x)^
3)/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)^3 \, dx &=-\frac{e^{-a-b x} (a+b x)^3}{b}+3 \int e^{-a-b x} (a+b x)^2 \, dx\\ &=-\frac{3 e^{-a-b x} (a+b x)^2}{b}-\frac{e^{-a-b x} (a+b x)^3}{b}+6 \int e^{-a-b x} (a+b x) \, dx\\ &=-\frac{6 e^{-a-b x} (a+b x)}{b}-\frac{3 e^{-a-b x} (a+b x)^2}{b}-\frac{e^{-a-b x} (a+b x)^3}{b}+6 \int e^{-a-b x} \, dx\\ &=-\frac{6 e^{-a-b x}}{b}-\frac{6 e^{-a-b x} (a+b x)}{b}-\frac{3 e^{-a-b x} (a+b x)^2}{b}-\frac{e^{-a-b x} (a+b x)^3}{b}\\ \end{align*}

Mathematica [A]  time = 0.046695, size = 41, normalized size = 0.51 \[ \frac{e^{-a-b x} \left (-(a+b x)^3-3 (a+b x)^2-6 (a+b x)-6\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(-a - b*x)*(a + b*x)^3,x]

[Out]

(E^(-a - b*x)*(-6 - 6*(a + b*x) - 3*(a + b*x)^2 - (a + b*x)^3))/b

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 68, normalized size = 0.9 \begin{align*} -{\frac{ \left ({b}^{3}{x}^{3}+3\,{b}^{2}{x}^{2}a+3\,{a}^{2}bx+3\,{b}^{2}{x}^{2}+{a}^{3}+6\,abx+3\,{a}^{2}+6\,bx+6\,a+6 \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)^3,x)

[Out]

-(b^3*x^3+3*a*b^2*x^2+3*a^2*b*x+3*b^2*x^2+a^3+6*a*b*x+3*a^2+6*b*x+6*a+6)*exp(-b*x-a)/b

________________________________________________________________________________________

Maxima [A]  time = 1.06245, size = 139, normalized size = 1.74 \begin{align*} -\frac{3 \,{\left (b x + 1\right )} a^{2} e^{\left (-b x - a\right )}}{b} - \frac{a^{3} e^{\left (-b x - a\right )}}{b} - \frac{3 \,{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a e^{\left (-b x - a\right )}}{b} - \frac{{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\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)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.47203, size = 128, normalized size = 1.6 \begin{align*} -\frac{{\left (b^{3} x^{3} + 3 \,{\left (a + 1\right )} b^{2} x^{2} + a^{3} + 3 \,{\left (a^{2} + 2 \, a + 2\right )} b x + 3 \, a^{2} + 6 \, a + 6\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)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.145624, size = 104, normalized size = 1.3 \begin{align*} \begin{cases} \frac{\left (- a^{3} - 3 a^{2} b x - 3 a^{2} - 3 a b^{2} x^{2} - 6 a b x - 6 a - b^{3} x^{3} - 3 b^{2} x^{2} - 6 b x - 6\right ) e^{- a - b x}}{b} & \text{for}\: b \neq 0 \\a^{3} x + \frac{3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac{b^{3} x^{4}}{4} & \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)**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.31471, size = 117, normalized size = 1.46 \begin{align*} -\frac{{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + 3 \, b^{5} x^{2} + a^{3} b^{3} + 6 \, a b^{4} x + 3 \, a^{2} b^{3} + 6 \, b^{4} x + 6 \, a b^{3} + 6 \, b^{3}\right )} e^{\left (-b x - a\right )}}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*(b*x+a)^3,x, algorithm="giac")

[Out]

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