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