Optimal. Leaf size=94 \[ e^{-a} a^3 (-b) \text {Ei}(-b x)-\frac {a^3 e^{-a-b x}}{x}+3 e^{-a} a^2 b \text {Ei}(-b x)-b^2 x e^{-a-b x}-3 a b e^{-a-b x}-b e^{-a-b x} \]
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Rubi [A] time = 0.16, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2199, 2194, 2177, 2178, 2176} \[ e^{-a} a^3 (-b) \text {Ei}(-b x)+3 e^{-a} a^2 b \text {Ei}(-b x)-\frac {a^3 e^{-a-b x}}{x}-b^2 x e^{-a-b x}-3 a b e^{-a-b x}-b e^{-a-b x} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rubi steps
\begin {align*} \int \frac {e^{-a-b x} (a+b x)^3}{x^2} \, dx &=\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*}
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Mathematica [A] time = 0.07, size = 54, normalized size = 0.57 \[ \frac {e^{-a-b x} \left (-a^3-(a-3) a^2 b x e^{b x} \text {Ei}(-b x)-3 a b x-b x (b x+1)\right )}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 56, normalized size = 0.60 \[ -\frac {{\left (a^{3} - 3 \, a^{2}\right )} b x {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + {\left (b^{2} x^{2} + a^{3} + {\left (3 \, a + 1\right )} b x\right )} e^{\left (-b x - a\right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 92, normalized size = 0.98 \[ -\frac {a^{3} b x {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 3 \, a^{2} b x {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + b^{2} x^{2} e^{\left (-b x - a\right )} + a^{3} e^{\left (-b x - a\right )} + 3 \, a b x e^{\left (-b x - a\right )} + b x e^{\left (-b x - a\right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 92, normalized size = 0.98 \[ \left (-3 a^{2} \Ei \left (1, b x \right ) {\mathrm e}^{-a}-\left (-\Ei \left (1, b x \right ) {\mathrm e}^{-a}+\frac {{\mathrm e}^{-b x -a}}{b x}\right ) a^{3}-2 a \,{\mathrm e}^{-b x -a}+\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right ) b \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 61, normalized size = 0.65 \[ -a^{3} b e^{\left (-a\right )} \Gamma \left (-1, b x\right ) + 3 \, a^{2} b {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - {\left (b x + 1\right )} b e^{\left (-b x - a\right )} - 3 \, a b e^{\left (-b x - a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.64, size = 72, normalized size = 0.77 \[ a^3\,b\,{\mathrm {e}}^{-a}\,\left (\mathrm {expint}\left (b\,x\right )-\frac {{\mathrm {e}}^{-b\,x}}{b\,x}\right )-3\,a\,b\,{\mathrm {e}}^{-a-b\,x}-b\,{\mathrm {e}}^{-a-b\,x}\,\left (b\,x+1\right )-3\,a^2\,b\,{\mathrm {e}}^{-a}\,\mathrm {expint}\left (b\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.87, size = 99, normalized size = 1.05 \[ - \frac {a^{3} e^{- a} \operatorname {E}_{2}\left (b x\right )}{x} + 3 a^{2} b e^{- a} \operatorname {Ei}{\left (- b x \right )} + 3 a b^{2} \left (\begin {cases} x & \text {for}\: b = 0 \\- \frac {e^{- b x}}{b} & \text {otherwise} \end {cases}\right ) e^{- a} + b^{3} x \left (\begin {cases} x & \text {for}\: b = 0 \\- \frac {e^{- b x}}{b} & \text {otherwise} \end {cases}\right ) e^{- a} - b^{3} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: b = 0 \\- \frac {\begin {cases} - \frac {e^{- b x}}{b} & \text {for}\: b \neq 0 \\x & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}\right ) e^{- a} \]
Verification of antiderivative is not currently implemented for this CAS.
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