Optimal. Leaf size=102 \[ -\frac {e^{-a-b x} (a+b x)^4}{b}-\frac {4 e^{-a-b x} (a+b x)^3}{b}-\frac {12 e^{-a-b x} (a+b x)^2}{b}-\frac {24 e^{-a-b x} (a+b x)}{b}-\frac {24 e^{-a-b x}}{b} \]
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Rubi [A] time = 0.10, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2176, 2194} \[ -\frac {e^{-a-b x} (a+b x)^4}{b}-\frac {4 e^{-a-b x} (a+b x)^3}{b}-\frac {12 e^{-a-b x} (a+b x)^2}{b}-\frac {24 e^{-a-b x} (a+b x)}{b}-\frac {24 e^{-a-b x}}{b} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rubi steps
\begin {align*} \int e^{-a-b x} (a+b x)^4 \, dx &=-\frac {e^{-a-b x} (a+b x)^4}{b}+4 \int e^{-a-b x} (a+b x)^3 \, dx\\ &=-\frac {4 e^{-a-b x} (a+b x)^3}{b}-\frac {e^{-a-b x} (a+b x)^4}{b}+12 \int e^{-a-b x} (a+b x)^2 \, dx\\ &=-\frac {12 e^{-a-b x} (a+b x)^2}{b}-\frac {4 e^{-a-b x} (a+b x)^3}{b}-\frac {e^{-a-b x} (a+b x)^4}{b}+24 \int e^{-a-b x} (a+b x) \, dx\\ &=-\frac {24 e^{-a-b x} (a+b x)}{b}-\frac {12 e^{-a-b x} (a+b x)^2}{b}-\frac {4 e^{-a-b x} (a+b x)^3}{b}-\frac {e^{-a-b x} (a+b x)^4}{b}+24 \int e^{-a-b x} \, dx\\ &=-\frac {24 e^{-a-b x}}{b}-\frac {24 e^{-a-b x} (a+b x)}{b}-\frac {12 e^{-a-b x} (a+b x)^2}{b}-\frac {4 e^{-a-b x} (a+b x)^3}{b}-\frac {e^{-a-b x} (a+b x)^4}{b}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 50, normalized size = 0.49 \[ \frac {e^{-a-b x} \left (-(a+b x)^4-4 (a+b x)^3-12 (a+b x)^2-24 (a+b x)-24\right )}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 83, normalized size = 0.81 \[ -\frac {{\left (b^{4} x^{4} + 4 \, {\left (a + 1\right )} b^{3} x^{3} + 6 \, {\left (a^{2} + 2 \, a + 2\right )} b^{2} x^{2} + a^{4} + 4 \, a^{3} + 4 \, {\left (a^{3} + 3 \, a^{2} + 6 \, a + 6\right )} b x + 12 \, a^{2} + 24 \, a + 24\right )} e^{\left (-b x - a\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 132, normalized size = 1.29 \[ -\frac {{\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, b^{7} x^{3} + 4 \, a^{3} b^{5} x + 12 \, a b^{6} x^{2} + a^{4} b^{4} + 12 \, a^{2} b^{5} x + 12 \, b^{6} x^{2} + 4 \, a^{3} b^{4} + 24 \, a b^{5} x + 12 \, a^{2} b^{4} + 24 \, b^{5} x + 24 \, a b^{4} + 24 \, b^{4}\right )} e^{\left (-b x - a\right )}}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 108, normalized size = 1.06 \[ -\frac {\left (b^{4} x^{4}+4 b^{3} x^{3} a +6 a^{2} b^{2} x^{2}+4 b^{3} x^{3}+4 a^{3} b x +12 a \,b^{2} x^{2}+a^{4}+12 a^{2} b x +12 b^{2} x^{2}+4 a^{3}+24 a b x +12 a^{2}+24 b x +24 a +24\right ) {\mathrm e}^{-b x -a}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 149, normalized size = 1.46 \[ -\frac {4 \, {\left (b x + 1\right )} a^{3} e^{\left (-b x - a\right )}}{b} - \frac {a^{4} e^{\left (-b x - a\right )}}{b} - \frac {6 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{2} e^{\left (-b x - a\right )}}{b} - \frac {4 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a e^{\left (-b x - a\right )}}{b} - \frac {{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} e^{\left (-b x - a\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 120, normalized size = 1.18 \[ -b^3\,x^4\,{\mathrm {e}}^{-a-b\,x}-x\,{\mathrm {e}}^{-a-b\,x}\,\left (4\,a^3+12\,a^2+24\,a+24\right )-\frac {{\mathrm {e}}^{-a-b\,x}\,\left (a^4+4\,a^3+12\,a^2+24\,a+24\right )}{b}-6\,b\,x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (a^2+2\,a+2\right )-4\,b^2\,x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (a+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 158, normalized size = 1.55 \[ \begin {cases} \frac {\left (- a^{4} - 4 a^{3} b x - 4 a^{3} - 6 a^{2} b^{2} x^{2} - 12 a^{2} b x - 12 a^{2} - 4 a b^{3} x^{3} - 12 a b^{2} x^{2} - 24 a b x - 24 a - b^{4} x^{4} - 4 b^{3} x^{3} - 12 b^{2} x^{2} - 24 b x - 24\right ) e^{- a - b x}}{b} & \text {for}\: b \neq 0 \\a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac {b^{4} x^{5}}{5} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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