Optimal. Leaf size=102 \[ e^{-a} a^3 \text {Ei}(-b x)-3 a^2 e^{-a-b x}-b^2 x^2 e^{-a-b x}-3 a e^{-a-b x}-3 a b x e^{-a-b x}-2 e^{-a-b x}-2 b x e^{-a-b x} \]
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Rubi [A] time = 0.16, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2199, 2194, 2178, 2176} \[ e^{-a} a^3 \text {Ei}(-b x)-3 a^2 e^{-a-b x}-b^2 x^2 e^{-a-b x}-3 a e^{-a-b x}-3 a b x e^{-a-b x}-2 e^{-a-b x}-2 b x e^{-a-b x} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2178
Rule 2194
Rule 2199
Rubi steps
\begin {align*} \int \frac {e^{-a-b x} (a+b x)^3}{x} \, dx &=\int \left (3 a^2 b e^{-a-b x}+\frac {a^3 e^{-a-b x}}{x}+3 a b^2 e^{-a-b x} x+b^3 e^{-a-b x} x^2\right ) \, dx\\ &=a^3 \int \frac {e^{-a-b x}}{x} \, dx+\left (3 a^2 b\right ) \int e^{-a-b x} \, dx+\left (3 a b^2\right ) \int e^{-a-b x} x \, dx+b^3 \int e^{-a-b x} x^2 \, dx\\ &=-3 a^2 e^{-a-b x}-3 a b e^{-a-b x} x-b^2 e^{-a-b x} x^2+a^3 e^{-a} \text {Ei}(-b x)+(3 a b) \int e^{-a-b x} \, dx+\left (2 b^2\right ) \int e^{-a-b x} x \, dx\\ &=-3 a e^{-a-b x}-3 a^2 e^{-a-b x}-2 b e^{-a-b x} x-3 a b e^{-a-b x} x-b^2 e^{-a-b x} x^2+a^3 e^{-a} \text {Ei}(-b x)+(2 b) \int e^{-a-b x} \, dx\\ &=-2 e^{-a-b x}-3 a e^{-a-b x}-3 a^2 e^{-a-b x}-2 b e^{-a-b x} x-3 a b e^{-a-b x} x-b^2 e^{-a-b x} x^2+a^3 e^{-a} \text {Ei}(-b x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 52, normalized size = 0.51 \[ e^{-a-b x} \left (a^3 e^{b x} \text {Ei}(-b x)-3 a^2-3 a (b x+1)-b^2 x^2-2 b x-2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 50, normalized size = 0.49 \[ a^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - {\left (b^{2} x^{2} + {\left (3 \, a + 2\right )} b x + 3 \, a^{2} + 3 \, a + 2\right )} e^{\left (-b x - a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 95, normalized size = 0.93 \[ -b^{2} x^{2} e^{\left (-b x - a\right )} + a^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 3 \, a b x e^{\left (-b x - a\right )} - 3 \, a^{2} e^{\left (-b x - a\right )} - 2 \, b x e^{\left (-b x - a\right )} - 3 \, a e^{\left (-b x - a\right )} - 2 \, e^{\left (-b x - a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 113, normalized size = 1.11 \[ -a^{3} \Ei \left (1, b x \right ) {\mathrm e}^{-a}-a^{2} {\mathrm e}^{-b x -a}+\left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right ) a -\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-2 \,{\mathrm e}^{-b x -a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 69, normalized size = 0.68 \[ a^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 3 \, {\left (b x + 1\right )} a e^{\left (-b x - a\right )} - 3 \, a^{2} e^{\left (-b x - a\right )} - {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.56, size = 69, normalized size = 0.68 \[ -{\mathrm {e}}^{-a-b\,x}\,\left (b^2\,x^2+2\,b\,x+2\right )-3\,a^2\,{\mathrm {e}}^{-a-b\,x}-3\,a\,{\mathrm {e}}^{-a-b\,x}\,\left (b\,x+1\right )-a^3\,{\mathrm {e}}^{-a}\,\mathrm {expint}\left (b\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.64, size = 70, normalized size = 0.69 \[ \left (a^{3} \operatorname {Ei}{\left (- b x \right )} - 3 a^{2} e^{- b x} - 3 a \left (b x e^{- b x} + e^{- b x}\right ) - b^{2} x^{2} e^{- b x} - 2 b x e^{- b x} - 2 e^{- b x}\right ) e^{- a} \]
Verification of antiderivative is not currently implemented for this CAS.
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