3.63 \(\int \frac{e^{-a-b x} (a+b x)^3}{x^4} \, dx\)

Optimal. Leaf size=198 \[ -\frac{1}{6} e^{-a} a^3 b^3 \text{Ei}(-b x)+\frac{3}{2} e^{-a} a^2 b^3 \text{Ei}(-b x)-\frac{a^3 b^2 e^{-a-b x}}{6 x}+\frac{3 a^2 b^2 e^{-a-b x}}{2 x}+\frac{a^3 b e^{-a-b x}}{6 x^2}-\frac{a^3 e^{-a-b x}}{3 x^3}-\frac{3 a^2 b e^{-a-b x}}{2 x^2}-3 e^{-a} a b^3 \text{Ei}(-b x)+e^{-a} b^3 \text{Ei}(-b x)-\frac{3 a b^2 e^{-a-b x}}{x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.290449, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2199, 2177, 2178} \[ -\frac{1}{6} e^{-a} a^3 b^3 \text{Ei}(-b x)+\frac{3}{2} e^{-a} a^2 b^3 \text{Ei}(-b x)-\frac{a^3 b^2 e^{-a-b x}}{6 x}+\frac{3 a^2 b^2 e^{-a-b x}}{2 x}+\frac{a^3 b e^{-a-b x}}{6 x^2}-\frac{a^3 e^{-a-b x}}{3 x^3}-\frac{3 a^2 b e^{-a-b x}}{2 x^2}-3 e^{-a} a b^3 \text{Ei}(-b x)+e^{-a} b^3 \text{Ei}(-b x)-\frac{3 a b^2 e^{-a-b x}}{x} \]

Antiderivative was successfully verified.

[In]

Int[(E^(-a - b*x)*(a + b*x)^3)/x^4,x]

[Out]

-(a^3*E^(-a - b*x))/(3*x^3) - (3*a^2*b*E^(-a - b*x))/(2*x^2) + (a^3*b*E^(-a - b*x))/(6*x^2) - (3*a*b^2*E^(-a -
b*x))/x + (3*a^2*b^2*E^(-a - b*x))/(2*x) - (a^3*b^2*E^(-a - b*x))/(6*x) + (b^3*ExpIntegralEi[-(b*x)])/E^a - (
3*a*b^3*ExpIntegralEi[-(b*x)])/E^a + (3*a^2*b^3*ExpIntegralEi[-(b*x)])/(2*E^a) - (a^3*b^3*ExpIntegralEi[-(b*x)
])/(6*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 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^4} \, dx &=\int \left (\frac{a^3 e^{-a-b x}}{x^4}+\frac{3 a^2 b e^{-a-b x}}{x^3}+\frac{3 a b^2 e^{-a-b x}}{x^2}+\frac{b^3 e^{-a-b x}}{x}\right ) \, dx\\ &=a^3 \int \frac{e^{-a-b x}}{x^4} \, dx+\left (3 a^2 b\right ) \int \frac{e^{-a-b x}}{x^3} \, dx+\left (3 a b^2\right ) \int \frac{e^{-a-b x}}{x^2} \, dx+b^3 \int \frac{e^{-a-b x}}{x} \, dx\\ &=-\frac{a^3 e^{-a-b x}}{3 x^3}-\frac{3 a^2 b e^{-a-b x}}{2 x^2}-\frac{3 a b^2 e^{-a-b x}}{x}+b^3 e^{-a} \text{Ei}(-b x)-\frac{1}{3} \left (a^3 b\right ) \int \frac{e^{-a-b x}}{x^3} \, dx-\frac{1}{2} \left (3 a^2 b^2\right ) \int \frac{e^{-a-b x}}{x^2} \, dx-\left (3 a b^3\right ) \int \frac{e^{-a-b x}}{x} \, dx\\ &=-\frac{a^3 e^{-a-b x}}{3 x^3}-\frac{3 a^2 b e^{-a-b x}}{2 x^2}+\frac{a^3 b e^{-a-b x}}{6 x^2}-\frac{3 a b^2 e^{-a-b x}}{x}+\frac{3 a^2 b^2 e^{-a-b x}}{2 x}+b^3 e^{-a} \text{Ei}(-b x)-3 a b^3 e^{-a} \text{Ei}(-b x)+\frac{1}{6} \left (a^3 b^2\right ) \int \frac{e^{-a-b x}}{x^2} \, dx+\frac{1}{2} \left (3 a^2 b^3\right ) \int \frac{e^{-a-b x}}{x} \, dx\\ &=-\frac{a^3 e^{-a-b x}}{3 x^3}-\frac{3 a^2 b e^{-a-b x}}{2 x^2}+\frac{a^3 b e^{-a-b x}}{6 x^2}-\frac{3 a b^2 e^{-a-b x}}{x}+\frac{3 a^2 b^2 e^{-a-b x}}{2 x}-\frac{a^3 b^2 e^{-a-b x}}{6 x}+b^3 e^{-a} \text{Ei}(-b x)-3 a b^3 e^{-a} \text{Ei}(-b x)+\frac{3}{2} a^2 b^3 e^{-a} \text{Ei}(-b x)-\frac{1}{6} \left (a^3 b^3\right ) \int \frac{e^{-a-b x}}{x} \, dx\\ &=-\frac{a^3 e^{-a-b x}}{3 x^3}-\frac{3 a^2 b e^{-a-b x}}{2 x^2}+\frac{a^3 b e^{-a-b x}}{6 x^2}-\frac{3 a b^2 e^{-a-b x}}{x}+\frac{3 a^2 b^2 e^{-a-b x}}{2 x}-\frac{a^3 b^2 e^{-a-b x}}{6 x}+b^3 e^{-a} \text{Ei}(-b x)-3 a b^3 e^{-a} \text{Ei}(-b x)+\frac{3}{2} a^2 b^3 e^{-a} \text{Ei}(-b x)-\frac{1}{6} a^3 b^3 e^{-a} \text{Ei}(-b x)\\ \end{align*}

Mathematica [A]  time = 0.105909, size = 81, normalized size = 0.41 \[ \frac{1}{6} e^{-a} \left (-\left (a^3-9 a^2+18 a-6\right ) b^3 \text{Ei}(-b x)-\frac{a e^{-b x} \left (a^2 \left (b^2 x^2-b x+2\right )-9 a b x (b x-1)+18 b^2 x^2\right )}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(-a - b*x)*(a + b*x)^3)/x^4,x]

[Out]

(-((a*(18*b^2*x^2 - 9*a*b*x*(-1 + b*x) + a^2*(2 - b*x + b^2*x^2)))/(E^(b*x)*x^3)) - (-6 + 18*a - 9*a^2 + a^3)*
b^3*ExpIntegralEi[-(b*x)])/(6*E^a)

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 167, normalized size = 0.8 \begin{align*}{b}^{3} \left ( -{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) -{a}^{3} \left ({\frac{{{\rm e}^{-bx-a}}}{3\,{b}^{3}{x}^{3}}}-{\frac{{{\rm e}^{-bx-a}}}{6\,{b}^{2}{x}^{2}}}+{\frac{{{\rm e}^{-bx-a}}}{6\,bx}}-{\frac{{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) }{6}} \right ) +3\,{a}^{2} \left ( -1/2\,{\frac{{{\rm e}^{-bx-a}}}{{b}^{2}{x}^{2}}}+1/2\,{\frac{{{\rm e}^{-bx-a}}}{bx}}-1/2\,{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) \right ) -3\,a \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^4,x)

[Out]

b^3*(-exp(-a)*Ei(1,b*x)-a^3*(1/3*exp(-b*x-a)/b^3/x^3-1/6*exp(-b*x-a)/b^2/x^2+1/6*exp(-b*x-a)/b/x-1/6*exp(-a)*E
i(1,b*x))+3*a^2*(-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*(exp(-b*x-a)/b/x-exp(
-a)*Ei(1,b*x)))

________________________________________________________________________________________

Maxima [A]  time = 1.23261, size = 85, normalized size = 0.43 \begin{align*} -a^{3} b^{3} e^{\left (-a\right )} \Gamma \left (-3, b x\right ) - 3 \, a^{2} b^{3} e^{\left (-a\right )} \Gamma \left (-2, b x\right ) - 3 \, a b^{3} e^{\left (-a\right )} \Gamma \left (-1, b x\right ) + b^{3}{\rm Ei}\left (-b x\right ) e^{\left (-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^4,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.44072, size = 182, normalized size = 0.92 \begin{align*} -\frac{{\left (a^{3} - 9 \, a^{2} + 18 \, a - 6\right )} b^{3} x^{3}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} +{\left ({\left (a^{3} - 9 \, a^{2} + 18 \, a\right )} b^{2} x^{2} + 2 \, a^{3} -{\left (a^{3} - 9 \, a^{2}\right )} b x\right )} e^{\left (-b x - a\right )}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*(b*x+a)^3/x^4,x, algorithm="fricas")

[Out]

-1/6*((a^3 - 9*a^2 + 18*a - 6)*b^3*x^3*Ei(-b*x)*e^(-a) + ((a^3 - 9*a^2 + 18*a)*b^2*x^2 + 2*a^3 - (a^3 - 9*a^2)
*b*x)*e^(-b*x - a))/x^3

________________________________________________________________________________________

Sympy [A]  time = 6.89928, size = 53, normalized size = 0.27 \begin{align*} \left (- \frac{a^{3} \operatorname{E}_{4}\left (b x\right )}{x^{3}} - \frac{3 a^{2} b \operatorname{E}_{3}\left (b x\right )}{x^{2}} - \frac{3 a b^{2} \operatorname{E}_{2}\left (b x\right )}{x} + b^{3} \operatorname{Ei}{\left (- b x \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**4,x)

[Out]

(-a**3*expint(4, b*x)/x**3 - 3*a**2*b*expint(3, b*x)/x**2 - 3*a*b**2*expint(2, b*x)/x + b**3*Ei(-b*x))*exp(-a)

________________________________________________________________________________________

Giac [A]  time = 1.36748, size = 247, normalized size = 1.25 \begin{align*} -\frac{a^{3} b^{3} x^{3}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 9 \, a^{2} b^{3} x^{3}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + 18 \, a b^{3} x^{3}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + a^{3} b^{2} x^{2} e^{\left (-b x - a\right )} - 6 \, b^{3} x^{3}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 9 \, a^{2} b^{2} x^{2} e^{\left (-b x - a\right )} - a^{3} b x e^{\left (-b x - a\right )} + 18 \, a b^{2} x^{2} e^{\left (-b x - a\right )} + 9 \, a^{2} b x e^{\left (-b x - a\right )} + 2 \, a^{3} e^{\left (-b x - a\right )}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*(b*x+a)^3/x^4,x, algorithm="giac")

[Out]

-1/6*(a^3*b^3*x^3*Ei(-b*x)*e^(-a) - 9*a^2*b^3*x^3*Ei(-b*x)*e^(-a) + 18*a*b^3*x^3*Ei(-b*x)*e^(-a) + a^3*b^2*x^2
*e^(-b*x - a) - 6*b^3*x^3*Ei(-b*x)*e^(-a) - 9*a^2*b^2*x^2*e^(-b*x - a) - a^3*b*x*e^(-b*x - a) + 18*a*b^2*x^2*e
^(-b*x - a) + 9*a^2*b*x*e^(-b*x - a) + 2*a^3*e^(-b*x - a))/x^3