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

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

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

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Rubi [A]  time = 0.460272, 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 \[ -\frac{1}{6} e^{-a} a^3 b^3 \text{ExpIntegralEi}(-b x)-\frac{a^3 b^2 e^{-a-b x}}{6 x}-\frac{a^3 e^{-a-b x}}{3 x^3}+\frac{a^3 b e^{-a-b x}}{6 x^2}+\frac{3}{2} e^{-a} a^2 b^3 \text{ExpIntegralEi}(-b x)+\frac{3 a^2 b^2 e^{-a-b x}}{2 x}-\frac{3 a^2 b e^{-a-b x}}{2 x^2}-3 e^{-a} a b^3 \text{ExpIntegralEi}(-b x)+e^{-a} b^3 \text{ExpIntegralEi}(-b x)-\frac{3 a b^2 e^{-a-b x}}{x} \]

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

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Rubi in Sympy [A]  time = 30.6649, size = 180, normalized size = 0.91 \[ - \frac{a^{3} b^{3} e^{- a} \operatorname{Ei}{\left (- b x \right )}}{6} - \frac{a^{3} b^{2} e^{- a - b x}}{6 x} + \frac{a^{3} b e^{- a - b x}}{6 x^{2}} - \frac{a^{3} e^{- a - b x}}{3 x^{3}} + \frac{3 a^{2} b^{3} e^{- a} \operatorname{Ei}{\left (- b x \right )}}{2} + \frac{3 a^{2} b^{2} e^{- a - b x}}{2 x} - \frac{3 a^{2} b e^{- a - b x}}{2 x^{2}} - 3 a b^{3} e^{- a} \operatorname{Ei}{\left (- b x \right )} - \frac{3 a b^{2} e^{- a - b x}}{x} + b^{3} e^{- a} \operatorname{Ei}{\left (- b x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(-b*x-a)*(b*x+a)**3/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0800223, size = 81, normalized size = 0.41 \[ \frac{1}{6} e^{-a} \left (-\frac{a e^{-b x} \left (a^2 \left (b^2 x^2-b x+2\right )-9 a b x (b x-1)+18 b^2 x^2\right )}{x^3}-\left (a^3-9 a^2+18 a-6\right ) b^3 \text{ExpIntegralEi}(-b x)\right ) \]

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)

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Maple [A]  time = 0.012, size = 167, normalized size = 0.8 \[{b}^{3} \left ( -{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) +3\,{a}^{2} \left ( -1/2\,{\frac{{{\rm e}^{-bx-a}}}{{b}^{2}{x}^{2}}}+1/2\,{\frac{{{\rm e}^{-bx-a}}}{bx}}-1/2\,{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) \right ) -3\,a \left ({\frac{{{\rm e}^{-bx-a}}}{bx}}-{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) \right ) -{a}^{3} \left ({\frac{{{\rm e}^{-bx-a}}}{3\,{b}^{3}{x}^{3}}}-{\frac{{{\rm e}^{-bx-a}}}{6\,{b}^{2}{x}^{2}}}+{\frac{{{\rm e}^{-bx-a}}}{6\,bx}}-{\frac{{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) }{6}} \right ) \right ) \]

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)+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))-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)*Ei(1,b*x)))

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Maxima [A]  time = 0.854917, size = 85, normalized size = 0.43 \[ -a^{3} b^{3} e^{\left (-a\right )} \Gamma \left (-3, b x\right ) - 3 \, a^{2} b^{3} e^{\left (-a\right )} \Gamma \left (-2, b x\right ) - 3 \, a b^{3} e^{\left (-a\right )} \Gamma \left (-1, b x\right ) + b^{3}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Fricas [A]  time = 0.253225, size = 112, normalized size = 0.57 \[ -\frac{{\left (a^{3} - 9 \, a^{2} + 18 \, a - 6\right )} b^{3} x^{3}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} +{\left ({\left (a^{3} - 9 \, a^{2} + 18 \, a\right )} b^{2} x^{2} + 2 \, a^{3} -{\left (a^{3} - 9 \, a^{2}\right )} b x\right )} e^{\left (-b x - a\right )}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 28.6214, size = 194, normalized size = 0.98 \[ - \frac{a^{3} b^{3} e^{- a} \operatorname{Ei}{\left (b x e^{i \pi } \right )}}{6} - \frac{a^{3} b^{2} e^{- a} e^{- b x}}{6 x} + \frac{a^{3} b e^{- a} e^{- b x}}{6 x^{2}} - \frac{a^{3} e^{- a} e^{- b x}}{3 x^{3}} + \frac{3 a^{2} b^{3} e^{- a} \operatorname{Ei}{\left (b x e^{i \pi } \right )}}{2} + \frac{3 a^{2} b^{2} e^{- a} e^{- b x}}{2 x} - \frac{3 a^{2} b e^{- a} e^{- b x}}{2 x^{2}} - 3 a b^{3} e^{- a} \operatorname{Ei}{\left (b x e^{i \pi } \right )} - \frac{3 a b^{2} e^{- a} e^{- b x}}{x} + b^{3} e^{- a} \operatorname{Ei}{\left (b x e^{i \pi } \right )} \]

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*b**3*exp(-a)*Ei(b*x*exp_polar(I*pi))/6 - a**3*b**2*exp(-a)*exp(-b*x)/(6*x)
+ a**3*b*exp(-a)*exp(-b*x)/(6*x**2) - a**3*exp(-a)*exp(-b*x)/(3*x**3) + 3*a**2*
b**3*exp(-a)*Ei(b*x*exp_polar(I*pi))/2 + 3*a**2*b**2*exp(-a)*exp(-b*x)/(2*x) - 3
*a**2*b*exp(-a)*exp(-b*x)/(2*x**2) - 3*a*b**3*exp(-a)*Ei(b*x*exp_polar(I*pi)) -
3*a*b**2*exp(-a)*exp(-b*x)/x + b**3*exp(-a)*Ei(b*x*exp_polar(I*pi))

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GIAC/XCAS [A]  time = 0.264831, size = 247, normalized size = 1.25 \[ -\frac{a^{3} b^{3} x^{3}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 9 \, a^{2} b^{3} x^{3}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + 18 \, a b^{3} x^{3}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + a^{3} b^{2} x^{2} e^{\left (-b x - a\right )} - 6 \, b^{3} x^{3}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 9 \, a^{2} b^{2} x^{2} e^{\left (-b x - a\right )} - a^{3} b x e^{\left (-b x - a\right )} + 18 \, a b^{2} x^{2} e^{\left (-b x - a\right )} + 9 \, a^{2} b x e^{\left (-b x - a\right )} + 2 \, a^{3} e^{\left (-b x - a\right )}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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