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

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

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

_______________________________________________________________________________________

Rubi [A]  time = 0.329603, 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 \[ \frac{1}{2} e^{-a} a^3 b^2 \text{ExpIntegralEi}(-b x)-\frac{a^3 e^{-a-b x}}{2 x^2}+\frac{a^3 b e^{-a-b x}}{2 x}-3 e^{-a} a^2 b^2 \text{ExpIntegralEi}(-b x)-\frac{3 a^2 b e^{-a-b x}}{x}+3 e^{-a} a b^2 \text{ExpIntegralEi}(-b x)-b^2 e^{-a-b x} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.9543, size = 116, normalized size = 0.89 \[ \frac{a^{3} b^{2} e^{- a} \operatorname{Ei}{\left (- b x \right )}}{2} + \frac{a^{3} b e^{- a - b x}}{2 x} - \frac{a^{3} e^{- a - b x}}{2 x^{2}} - 3 a^{2} b^{2} e^{- a} \operatorname{Ei}{\left (- b x \right )} - \frac{3 a^{2} b e^{- a - b x}}{x} + 3 a b^{2} e^{- a} \operatorname{Ei}{\left (- b x \right )} - b^{2} e^{- a - b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.04669, size = 68, normalized size = 0.52 \[ \frac{e^{-a-b x} \left (a^3 (b x-1)+\left (a^2-6 a+6\right ) a b^2 x^2 e^{b x} \text{ExpIntegralEi}(-b x)-6 a^2 b x-2 b^2 x^2\right )}{2 x^2} \]

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 = 0.013, size = 112, normalized size = 0.9 \[ -{b}^{2} \left ({{\rm e}^{-bx-a}}+3\,a{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) -{a}^{3} \left ( -{\frac{{{\rm e}^{-bx-a}}}{2\,{b}^{2}{x}^{2}}}+{\frac{{{\rm e}^{-bx-a}}}{2\,bx}}-{\frac{{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) }{2}} \right ) +3\,{a}^{2} \left ({\frac{{{\rm e}^{-bx-a}}}{bx}}-{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) \right ) \right ) \]

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 = 0.875049, size = 86, normalized size = 0.66 \[ -a^{3} b^{2} e^{\left (-a\right )} \Gamma \left (-2, b x\right ) - 3 \, a^{2} b^{2} e^{\left (-a\right )} \Gamma \left (-1, b x\right ) + 3 \, a b^{2}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - b^{2} e^{\left (-b x - a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*e^(-b*x - a)/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 = 0.243611, size = 95, normalized size = 0.73 \[ \frac{{\left (a^{3} - 6 \, a^{2} + 6 \, a\right )} b^{2} x^{2}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} -{\left (2 \, b^{2} x^{2} + a^{3} -{\left (a^{3} - 6 \, a^{2}\right )} b x\right )} e^{\left (-b x - a\right )}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*e^(-b*x - a)/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 = 22.6113, size = 133, normalized size = 1.02 \[ \frac{a^{3} b^{2} e^{- a} \operatorname{Ei}{\left (b x e^{i \pi } \right )}}{2} + \frac{a^{3} b e^{- a} e^{- b x}}{2 x} - \frac{a^{3} e^{- a} e^{- b x}}{2 x^{2}} - 3 a^{2} b^{2} e^{- a} \operatorname{Ei}{\left (b x e^{i \pi } \right )} - \frac{3 a^{2} b e^{- a} e^{- b x}}{x} + 3 a b^{2} e^{- a} \operatorname{Ei}{\left (b x e^{i \pi } \right )} + b^{3} \left (\begin{cases} x & \text{for}\: b = 0 \\- \frac{e^{- b x}}{b} & \text{otherwise} \end{cases}\right ) e^{- a} \]

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*b**2*exp(-a)*Ei(b*x*exp_polar(I*pi))/2 + a**3*b*exp(-a)*exp(-b*x)/(2*x) - a
**3*exp(-a)*exp(-b*x)/(2*x**2) - 3*a**2*b**2*exp(-a)*Ei(b*x*exp_polar(I*pi)) - 3
*a**2*b*exp(-a)*exp(-b*x)/x + 3*a*b**2*exp(-a)*Ei(b*x*exp_polar(I*pi)) + b**3*Pi
ecewise((x, Eq(b, 0)), (-exp(-b*x)/b, True))*exp(-a)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.266126, size = 169, normalized size = 1.3 \[ \frac{a^{3} b^{2} x^{2}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 6 \, a^{2} b^{2} x^{2}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + 6 \, a b^{2} x^{2}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + a^{3} b x e^{\left (-b x - a\right )} - 6 \, a^{2} b x e^{\left (-b x - a\right )} - 2 \, b^{2} x^{2} e^{\left (-b x - a\right )} - a^{3} e^{\left (-b x - a\right )}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*e^(-b*x - a)/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*E
i(-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