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

Optimal. Leaf size=184 \[ -\frac{e^{-a-b x} (a+b x)^4}{b^2}+\frac{a e^{-a-b x} (a+b x)^3}{b^2}-\frac{4 e^{-a-b x} (a+b x)^3}{b^2}+\frac{3 a e^{-a-b x} (a+b x)^2}{b^2}-\frac{12 e^{-a-b x} (a+b x)^2}{b^2}+\frac{6 a e^{-a-b x} (a+b x)}{b^2}-\frac{24 e^{-a-b x} (a+b x)}{b^2}+\frac{6 a e^{-a-b x}}{b^2}-\frac{24 e^{-a-b x}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.241916, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2196, 2176, 2194} \[ -\frac{e^{-a-b x} (a+b x)^4}{b^2}+\frac{a e^{-a-b x} (a+b x)^3}{b^2}-\frac{4 e^{-a-b x} (a+b x)^3}{b^2}+\frac{3 a e^{-a-b x} (a+b x)^2}{b^2}-\frac{12 e^{-a-b x} (a+b x)^2}{b^2}+\frac{6 a e^{-a-b x} (a+b x)}{b^2}-\frac{24 e^{-a-b x} (a+b x)}{b^2}+\frac{6 a e^{-a-b x}}{b^2}-\frac{24 e^{-a-b x}}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(-a - b*x)*x*(a + b*x)^3,x]

[Out]

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

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

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} x (a+b x)^3 \, dx &=\int \left (-\frac{a e^{-a-b x} (a+b x)^3}{b}+\frac{e^{-a-b x} (a+b x)^4}{b}\right ) \, dx\\ &=\frac{\int e^{-a-b x} (a+b x)^4 \, dx}{b}-\frac{a \int e^{-a-b x} (a+b x)^3 \, dx}{b}\\ &=\frac{a e^{-a-b x} (a+b x)^3}{b^2}-\frac{e^{-a-b x} (a+b x)^4}{b^2}+\frac{4 \int e^{-a-b x} (a+b x)^3 \, dx}{b}-\frac{(3 a) \int e^{-a-b x} (a+b x)^2 \, dx}{b}\\ &=\frac{3 a e^{-a-b x} (a+b x)^2}{b^2}-\frac{4 e^{-a-b x} (a+b x)^3}{b^2}+\frac{a e^{-a-b x} (a+b x)^3}{b^2}-\frac{e^{-a-b x} (a+b x)^4}{b^2}+\frac{12 \int e^{-a-b x} (a+b x)^2 \, dx}{b}-\frac{(6 a) \int e^{-a-b x} (a+b x) \, dx}{b}\\ &=\frac{6 a e^{-a-b x} (a+b x)}{b^2}-\frac{12 e^{-a-b x} (a+b x)^2}{b^2}+\frac{3 a e^{-a-b x} (a+b x)^2}{b^2}-\frac{4 e^{-a-b x} (a+b x)^3}{b^2}+\frac{a e^{-a-b x} (a+b x)^3}{b^2}-\frac{e^{-a-b x} (a+b x)^4}{b^2}+\frac{24 \int e^{-a-b x} (a+b x) \, dx}{b}-\frac{(6 a) \int e^{-a-b x} \, dx}{b}\\ &=\frac{6 a e^{-a-b x}}{b^2}-\frac{24 e^{-a-b x} (a+b x)}{b^2}+\frac{6 a e^{-a-b x} (a+b x)}{b^2}-\frac{12 e^{-a-b x} (a+b x)^2}{b^2}+\frac{3 a e^{-a-b x} (a+b x)^2}{b^2}-\frac{4 e^{-a-b x} (a+b x)^3}{b^2}+\frac{a e^{-a-b x} (a+b x)^3}{b^2}-\frac{e^{-a-b x} (a+b x)^4}{b^2}+\frac{24 \int e^{-a-b x} \, dx}{b}\\ &=-\frac{24 e^{-a-b x}}{b^2}+\frac{6 a e^{-a-b x}}{b^2}-\frac{24 e^{-a-b x} (a+b x)}{b^2}+\frac{6 a e^{-a-b x} (a+b x)}{b^2}-\frac{12 e^{-a-b x} (a+b x)^2}{b^2}+\frac{3 a e^{-a-b x} (a+b x)^2}{b^2}-\frac{4 e^{-a-b x} (a+b x)^3}{b^2}+\frac{a e^{-a-b x} (a+b x)^3}{b^2}-\frac{e^{-a-b x} (a+b x)^4}{b^2}\\ \end{align*}

Mathematica [A]  time = 0.121952, size = 96, normalized size = 0.52 \[ \frac{e^{-a-b x} \left (-3 a^2 \left (b^2 x^2+2 b x+2\right )-a^3 (b x+1)-3 a \left (b^3 x^3+3 b^2 x^2+6 b x+6\right )-b^4 x^4-4 b^3 x^3-12 b^2 x^2-24 b x-24\right )}{b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(-a - b*x)*x*(a + b*x)^3,x]

[Out]

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

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Maple [A]  time = 0.004, size = 102, normalized size = 0.6 \begin{align*} -{\frac{ \left ({b}^{4}{x}^{4}+3\,{b}^{3}{x}^{3}a+3\,{a}^{2}{b}^{2}{x}^{2}+4\,{b}^{3}{x}^{3}+{a}^{3}bx+9\,a{b}^{2}{x}^{2}+6\,{a}^{2}bx+12\,{b}^{2}{x}^{2}+{a}^{3}+18\,abx+6\,{a}^{2}+24\,bx+18\,a+24 \right ){{\rm e}^{-bx-a}}}{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^4*x^4+3*a*b^3*x^3+3*a^2*b^2*x^2+4*b^3*x^3+a^3*b*x+9*a*b^2*x^2+6*a^2*b*x+12*b^2*x^2+a^3+18*a*b*x+6*a^2+24*b
*x+18*a+24)*exp(-b*x-a)/b^2

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Maxima [A]  time = 1.09198, size = 178, normalized size = 0.97 \begin{align*} -\frac{{\left (b x + 1\right )} a^{3} e^{\left (-b x - a\right )}}{b^{2}} - \frac{3 \,{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{2} e^{\left (-b x - a\right )}}{b^{2}} - \frac{3 \,{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a e^{\left (-b x - a\right )}}{b^{2}} - \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^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.50241, size = 182, normalized size = 0.99 \begin{align*} -\frac{{\left (b^{4} x^{4} +{\left (3 \, a + 4\right )} b^{3} x^{3} + 3 \,{\left (a^{2} + 3 \, a + 4\right )} b^{2} x^{2} + a^{3} +{\left (a^{3} + 6 \, a^{2} + 18 \, a + 24\right )} b x + 6 \, a^{2} + 18 \, a + 24\right )} e^{\left (-b x - a\right )}}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^4*x^4 + (3*a + 4)*b^3*x^3 + 3*(a^2 + 3*a + 4)*b^2*x^2 + a^3 + (a^3 + 6*a^2 + 18*a + 24)*b*x + 6*a^2 + 18*a
+ 24)*e^(-b*x - a)/b^2

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Sympy [A]  time = 0.163471, size = 148, normalized size = 0.8 \begin{align*} \begin{cases} \frac{\left (- a^{3} b x - a^{3} - 3 a^{2} b^{2} x^{2} - 6 a^{2} b x - 6 a^{2} - 3 a b^{3} x^{3} - 9 a b^{2} x^{2} - 18 a b x - 18 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^{2}} & \text{for}\: b^{2} \neq 0 \\\frac{a^{3} x^{2}}{2} + a^{2} b x^{3} + \frac{3 a b^{2} x^{4}}{4} + \frac{b^{3} 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)*x*(b*x+a)**3,x)

[Out]

Piecewise(((-a**3*b*x - a**3 - 3*a**2*b**2*x**2 - 6*a**2*b*x - 6*a**2 - 3*a*b**3*x**3 - 9*a*b**2*x**2 - 18*a*b
*x - 18*a - b**4*x**4 - 4*b**3*x**3 - 12*b**2*x**2 - 24*b*x - 24)*exp(-a - b*x)/b**2, Ne(b**2, 0)), (a**3*x**2
/2 + a**2*b*x**3 + 3*a*b**2*x**4/4 + b**3*x**5/5, True))

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Giac [A]  time = 1.35255, size = 166, normalized size = 0.9 \begin{align*} -\frac{{\left (b^{7} x^{4} + 3 \, a b^{6} x^{3} + 3 \, a^{2} b^{5} x^{2} + 4 \, b^{6} x^{3} + a^{3} b^{4} x + 9 \, a b^{5} x^{2} + 6 \, a^{2} b^{4} x + 12 \, b^{5} x^{2} + a^{3} b^{3} + 18 \, a b^{4} x + 6 \, a^{2} b^{3} + 24 \, b^{4} x + 18 \, a b^{3} + 24 \, b^{3}\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)*x*(b*x+a)^3,x, algorithm="giac")

[Out]

-(b^7*x^4 + 3*a*b^6*x^3 + 3*a^2*b^5*x^2 + 4*b^6*x^3 + a^3*b^4*x + 9*a*b^5*x^2 + 6*a^2*b^4*x + 12*b^5*x^2 + a^3
*b^3 + 18*a*b^4*x + 6*a^2*b^3 + 24*b^4*x + 18*a*b^3 + 24*b^3)*e^(-b*x - a)/b^5