Optimal. Leaf size=271 \[ -\frac{e^{-a-b x} (a+b x)^4 (b c-a d)}{b^2}-\frac{4 e^{-a-b x} (a+b x)^3 (b c-a d)}{b^2}-\frac{12 e^{-a-b x} (a+b x)^2 (b c-a d)}{b^2}-\frac{24 e^{-a-b x} (a+b x) (b c-a d)}{b^2}-\frac{24 e^{-a-b x} (b c-a d)}{b^2}-\frac{d e^{-a-b x} (a+b x)^5}{b^2}-\frac{5 d e^{-a-b x} (a+b x)^4}{b^2}-\frac{20 d e^{-a-b x} (a+b x)^3}{b^2}-\frac{60 d e^{-a-b x} (a+b x)^2}{b^2}-\frac{120 d e^{-a-b x} (a+b x)}{b^2}-\frac{120 d e^{-a-b x}}{b^2} \]
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Rubi [A] time = 0.338639, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2196, 2176, 2194} \[ -\frac{e^{-a-b x} (a+b x)^4 (b c-a d)}{b^2}-\frac{4 e^{-a-b x} (a+b x)^3 (b c-a d)}{b^2}-\frac{12 e^{-a-b x} (a+b x)^2 (b c-a d)}{b^2}-\frac{24 e^{-a-b x} (a+b x) (b c-a d)}{b^2}-\frac{24 e^{-a-b x} (b c-a d)}{b^2}-\frac{d e^{-a-b x} (a+b x)^5}{b^2}-\frac{5 d e^{-a-b x} (a+b x)^4}{b^2}-\frac{20 d e^{-a-b x} (a+b x)^3}{b^2}-\frac{60 d e^{-a-b x} (a+b x)^2}{b^2}-\frac{120 d e^{-a-b x} (a+b x)}{b^2}-\frac{120 d e^{-a-b x}}{b^2} \]
Antiderivative was successfully verified.
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Rule 2196
Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{-a-b x} (a+b x)^4 (c+d x) \, dx &=\int \left (\frac{(b c-a d) e^{-a-b x} (a+b x)^4}{b}+\frac{d e^{-a-b x} (a+b x)^5}{b}\right ) \, dx\\ &=\frac{d \int e^{-a-b x} (a+b x)^5 \, dx}{b}+\frac{(b c-a d) \int e^{-a-b x} (a+b x)^4 \, dx}{b}\\ &=-\frac{(b c-a d) e^{-a-b x} (a+b x)^4}{b^2}-\frac{d e^{-a-b x} (a+b x)^5}{b^2}+\frac{(5 d) \int e^{-a-b x} (a+b x)^4 \, dx}{b}+\frac{(4 (b c-a d)) \int e^{-a-b x} (a+b x)^3 \, dx}{b}\\ &=-\frac{4 (b c-a d) e^{-a-b x} (a+b x)^3}{b^2}-\frac{5 d e^{-a-b x} (a+b x)^4}{b^2}-\frac{(b c-a d) e^{-a-b x} (a+b x)^4}{b^2}-\frac{d e^{-a-b x} (a+b x)^5}{b^2}+\frac{(20 d) \int e^{-a-b x} (a+b x)^3 \, dx}{b}+\frac{(12 (b c-a d)) \int e^{-a-b x} (a+b x)^2 \, dx}{b}\\ &=-\frac{12 (b c-a d) e^{-a-b x} (a+b x)^2}{b^2}-\frac{20 d e^{-a-b x} (a+b x)^3}{b^2}-\frac{4 (b c-a d) e^{-a-b x} (a+b x)^3}{b^2}-\frac{5 d e^{-a-b x} (a+b x)^4}{b^2}-\frac{(b c-a d) e^{-a-b x} (a+b x)^4}{b^2}-\frac{d e^{-a-b x} (a+b x)^5}{b^2}+\frac{(60 d) \int e^{-a-b x} (a+b x)^2 \, dx}{b}+\frac{(24 (b c-a d)) \int e^{-a-b x} (a+b x) \, dx}{b}\\ &=-\frac{24 (b c-a d) e^{-a-b x} (a+b x)}{b^2}-\frac{60 d e^{-a-b x} (a+b x)^2}{b^2}-\frac{12 (b c-a d) e^{-a-b x} (a+b x)^2}{b^2}-\frac{20 d e^{-a-b x} (a+b x)^3}{b^2}-\frac{4 (b c-a d) e^{-a-b x} (a+b x)^3}{b^2}-\frac{5 d e^{-a-b x} (a+b x)^4}{b^2}-\frac{(b c-a d) e^{-a-b x} (a+b x)^4}{b^2}-\frac{d e^{-a-b x} (a+b x)^5}{b^2}+\frac{(120 d) \int e^{-a-b x} (a+b x) \, dx}{b}+\frac{(24 (b c-a d)) \int e^{-a-b x} \, dx}{b}\\ &=-\frac{24 (b c-a d) e^{-a-b x}}{b^2}-\frac{120 d e^{-a-b x} (a+b x)}{b^2}-\frac{24 (b c-a d) e^{-a-b x} (a+b x)}{b^2}-\frac{60 d e^{-a-b x} (a+b x)^2}{b^2}-\frac{12 (b c-a d) e^{-a-b x} (a+b x)^2}{b^2}-\frac{20 d e^{-a-b x} (a+b x)^3}{b^2}-\frac{4 (b c-a d) e^{-a-b x} (a+b x)^3}{b^2}-\frac{5 d e^{-a-b x} (a+b x)^4}{b^2}-\frac{(b c-a d) e^{-a-b x} (a+b x)^4}{b^2}-\frac{d e^{-a-b x} (a+b x)^5}{b^2}+\frac{(120 d) \int e^{-a-b x} \, dx}{b}\\ &=-\frac{120 d e^{-a-b x}}{b^2}-\frac{24 (b c-a d) e^{-a-b x}}{b^2}-\frac{120 d e^{-a-b x} (a+b x)}{b^2}-\frac{24 (b c-a d) e^{-a-b x} (a+b x)}{b^2}-\frac{60 d e^{-a-b x} (a+b x)^2}{b^2}-\frac{12 (b c-a d) e^{-a-b x} (a+b x)^2}{b^2}-\frac{20 d e^{-a-b x} (a+b x)^3}{b^2}-\frac{4 (b c-a d) e^{-a-b x} (a+b x)^3}{b^2}-\frac{5 d e^{-a-b x} (a+b x)^4}{b^2}-\frac{(b c-a d) e^{-a-b x} (a+b x)^4}{b^2}-\frac{d e^{-a-b x} (a+b x)^5}{b^2}\\ \end{align*}
Mathematica [A] time = 0.252916, size = 191, normalized size = 0.7 \[ \frac{e^{-a-b x} \left (-2 b^3 x^2 \left (3 \left (a^2+2 a+2\right ) c+\left (3 a^2+8 a+10\right ) d x\right )-2 b^2 x \left (2 \left (a^3+3 a^2+6 a+6\right ) c+\left (2 a^3+9 a^2+24 a+30\right ) d x\right )-b \left (\left (a^4+4 a^3+12 a^2+24 a+24\right ) c+\left (a^4+8 a^3+36 a^2+96 a+120\right ) d x\right )-\left (a^4+8 a^3+36 a^2+96 a+120\right ) d-b^4 x^3 (4 (a+1) c+(4 a+5) d x)+b^5 \left (-x^4\right ) (c+d x)\right )}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 297, normalized size = 1.1 \begin{align*} -{\frac{ \left ({b}^{5}d{x}^{5}+4\,a{b}^{4}d{x}^{4}+{b}^{5}c{x}^{4}+6\,{a}^{2}{b}^{3}d{x}^{3}+4\,a{b}^{4}c{x}^{3}+5\,{b}^{4}d{x}^{4}+4\,{a}^{3}{b}^{2}d{x}^{2}+6\,{a}^{2}{b}^{3}c{x}^{2}+16\,a{b}^{3}d{x}^{3}+4\,{b}^{4}c{x}^{3}+{a}^{4}bdx+4\,{a}^{3}{b}^{2}cx+18\,{a}^{2}{b}^{2}d{x}^{2}+12\,a{b}^{3}c{x}^{2}+20\,{b}^{3}d{x}^{3}+c{a}^{4}b+8\,{a}^{3}bdx+12\,{a}^{2}{b}^{2}cx+48\,a{b}^{2}d{x}^{2}+12\,{b}^{3}c{x}^{2}+d{a}^{4}+4\,c{a}^{3}b+36\,{a}^{2}bdx+24\,a{b}^{2}cx+60\,{b}^{2}d{x}^{2}+8\,{a}^{3}d+12\,{a}^{2}bc+96\,abdx+24\,{b}^{2}cx+36\,{a}^{2}d+24\,abc+120\,bdx+96\,ad+24\,bc+120\,d \right ){{\rm e}^{-bx-a}}}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08597, size = 464, normalized size = 1.71 \begin{align*} -\frac{4 \,{\left (b x + 1\right )} a^{3} c e^{\left (-b x - a\right )}}{b} - \frac{a^{4} c e^{\left (-b x - a\right )}}{b} - \frac{{\left (b x + 1\right )} a^{4} d e^{\left (-b x - a\right )}}{b^{2}} - \frac{6 \,{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{2} c e^{\left (-b x - a\right )}}{b} - \frac{4 \,{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{3} d e^{\left (-b x - a\right )}}{b^{2}} - \frac{4 \,{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a c e^{\left (-b x - a\right )}}{b} - \frac{6 \,{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{2} d e^{\left (-b x - a\right )}}{b^{2}} - \frac{{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} c e^{\left (-b x - a\right )}}{b} - \frac{4 \,{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a d e^{\left (-b x - a\right )}}{b^{2}} - \frac{{\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} d e^{\left (-b x - a\right )}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50285, size = 468, normalized size = 1.73 \begin{align*} -\frac{{\left (b^{5} d x^{5} +{\left (b^{5} c +{\left (4 \, a + 5\right )} b^{4} d\right )} x^{4} + 2 \,{\left (2 \,{\left (a + 1\right )} b^{4} c +{\left (3 \, a^{2} + 8 \, a + 10\right )} b^{3} d\right )} x^{3} +{\left (a^{4} + 4 \, a^{3} + 12 \, a^{2} + 24 \, a + 24\right )} b c + 2 \,{\left (3 \,{\left (a^{2} + 2 \, a + 2\right )} b^{3} c +{\left (2 \, a^{3} + 9 \, a^{2} + 24 \, a + 30\right )} b^{2} d\right )} x^{2} +{\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} d +{\left (4 \,{\left (a^{3} + 3 \, a^{2} + 6 \, a + 6\right )} b^{2} c +{\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} b d\right )} x\right )} e^{\left (-b x - a\right )}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.245281, size = 447, normalized size = 1.65 \begin{align*} \begin{cases} \frac{\left (- a^{4} b c - a^{4} b d x - a^{4} d - 4 a^{3} b^{2} c x - 4 a^{3} b^{2} d x^{2} - 4 a^{3} b c - 8 a^{3} b d x - 8 a^{3} d - 6 a^{2} b^{3} c x^{2} - 6 a^{2} b^{3} d x^{3} - 12 a^{2} b^{2} c x - 18 a^{2} b^{2} d x^{2} - 12 a^{2} b c - 36 a^{2} b d x - 36 a^{2} d - 4 a b^{4} c x^{3} - 4 a b^{4} d x^{4} - 12 a b^{3} c x^{2} - 16 a b^{3} d x^{3} - 24 a b^{2} c x - 48 a b^{2} d x^{2} - 24 a b c - 96 a b d x - 96 a d - b^{5} c x^{4} - b^{5} d x^{5} - 4 b^{4} c x^{3} - 5 b^{4} d x^{4} - 12 b^{3} c x^{2} - 20 b^{3} d x^{3} - 24 b^{2} c x - 60 b^{2} d x^{2} - 24 b c - 120 b d x - 120 d\right ) e^{- a - b x}}{b^{2}} & \text{for}\: b^{2} \neq 0 \\a^{4} c x + \frac{b^{4} d x^{6}}{6} + x^{5} \left (\frac{4 a b^{3} d}{5} + \frac{b^{4} c}{5}\right ) + x^{4} \left (\frac{3 a^{2} b^{2} d}{2} + a b^{3} c\right ) + x^{3} \left (\frac{4 a^{3} b d}{3} + 2 a^{2} b^{2} c\right ) + x^{2} \left (\frac{a^{4} d}{2} + 2 a^{3} b c\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24089, size = 447, normalized size = 1.65 \begin{align*} -\frac{{\left (b^{9} d x^{5} + b^{9} c x^{4} + 4 \, a b^{8} d x^{4} + 4 \, a b^{8} c x^{3} + 6 \, a^{2} b^{7} d x^{3} + 5 \, b^{8} d x^{4} + 6 \, a^{2} b^{7} c x^{2} + 4 \, a^{3} b^{6} d x^{2} + 4 \, b^{8} c x^{3} + 16 \, a b^{7} d x^{3} + 4 \, a^{3} b^{6} c x + a^{4} b^{5} d x + 12 \, a b^{7} c x^{2} + 18 \, a^{2} b^{6} d x^{2} + 20 \, b^{7} d x^{3} + a^{4} b^{5} c + 12 \, a^{2} b^{6} c x + 8 \, a^{3} b^{5} d x + 12 \, b^{7} c x^{2} + 48 \, a b^{6} d x^{2} + 4 \, a^{3} b^{5} c + a^{4} b^{4} d + 24 \, a b^{6} c x + 36 \, a^{2} b^{5} d x + 60 \, b^{6} d x^{2} + 12 \, a^{2} b^{5} c + 8 \, a^{3} b^{4} d + 24 \, b^{6} c x + 96 \, a b^{5} d x + 24 \, a b^{5} c + 36 \, a^{2} b^{4} d + 120 \, b^{5} d x + 24 \, b^{5} c + 96 \, a b^{4} d + 120 \, b^{4} d\right )} e^{\left (-b x - a\right )}}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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