Integrand size = 21, antiderivative size = 94 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^2} \, dx=-b e^{-a-b x}-3 a b e^{-a-b x}-\frac {a^3 e^{-a-b x}}{x}-b^2 e^{-a-b x} x+3 a^2 b e^{-a} \operatorname {ExpIntegralEi}(-b x)-a^3 b e^{-a} \operatorname {ExpIntegralEi}(-b x) \]
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Time = 0.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2230, 2225, 2208, 2209, 2207} \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^2} \, dx=e^{-a} a^3 (-b) \operatorname {ExpIntegralEi}(-b x)-\frac {a^3 e^{-a-b x}}{x}+3 e^{-a} a^2 b \operatorname {ExpIntegralEi}(-b x)-b^2 x e^{-a-b x}-3 a b e^{-a-b x}-b e^{-a-b x} \]
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Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps \begin{align*} \text {integral}& = \int \left (3 a b^2 e^{-a-b x}+\frac {a^3 e^{-a-b x}}{x^2}+\frac {3 a^2 b e^{-a-b x}}{x}+b^3 e^{-a-b x} x\right ) \, dx \\ & = a^3 \int \frac {e^{-a-b x}}{x^2} \, dx+\left (3 a^2 b\right ) \int \frac {e^{-a-b x}}{x} \, dx+\left (3 a b^2\right ) \int e^{-a-b x} \, dx+b^3 \int e^{-a-b x} x \, dx \\ & = -3 a b e^{-a-b x}-\frac {a^3 e^{-a-b x}}{x}-b^2 e^{-a-b x} x+3 a^2 b e^{-a} \text {Ei}(-b x)-\left (a^3 b\right ) \int \frac {e^{-a-b x}}{x} \, dx+b^2 \int e^{-a-b x} \, dx \\ & = -b e^{-a-b x}-3 a b e^{-a-b x}-\frac {a^3 e^{-a-b x}}{x}-b^2 e^{-a-b x} x+3 a^2 b e^{-a} \text {Ei}(-b x)-a^3 b e^{-a} \text {Ei}(-b x) \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.57 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^2} \, dx=\frac {e^{-a-b x} \left (-a^3-3 a b x-b x (1+b x)-(-3+a) a^2 b e^{b x} x \operatorname {ExpIntegralEi}(-b x)\right )}{x} \]
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Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.94
method | result | size |
risch | \(-3 a b \,{\mathrm e}^{-b x -a}-b^{2} {\mathrm e}^{-b x -a} x -b \,{\mathrm e}^{-b x -a}-\frac {a^{3} {\mathrm e}^{-b x -a}}{x}+b \,a^{3} {\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )-3 b \,a^{2} {\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )\) | \(88\) |
derivativedivides | \(b \left (-2 a \,{\mathrm e}^{-b x -a}+\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}-a^{3} \left (\frac {{\mathrm e}^{-b x -a}}{b x}-{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )\right )-3 a^{2} {\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )\right )\) | \(92\) |
default | \(b \left (-2 a \,{\mathrm e}^{-b x -a}+\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}-a^{3} \left (\frac {{\mathrm e}^{-b x -a}}{b x}-{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )\right )-3 a^{2} {\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )\right )\) | \(92\) |
meijerg | \(b \,{\mathrm e}^{-a} \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )+3 \,{\mathrm e}^{-a} b a \left (1-{\mathrm e}^{-b x}\right )+3 b \,{\mathrm e}^{-a} a^{2} \left (\ln \left (x \right )+\ln \left (b \right )-\ln \left (b x \right )-\operatorname {Ei}_{1}\left (b x \right )\right )+{\mathrm e}^{-a} a^{3} b \left (-\frac {1}{b x}+1-\ln \left (x \right )-\ln \left (b \right )+\frac {-2 b x +2}{2 b x}-\frac {{\mathrm e}^{-b x}}{b x}+\ln \left (b x \right )+\operatorname {Ei}_{1}\left (b x \right )\right )\) | \(131\) |
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Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.60 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^2} \, dx=-\frac {{\left (a^{3} - 3 \, a^{2}\right )} b x {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + {\left (b^{2} x^{2} + a^{3} + {\left (3 \, a + 1\right )} b x\right )} e^{\left (-b x - a\right )}}{x} \]
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Time = 1.47 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^2} \, dx=- \frac {a^{3} e^{- a} \operatorname {E}_{2}\left (b x\right )}{x} + 3 a^{2} b e^{- a} \operatorname {Ei}{\left (- b x \right )} + 3 a b^{2} \left (\begin {cases} x & \text {for}\: b = 0 \\- \frac {e^{- b x}}{b} & \text {otherwise} \end {cases}\right ) e^{- a} + b^{3} x \left (\begin {cases} x & \text {for}\: b = 0 \\- \frac {e^{- b x}}{b} & \text {otherwise} \end {cases}\right ) e^{- a} - b^{3} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: b = 0 \\- \frac {\begin {cases} - \frac {e^{- b x}}{b} & \text {for}\: b \neq 0 \\x & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}\right ) e^{- a} \]
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Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.65 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^2} \, dx=-a^{3} b e^{\left (-a\right )} \Gamma \left (-1, b x\right ) + 3 \, a^{2} b {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - {\left (b x + 1\right )} b e^{\left (-b x - a\right )} - 3 \, a b e^{\left (-b x - a\right )} \]
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Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.98 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^2} \, dx=-\frac {a^{3} b x {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 3 \, a^{2} b x {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + b^{2} x^{2} e^{\left (-b x - a\right )} + a^{3} e^{\left (-b x - a\right )} + 3 \, a b x e^{\left (-b x - a\right )} + b x e^{\left (-b x - a\right )}}{x} \]
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Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^2} \, dx=a^3\,b\,{\mathrm {e}}^{-a}\,\left (\mathrm {expint}\left (b\,x\right )-\frac {{\mathrm {e}}^{-b\,x}}{b\,x}\right )-3\,a\,b\,{\mathrm {e}}^{-a-b\,x}-b\,{\mathrm {e}}^{-a-b\,x}\,\left (b\,x+1\right )-3\,a^2\,b\,{\mathrm {e}}^{-a}\,\mathrm {expint}\left (b\,x\right ) \]
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