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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]