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

Optimal. Leaf size=130 \[ \frac{1}{2} e^{-a} a^3 b^2 \text{Ei}(-b x)-3 e^{-a} a^2 b^2 \text{Ei}(-b x)-\frac{a^3 e^{-a-b x}}{2 x^2}+\frac{a^3 b e^{-a-b x}}{2 x}-\frac{3 a^2 b e^{-a-b x}}{x}+3 e^{-a} a b^2 \text{Ei}(-b x)-b^2 e^{-a-b x} \]

[Out]

-(b^2*E^(-a - b*x)) - (a^3*E^(-a - b*x))/(2*x^2) - (3*a^2*b*E^(-a - b*x))/x + (a^3*b*E^(-a - b*x))/(2*x) + (3*
a*b^2*ExpIntegralEi[-(b*x)])/E^a - (3*a^2*b^2*ExpIntegralEi[-(b*x)])/E^a + (a^3*b^2*ExpIntegralEi[-(b*x)])/(2*
E^a)

________________________________________________________________________________________

Rubi [A]  time = 0.213394, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2199, 2194, 2177, 2178} \[ \frac{1}{2} e^{-a} a^3 b^2 \text{Ei}(-b x)-3 e^{-a} a^2 b^2 \text{Ei}(-b x)-\frac{a^3 e^{-a-b x}}{2 x^2}+\frac{a^3 b e^{-a-b x}}{2 x}-\frac{3 a^2 b e^{-a-b x}}{x}+3 e^{-a} a b^2 \text{Ei}(-b x)-b^2 e^{-a-b x} \]

Antiderivative was successfully verified.

[In]

Int[(E^(-a - b*x)*(a + b*x)^3)/x^3,x]

[Out]

-(b^2*E^(-a - b*x)) - (a^3*E^(-a - b*x))/(2*x^2) - (3*a^2*b*E^(-a - b*x))/x + (a^3*b*E^(-a - b*x))/(2*x) + (3*
a*b^2*ExpIntegralEi[-(b*x)])/E^a - (3*a^2*b^2*ExpIntegralEi[-(b*x)])/E^a + (a^3*b^2*ExpIntegralEi[-(b*x)])/(2*
E^a)

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[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]

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), 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] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

\begin{align*} \int \frac{e^{-a-b x} (a+b x)^3}{x^3} \, dx &=\int \left (b^3 e^{-a-b x}+\frac{a^3 e^{-a-b x}}{x^3}+\frac{3 a^2 b e^{-a-b x}}{x^2}+\frac{3 a b^2 e^{-a-b x}}{x}\right ) \, dx\\ &=a^3 \int \frac{e^{-a-b x}}{x^3} \, dx+\left (3 a^2 b\right ) \int \frac{e^{-a-b x}}{x^2} \, dx+\left (3 a b^2\right ) \int \frac{e^{-a-b x}}{x} \, dx+b^3 \int e^{-a-b x} \, dx\\ &=-b^2 e^{-a-b x}-\frac{a^3 e^{-a-b x}}{2 x^2}-\frac{3 a^2 b e^{-a-b x}}{x}+3 a b^2 e^{-a} \text{Ei}(-b x)-\frac{1}{2} \left (a^3 b\right ) \int \frac{e^{-a-b x}}{x^2} \, dx-\left (3 a^2 b^2\right ) \int \frac{e^{-a-b x}}{x} \, dx\\ &=-b^2 e^{-a-b x}-\frac{a^3 e^{-a-b x}}{2 x^2}-\frac{3 a^2 b e^{-a-b x}}{x}+\frac{a^3 b e^{-a-b x}}{2 x}+3 a b^2 e^{-a} \text{Ei}(-b x)-3 a^2 b^2 e^{-a} \text{Ei}(-b x)+\frac{1}{2} \left (a^3 b^2\right ) \int \frac{e^{-a-b x}}{x} \, dx\\ &=-b^2 e^{-a-b x}-\frac{a^3 e^{-a-b x}}{2 x^2}-\frac{3 a^2 b e^{-a-b x}}{x}+\frac{a^3 b e^{-a-b x}}{2 x}+3 a b^2 e^{-a} \text{Ei}(-b x)-3 a^2 b^2 e^{-a} \text{Ei}(-b x)+\frac{1}{2} a^3 b^2 e^{-a} \text{Ei}(-b x)\\ \end{align*}

Mathematica [A]  time = 0.0724097, size = 68, normalized size = 0.52 \[ \frac{e^{-a-b x} \left (\left (a^2-6 a+6\right ) a b^2 x^2 e^{b x} \text{Ei}(-b x)+a^3 (b x-1)-6 a^2 b x-2 b^2 x^2\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(-a - b*x)*(-6*a^2*b*x - 2*b^2*x^2 + a^3*(-1 + b*x) + a*(6 - 6*a + a^2)*b^2*E^(b*x)*x^2*ExpIntegralEi[-(b*x
)]))/(2*x^2)

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 112, normalized size = 0.9 \begin{align*} -{b}^{2} \left ({{\rm e}^{-bx-a}}+3\,a{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) -{a}^{3} \left ( -{\frac{{{\rm e}^{-bx-a}}}{2\,{b}^{2}{x}^{2}}}+{\frac{{{\rm e}^{-bx-a}}}{2\,bx}}-{\frac{{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) }{2}} \right ) +3\,{a}^{2} \left ({\frac{{{\rm e}^{-bx-a}}}{bx}}-{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.2718, size = 86, normalized size = 0.66 \begin{align*} -a^{3} b^{2} e^{\left (-a\right )} \Gamma \left (-2, b x\right ) - 3 \, a^{2} b^{2} e^{\left (-a\right )} \Gamma \left (-1, b x\right ) + 3 \, a b^{2}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - b^{2} e^{\left (-b x - a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^3*b^2*e^(-a)*gamma(-2, b*x) - 3*a^2*b^2*e^(-a)*gamma(-1, b*x) + 3*a*b^2*Ei(-b*x)*e^(-a) - b^2*e^(-b*x - a)

________________________________________________________________________________________

Fricas [A]  time = 1.47975, size = 146, normalized size = 1.12 \begin{align*} \frac{{\left (a^{3} - 6 \, a^{2} + 6 \, a\right )} b^{2} x^{2}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} -{\left (2 \, b^{2} x^{2} + a^{3} -{\left (a^{3} - 6 \, a^{2}\right )} b x\right )} e^{\left (-b x - a\right )}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 6.58302, size = 56, normalized size = 0.43 \begin{align*} \left (- \frac{a^{3} \operatorname{E}_{3}\left (b x\right )}{x^{2}} - \frac{3 a^{2} b \operatorname{E}_{2}\left (b x\right )}{x} + 3 a b^{2} \operatorname{Ei}{\left (- b x \right )} + b^{3} \left (\begin{cases} x & \text{for}\: b = 0 \\- \frac{e^{- b x}}{b} & \text{otherwise} \end{cases}\right )\right ) e^{- a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*(b*x+a)**3/x**3,x)

[Out]

(-a**3*expint(3, b*x)/x**2 - 3*a**2*b*expint(2, b*x)/x + 3*a*b**2*Ei(-b*x) + b**3*Piecewise((x, Eq(b, 0)), (-e
xp(-b*x)/b, True)))*exp(-a)

________________________________________________________________________________________

Giac [A]  time = 1.33416, size = 169, normalized size = 1.3 \begin{align*} \frac{a^{3} b^{2} x^{2}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 6 \, a^{2} b^{2} x^{2}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + 6 \, a b^{2} x^{2}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + a^{3} b x e^{\left (-b x - a\right )} - 6 \, a^{2} b x e^{\left (-b x - a\right )} - 2 \, b^{2} x^{2} e^{\left (-b x - a\right )} - a^{3} e^{\left (-b x - a\right )}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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