Optimal. Leaf size=130 \[ -b^2 e^{-a-b x}-\frac {a^3 e^{-a-b x}}{2 x^2}-\frac {3 a^2 b e^{-a-b x}}{x}+\frac {a^3 b e^{-a-b x}}{2 x}+3 a b^2 e^{-a} \text {Ei}(-b x)-3 a^2 b^2 e^{-a} \text {Ei}(-b x)+\frac {1}{2} a^3 b^2 e^{-a} \text {Ei}(-b x) \]
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Rubi [A]
time = 0.15, antiderivative size = 130, 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 = {2230, 2225,
2208, 2209} \begin {gather*} \frac {1}{2} e^{-a} a^3 b^2 \text {Ei}(-b x)-\frac {a^3 e^{-a-b x}}{2 x^2}+\frac {a^3 b e^{-a-b x}}{2 x}-3 e^{-a} a^2 b^2 \text {Ei}(-b x)-\frac {3 a^2 b e^{-a-b x}}{x}+3 e^{-a} a b^2 \text {Ei}(-b x)-b^2 e^{-a-b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps
\begin {align*} \int \frac {e^{-a-b x} (a+b x)^3}{x^3} \, dx &=\int \left (b^3 e^{-a-b x}+\frac {a^3 e^{-a-b x}}{x^3}+\frac {3 a^2 b e^{-a-b x}}{x^2}+\frac {3 a b^2 e^{-a-b x}}{x}\right ) \, dx\\ &=a^3 \int \frac {e^{-a-b x}}{x^3} \, dx+\left (3 a^2 b\right ) \int \frac {e^{-a-b x}}{x^2} \, dx+\left (3 a b^2\right ) \int \frac {e^{-a-b x}}{x} \, dx+b^3 \int e^{-a-b x} \, dx\\ &=-b^2 e^{-a-b x}-\frac {a^3 e^{-a-b x}}{2 x^2}-\frac {3 a^2 b e^{-a-b x}}{x}+3 a b^2 e^{-a} \text {Ei}(-b x)-\frac {1}{2} \left (a^3 b\right ) \int \frac {e^{-a-b x}}{x^2} \, dx-\left (3 a^2 b^2\right ) \int \frac {e^{-a-b x}}{x} \, dx\\ &=-b^2 e^{-a-b x}-\frac {a^3 e^{-a-b x}}{2 x^2}-\frac {3 a^2 b e^{-a-b x}}{x}+\frac {a^3 b e^{-a-b x}}{2 x}+3 a b^2 e^{-a} \text {Ei}(-b x)-3 a^2 b^2 e^{-a} \text {Ei}(-b x)+\frac {1}{2} \left (a^3 b^2\right ) \int \frac {e^{-a-b x}}{x} \, dx\\ &=-b^2 e^{-a-b x}-\frac {a^3 e^{-a-b x}}{2 x^2}-\frac {3 a^2 b e^{-a-b x}}{x}+\frac {a^3 b e^{-a-b x}}{2 x}+3 a b^2 e^{-a} \text {Ei}(-b x)-3 a^2 b^2 e^{-a} \text {Ei}(-b x)+\frac {1}{2} a^3 b^2 e^{-a} \text {Ei}(-b x)\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 68, normalized size = 0.52 \begin {gather*} \frac {e^{-a-b x} \left (-6 a^2 b x-2 b^2 x^2+a^3 (-1+b x)+a \left (6-6 a+a^2\right ) b^2 e^{b x} x^2 \text {Ei}(-b x)\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 112, normalized size = 0.86
method | result | size |
derivativedivides | \(-b^{2} \left ({\mathrm e}^{-b x -a}-a^{3} \left (-\frac {{\mathrm e}^{-b x -a}}{2 b^{2} x^{2}}+\frac {{\mathrm e}^{-b x -a}}{2 b x}-\frac {{\mathrm e}^{-a} \expIntegral \left (1, b x \right )}{2}\right )+3 a^{2} \left (\frac {{\mathrm e}^{-b x -a}}{b x}-{\mathrm e}^{-a} \expIntegral \left (1, b x \right )\right )+3 a \,{\mathrm e}^{-a} \expIntegral \left (1, b x \right )\right )\) | \(112\) |
default | \(-b^{2} \left ({\mathrm e}^{-b x -a}-a^{3} \left (-\frac {{\mathrm e}^{-b x -a}}{2 b^{2} x^{2}}+\frac {{\mathrm e}^{-b x -a}}{2 b x}-\frac {{\mathrm e}^{-a} \expIntegral \left (1, b x \right )}{2}\right )+3 a^{2} \left (\frac {{\mathrm e}^{-b x -a}}{b x}-{\mathrm e}^{-a} \expIntegral \left (1, b x \right )\right )+3 a \,{\mathrm e}^{-a} \expIntegral \left (1, b x \right )\right )\) | \(112\) |
risch | \(-b^{2} {\mathrm e}^{-b x -a}-\frac {a^{3} {\mathrm e}^{-b x -a}}{2 x^{2}}+\frac {a^{3} b \,{\mathrm e}^{-b x -a}}{2 x}-\frac {b^{2} a^{3} {\mathrm e}^{-a} \expIntegral \left (1, b x \right )}{2}-\frac {3 a^{2} b \,{\mathrm e}^{-b x -a}}{x}+3 b^{2} a^{2} {\mathrm e}^{-a} \expIntegral \left (1, b x \right )-3 b^{2} a \,{\mathrm e}^{-a} \expIntegral \left (1, b x \right )\) | \(118\) |
meijerg | \({\mathrm e}^{-a} b^{2} \left (1-{\mathrm e}^{-b x}\right )+3 b^{2} {\mathrm e}^{-a} a \left (-\ln \left (b x \right )-\expIntegral \left (1, b x \right )+\ln \left (x \right )+\ln \left (b \right )\right )+3 b^{2} {\mathrm e}^{-a} a^{2} \left (\frac {-2 b x +2}{2 b x}-\frac {{\mathrm e}^{-b x}}{b x}+\ln \left (b x \right )+\expIntegral \left (1, b x \right )+1-\ln \left (x \right )-\ln \left (b \right )-\frac {1}{b x}\right )+{\mathrm e}^{-a} a^{3} b^{2} \left (\frac {9 b^{2} x^{2}-12 b x +6}{12 b^{2} x^{2}}-\frac {\left (-3 b x +3\right ) {\mathrm e}^{-b x}}{6 b^{2} x^{2}}-\frac {\ln \left (b x \right )}{2}-\frac {\expIntegral \left (1, b x \right )}{2}-\frac {3}{4}+\frac {\ln \left (x \right )}{2}+\frac {\ln \left (b \right )}{2}-\frac {1}{2 b^{2} x^{2}}+\frac {1}{b x}\right )\) | \(203\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 64, normalized size = 0.49 \begin {gather*} -a^{3} b^{2} e^{\left (-a\right )} \Gamma \left (-2, b x\right ) - 3 \, a^{2} b^{2} e^{\left (-a\right )} \Gamma \left (-1, b x\right ) + 3 \, a b^{2} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - b^{2} e^{\left (-b x - a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 70, normalized size = 0.54 \begin {gather*} \frac {{\left (a^{3} - 6 \, a^{2} + 6 \, a\right )} b^{2} x^{2} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - {\left (2 \, b^{2} x^{2} + a^{3} - {\left (a^{3} - 6 \, a^{2}\right )} b x\right )} e^{\left (-b x - a\right )}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.75, size = 56, normalized size = 0.43 \begin {gather*} \left (- \frac {a^{3} \operatorname {E}_{3}\left (b x\right )}{x^{2}} - \frac {3 a^{2} b \operatorname {E}_{2}\left (b x\right )}{x} + 3 a b^{2} \operatorname {Ei}{\left (- b x \right )} + b^{3} \left (\begin {cases} x & \text {for}\: b = 0 \\- \frac {e^{- b x}}{b} & \text {otherwise} \end {cases}\right )\right ) e^{- a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.79, size = 125, normalized size = 0.96 \begin {gather*} \frac {a^{3} b^{2} x^{2} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 6 \, a^{2} b^{2} x^{2} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + 6 \, a b^{2} x^{2} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + a^{3} b x e^{\left (-b x - a\right )} - 6 \, a^{2} b x e^{\left (-b x - a\right )} - 2 \, b^{2} x^{2} e^{\left (-b x - a\right )} - a^{3} e^{\left (-b x - a\right )}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.64, size = 100, normalized size = 0.77 \begin {gather*} 3\,a^2\,b^2\,{\mathrm {e}}^{-a}\,\left (\mathrm {expint}\left (b\,x\right )-\frac {{\mathrm {e}}^{-b\,x}}{b\,x}\right )-3\,a\,b^2\,{\mathrm {e}}^{-a}\,\mathrm {expint}\left (b\,x\right )-b^2\,{\mathrm {e}}^{-a-b\,x}+a^3\,b^2\,{\mathrm {e}}^{-a}\,\left ({\mathrm {e}}^{-b\,x}\,\left (\frac {1}{2\,b\,x}-\frac {1}{2\,b^2\,x^2}\right )-\frac {\mathrm {expint}\left (b\,x\right )}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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