3.1.59 \(\int e^{-a-b x} (a+b x)^3 \, dx\) [59]

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

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, 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 = {2207, 2225} \begin {gather*} -\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} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.15, size = 41, normalized size = 0.51 \begin {gather*} \frac {e^{-a-b x} \left (-6-6 (a+b x)-3 (a+b x)^2-(a+b x)^3\right )}{b} \end {gather*}

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 = 0.06, size = 77, normalized size = 0.96

method	result	size
gosper	\(-\frac {\left (b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +3 b^{2} x^{2}+a^{3}+6 a b x +3 a^{2}+6 b x +6 a +6\right ) {\mathrm e}^{-b x -a}}{b}\)	\(68\)
risch	\(-\frac {\left (b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +3 b^{2} x^{2}+a^{3}+6 a b x +3 a^{2}+6 b x +6 a +6\right ) {\mathrm e}^{-b x -a}}{b}\)	\(68\)
derivativedivides	\(\frac {{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}}{b}\)	\(77\)
default	\(\frac {{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}}{b}\)	\(77\)
norman	\(\left (-3 a b -3 b \right ) x^{2} {\mathrm e}^{-b x -a}+\left (-3 a^{2}-6 a -6\right ) x \,{\mathrm e}^{-b x -a}-b^{2} x^{3} {\mathrm e}^{-b x -a}-\frac {\left (a^{3}+3 a^{2}+6 a +6\right ) {\mathrm e}^{-b x -a}}{b}\)	\(88\)
meijerg	\(\frac {{\mathrm e}^{-a} \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b}+\frac {3 \,{\mathrm e}^{-a} a \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b}+\frac {3 \,{\mathrm e}^{-a} a^{2} \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b}+\frac {{\mathrm e}^{-a} a^{3} \left (1-{\mathrm e}^{-b x}\right )}{b}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-b*x-a)*(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 103, normalized size = 1.29 \begin {gather*} -\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 {gather*}

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

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Fricas [A]
time = 0.41, size = 57, normalized size = 0.71 \begin {gather*} -\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 {gather*}

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

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Sympy [A]
time = 0.06, size = 104, normalized size = 1.30 \begin {gather*} \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 {gather*}

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 [A]
time = 3.56, size = 87, normalized size = 1.09 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 0.11, size = 66, normalized size = 0.82 \begin {gather*} -x\,{\mathrm {e}}^{-a-b\,x}\,\left (3\,a^2+3\,a\,b\,x+6\,a+b^2\,x^2+3\,b\,x+6\right )-\frac {{\mathrm {e}}^{-a-b\,x}\,\left (a^3+3\,a^2+6\,a+6\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(- a - b*x)*(a + b*x)^3,x)

[Out]

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

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