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

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Rubi [A]  time = 0.110551, 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 \[ -\frac{e^{-a-b x} (a+b x)^3}{b}-\frac{3 e^{-a-b x} (a+b x)^2}{b}-\frac{6 e^{-a-b x} (a+b x)}{b}-\frac{6 e^{-a-b x}}{b} \]

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

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Rubi in Sympy [A]  time = 11.9732, size = 65, normalized size = 0.81 \[ - \frac{\left (a + b x\right )^{3} e^{- a - b x}}{b} - \frac{3 \left (a + b x\right )^{2} e^{- a - b x}}{b} - \frac{6 \left (a + b x\right ) e^{- a - b x}}{b} - \frac{6 e^{- a - b x}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(-b*x-a)*(b*x+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0164372, 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

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Maple [A]  time = 0.006, size = 68, normalized size = 0.9 \[ -{\frac{ \left ({b}^{3}{x}^{3}+3\,{b}^{2}{x}^{2}a+3\,{a}^{2}bx+3\,{b}^{2}{x}^{2}+{a}^{3}+6\,abx+3\,{a}^{2}+6\,bx+6\,a+6 \right ){{\rm e}^{-bx-a}}}{b}} \]

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

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Maxima [A]  time = 0.837064, size = 139, normalized size = 1.74 \[ -\frac{3 \,{\left (b x + 1\right )} a^{2} e^{\left (-b x - a\right )}}{b} - \frac{a^{3} e^{\left (-b x - a\right )}}{b} - \frac{3 \,{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a e^{\left (-b x - a\right )}}{b} - \frac{{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*e^(-b*x - a),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

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Fricas [A]  time = 0.241586, size = 77, normalized size = 0.96 \[ -\frac{{\left (b^{3} x^{3} + 3 \,{\left (a + 1\right )} b^{2} x^{2} + a^{3} + 3 \,{\left (a^{2} + 2 \, a + 2\right )} b x + 3 \, a^{2} + 6 \, a + 6\right )} e^{\left (-b x - a\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*e^(-b*x - a),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

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Sympy [A]  time = 0.355251, size = 104, normalized size = 1.3 \[ \begin{cases} \frac{\left (- a^{3} - 3 a^{2} b x - 3 a^{2} - 3 a b^{2} x^{2} - 6 a b x - 6 a - b^{3} x^{3} - 3 b^{2} x^{2} - 6 b x - 6\right ) e^{- a - b x}}{b} & \text{for}\: b \neq 0 \\a^{3} x + \frac{3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac{b^{3} x^{4}}{4} & \text{otherwise} \end{cases} \]

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))

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GIAC/XCAS [A]  time = 0.250249, size = 117, normalized size = 1.46 \[ -\frac{{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + 3 \, b^{5} x^{2} + a^{3} b^{3} + 6 \, a b^{4} x + 3 \, a^{2} b^{3} + 6 \, b^{4} x + 6 \, a b^{3} + 6 \, b^{3}\right )} e^{\left (-b x - a\right )}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*e^(-b*x - a),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