Integrand size = 21, antiderivative size = 102 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x} \, dx=-2 e^{-a-b x}-3 a e^{-a-b x}-3 a^2 e^{-a-b x}-2 b e^{-a-b x} x-3 a b e^{-a-b x} x-b^2 e^{-a-b x} x^2+a^3 e^{-a} \operatorname {ExpIntegralEi}(-b x) \]
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Time = 0.10 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2230, 2225, 2209, 2207} \[ \int \frac {e^{-a-b x} (a+b x)^3}{x} \, dx=e^{-a} a^3 \operatorname {ExpIntegralEi}(-b x)-3 a^2 e^{-a-b x}-b^2 x^2 e^{-a-b x}-3 a e^{-a-b x}-3 a b x e^{-a-b x}-2 e^{-a-b x}-2 b x e^{-a-b x} \]
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Rule 2207
Rule 2209
Rule 2225
Rule 2230
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.18 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.51 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x} \, dx=e^{-a-b x} \left (-2-3 a^2-2 b x-b^2 x^2-3 a (1+b x)+a^3 e^{b x} \operatorname {ExpIntegralEi}(-b x)\right ) \]
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Time = 0.21 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.94
method | result | size |
meijerg | \({\mathrm e}^{-a} \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )+3 \,{\mathrm e}^{-a} a \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )+3 \,{\mathrm e}^{-a} a^{2} \left (1-{\mathrm e}^{-b x}\right )+{\mathrm e}^{-a} a^{3} \left (\ln \left (x \right )+\ln \left (b \right )-\ln \left (b x \right )-\operatorname {Ei}_{1}\left (b x \right )\right )\) | \(96\) |
risch | \(-b^{2} {\mathrm e}^{-b x -a} x^{2}-a^{3} {\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )-3 a b \,{\mathrm e}^{-b x -a} x -3 a^{2} {\mathrm e}^{-b x -a}-2 b \,{\mathrm e}^{-b x -a} x -3 a \,{\mathrm e}^{-b x -a}-2 \,{\mathrm e}^{-b x -a}\) | \(97\) |
derivativedivides | \(-a^{2} {\mathrm e}^{-b x -a}+a \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )-\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-2 \,{\mathrm e}^{-b x -a}-a^{3} {\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )\) | \(113\) |
default | \(-a^{2} {\mathrm e}^{-b x -a}+a \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )-\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-2 \,{\mathrm e}^{-b x -a}-a^{3} {\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )\) | \(113\) |
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Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.49 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x} \, dx=a^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - {\left (b^{2} x^{2} + {\left (3 \, a + 2\right )} b x + 3 \, a^{2} + 3 \, a + 2\right )} e^{\left (-b x - a\right )} \]
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Time = 3.43 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.69 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x} \, dx=\left (a^{3} \operatorname {Ei}{\left (- b x \right )} - 3 a^{2} e^{- b x} - 3 a \left (b x e^{- b x} + e^{- b x}\right ) - b^{2} x^{2} e^{- b x} - 2 b x e^{- b x} - 2 e^{- b x}\right ) e^{- a} \]
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Time = 0.23 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.68 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x} \, dx=a^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 3 \, {\left (b x + 1\right )} a e^{\left (-b x - a\right )} - 3 \, a^{2} e^{\left (-b x - a\right )} - {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )} \]
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Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x} \, dx=-b^{2} x^{2} e^{\left (-b x - a\right )} + a^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 3 \, a b x e^{\left (-b x - a\right )} - 3 \, a^{2} e^{\left (-b x - a\right )} - 2 \, b x e^{\left (-b x - a\right )} - 3 \, a e^{\left (-b x - a\right )} - 2 \, e^{\left (-b x - a\right )} \]
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Time = 0.23 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.68 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x} \, dx=-{\mathrm {e}}^{-a-b\,x}\,\left (b^2\,x^2+2\,b\,x+2\right )-3\,a^2\,{\mathrm {e}}^{-a-b\,x}-3\,a\,{\mathrm {e}}^{-a-b\,x}\,\left (b\,x+1\right )-a^3\,{\mathrm {e}}^{-a}\,\mathrm {expint}\left (b\,x\right ) \]
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