Optimal. Leaf size=80 \[ -\frac{e^{-a-b x} (a+b x)^3}{b}-\frac{3 e^{-a-b x} (a+b x)^2}{b}-\frac{6 e^{-a-b x} (a+b x)}{b}-\frac{6 e^{-a-b x}}{b} \]
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Rubi [A] time = 0.0683236, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2176, 2194} \[ -\frac{e^{-a-b x} (a+b x)^3}{b}-\frac{3 e^{-a-b x} (a+b x)^2}{b}-\frac{6 e^{-a-b x} (a+b x)}{b}-\frac{6 e^{-a-b x}}{b} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{-a-b x} (a+b x)^3 \, dx &=-\frac{e^{-a-b x} (a+b x)^3}{b}+3 \int e^{-a-b x} (a+b x)^2 \, dx\\ &=-\frac{3 e^{-a-b x} (a+b x)^2}{b}-\frac{e^{-a-b x} (a+b x)^3}{b}+6 \int e^{-a-b x} (a+b x) \, dx\\ &=-\frac{6 e^{-a-b x} (a+b x)}{b}-\frac{3 e^{-a-b x} (a+b x)^2}{b}-\frac{e^{-a-b x} (a+b x)^3}{b}+6 \int e^{-a-b x} \, dx\\ &=-\frac{6 e^{-a-b x}}{b}-\frac{6 e^{-a-b x} (a+b x)}{b}-\frac{3 e^{-a-b x} (a+b x)^2}{b}-\frac{e^{-a-b x} (a+b x)^3}{b}\\ \end{align*}
Mathematica [A] time = 0.046695, size = 41, normalized size = 0.51 \[ \frac{e^{-a-b x} \left (-(a+b x)^3-3 (a+b x)^2-6 (a+b x)-6\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 68, normalized size = 0.9 \begin{align*} -{\frac{ \left ({b}^{3}{x}^{3}+3\,{b}^{2}{x}^{2}a+3\,{a}^{2}bx+3\,{b}^{2}{x}^{2}+{a}^{3}+6\,abx+3\,{a}^{2}+6\,bx+6\,a+6 \right ){{\rm e}^{-bx-a}}}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06245, size = 139, normalized size = 1.74 \begin{align*} -\frac{3 \,{\left (b x + 1\right )} a^{2} e^{\left (-b x - a\right )}}{b} - \frac{a^{3} e^{\left (-b x - a\right )}}{b} - \frac{3 \,{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a e^{\left (-b x - a\right )}}{b} - \frac{{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47203, size = 128, normalized size = 1.6 \begin{align*} -\frac{{\left (b^{3} x^{3} + 3 \,{\left (a + 1\right )} b^{2} x^{2} + a^{3} + 3 \,{\left (a^{2} + 2 \, a + 2\right )} b x + 3 \, a^{2} + 6 \, a + 6\right )} e^{\left (-b x - a\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.145624, size = 104, normalized size = 1.3 \begin{align*} \begin{cases} \frac{\left (- a^{3} - 3 a^{2} b x - 3 a^{2} - 3 a b^{2} x^{2} - 6 a b x - 6 a - b^{3} x^{3} - 3 b^{2} x^{2} - 6 b x - 6\right ) e^{- a - b x}}{b} & \text{for}\: b \neq 0 \\a^{3} x + \frac{3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac{b^{3} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31471, size = 117, normalized size = 1.46 \begin{align*} -\frac{{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + 3 \, b^{5} x^{2} + a^{3} b^{3} + 6 \, a b^{4} x + 3 \, a^{2} b^{3} + 6 \, b^{4} x + 6 \, a b^{3} + 6 \, b^{3}\right )} e^{\left (-b x - a\right )}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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