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 = 0.34, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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 2176
Rule 2194
Rule 2196
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*}
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Mathematica [A] time = 0.27, size = 191, normalized size = 0.70 \[ \frac {e^{-a-b x} \left (-2 b^3 x^2 \left (3 \left (a^2+2 a+2\right ) c+\left (3 a^2+8 a+10\right ) d x\right )-2 b^2 x \left (2 \left (a^3+3 a^2+6 a+6\right ) c+\left (2 a^3+9 a^2+24 a+30\right ) d x\right )-b \left (\left (a^4+4 a^3+12 a^2+24 a+24\right ) c+\left (a^4+8 a^3+36 a^2+96 a+120\right ) d x\right )-\left (a^4+8 a^3+36 a^2+96 a+120\right ) d-b^4 x^3 (4 (a+1) c+(4 a+5) d x)+b^5 \left (-x^4\right ) (c+d x)\right )}{b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 197, normalized size = 0.73 \[ -\frac {{\left (b^{5} d x^{5} + {\left (b^{5} c + {\left (4 \, a + 5\right )} b^{4} d\right )} x^{4} + 2 \, {\left (2 \, {\left (a + 1\right )} b^{4} c + {\left (3 \, a^{2} + 8 \, a + 10\right )} b^{3} d\right )} x^{3} + {\left (a^{4} + 4 \, a^{3} + 12 \, a^{2} + 24 \, a + 24\right )} b c + 2 \, {\left (3 \, {\left (a^{2} + 2 \, a + 2\right )} b^{3} c + {\left (2 \, a^{3} + 9 \, a^{2} + 24 \, a + 30\right )} b^{2} d\right )} x^{2} + {\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} d + {\left (4 \, {\left (a^{3} + 3 \, a^{2} + 6 \, a + 6\right )} b^{2} c + {\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} b d\right )} x\right )} e^{\left (-b x - a\right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 331, normalized size = 1.22 \[ -\frac {{\left (b^{9} d x^{5} + b^{9} c x^{4} + 4 \, a b^{8} d x^{4} + 4 \, a b^{8} c x^{3} + 6 \, a^{2} b^{7} d x^{3} + 5 \, b^{8} d x^{4} + 6 \, a^{2} b^{7} c x^{2} + 4 \, a^{3} b^{6} d x^{2} + 4 \, b^{8} c x^{3} + 16 \, a b^{7} d x^{3} + 4 \, a^{3} b^{6} c x + a^{4} b^{5} d x + 12 \, a b^{7} c x^{2} + 18 \, a^{2} b^{6} d x^{2} + 20 \, b^{7} d x^{3} + a^{4} b^{5} c + 12 \, a^{2} b^{6} c x + 8 \, a^{3} b^{5} d x + 12 \, b^{7} c x^{2} + 48 \, a b^{6} d x^{2} + 4 \, a^{3} b^{5} c + a^{4} b^{4} d + 24 \, a b^{6} c x + 36 \, a^{2} b^{5} d x + 60 \, b^{6} d x^{2} + 12 \, a^{2} b^{5} c + 8 \, a^{3} b^{4} d + 24 \, b^{6} c x + 96 \, a b^{5} d x + 24 \, a b^{5} c + 36 \, a^{2} b^{4} d + 120 \, b^{5} d x + 24 \, b^{5} c + 96 \, a b^{4} d + 120 \, b^{4} d\right )} e^{\left (-b x - a\right )}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 297, normalized size = 1.10 \[ -\frac {\left (d \,b^{5} x^{5}+4 a \,b^{4} d \,x^{4}+b^{5} c \,x^{4}+6 a^{2} b^{3} d \,x^{3}+4 a \,b^{4} c \,x^{3}+5 b^{4} d \,x^{4}+4 a^{3} b^{2} d \,x^{2}+6 a^{2} b^{3} c \,x^{2}+16 a \,b^{3} d \,x^{3}+4 b^{4} c \,x^{3}+a^{4} b d x +4 a^{3} b^{2} c x +18 a^{2} b^{2} d \,x^{2}+12 a \,b^{3} c \,x^{2}+20 b^{3} d \,x^{3}+c \,a^{4} b +8 a^{3} b d x +12 a^{2} b^{2} c x +48 a \,b^{2} d \,x^{2}+12 b^{3} c \,x^{2}+d \,a^{4}+4 c \,a^{3} b +36 a^{2} b d x +24 a \,b^{2} c x +60 b^{2} d \,x^{2}+8 a^{3} d +12 a^{2} b c +96 a b d x +24 b^{2} c x +36 a^{2} d +24 a b c +120 b d x +96 a d +24 b c +120 d \right ) {\mathrm e}^{-b x -a}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 344, normalized size = 1.27 \[ -\frac {4 \, {\left (b x + 1\right )} a^{3} c e^{\left (-b x - a\right )}}{b} - \frac {a^{4} c e^{\left (-b x - a\right )}}{b} - \frac {{\left (b x + 1\right )} a^{4} d e^{\left (-b x - a\right )}}{b^{2}} - \frac {6 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{2} c e^{\left (-b x - a\right )}}{b} - \frac {4 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{3} d e^{\left (-b x - a\right )}}{b^{2}} - \frac {4 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a c e^{\left (-b x - a\right )}}{b} - \frac {6 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{2} d e^{\left (-b x - a\right )}}{b^{2}} - \frac {{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} c e^{\left (-b x - a\right )}}{b} - \frac {4 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a d e^{\left (-b x - a\right )}}{b^{2}} - \frac {{\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} d e^{\left (-b x - a\right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 264, normalized size = 0.97 \[ -\frac {{\mathrm {e}}^{-a-b\,x}\,\left (120\,d+96\,a\,d+24\,b\,c+36\,a^2\,d+8\,a^3\,d+a^4\,d+24\,a\,b\,c+12\,a^2\,b\,c+4\,a^3\,b\,c+a^4\,b\,c\right )}{b^2}-x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (60\,d+48\,a\,d+12\,b\,c+18\,a^2\,d+4\,a^3\,d+12\,a\,b\,c+6\,a^2\,b\,c\right )-x\,{\mathrm {e}}^{-a-b\,x}\,\left (24\,c+24\,a\,c+12\,a^2\,c+4\,a^3\,c+\frac {d\,a^4+8\,d\,a^3+36\,d\,a^2+96\,d\,a+120\,d}{b}\right )-b^3\,d\,x^5\,{\mathrm {e}}^{-a-b\,x}-b^2\,x^4\,{\mathrm {e}}^{-a-b\,x}\,\left (5\,d+4\,a\,d+b\,c\right )-2\,b\,x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (10\,d+8\,a\,d+2\,b\,c+3\,a^2\,d+2\,a\,b\,c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 447, normalized size = 1.65 \[ \begin {cases} \frac {\left (- a^{4} b c - a^{4} b d x - a^{4} d - 4 a^{3} b^{2} c x - 4 a^{3} b^{2} d x^{2} - 4 a^{3} b c - 8 a^{3} b d x - 8 a^{3} d - 6 a^{2} b^{3} c x^{2} - 6 a^{2} b^{3} d x^{3} - 12 a^{2} b^{2} c x - 18 a^{2} b^{2} d x^{2} - 12 a^{2} b c - 36 a^{2} b d x - 36 a^{2} d - 4 a b^{4} c x^{3} - 4 a b^{4} d x^{4} - 12 a b^{3} c x^{2} - 16 a b^{3} d x^{3} - 24 a b^{2} c x - 48 a b^{2} d x^{2} - 24 a b c - 96 a b d x - 96 a d - b^{5} c x^{4} - b^{5} d x^{5} - 4 b^{4} c x^{3} - 5 b^{4} d x^{4} - 12 b^{3} c x^{2} - 20 b^{3} d x^{3} - 24 b^{2} c x - 60 b^{2} d x^{2} - 24 b c - 120 b d x - 120 d\right ) e^{- a - b x}}{b^{2}} & \text {for}\: b^{2} \neq 0 \\a^{4} c x + \frac {b^{4} d x^{6}}{6} + x^{5} \left (\frac {4 a b^{3} d}{5} + \frac {b^{4} c}{5}\right ) + x^{4} \left (\frac {3 a^{2} b^{2} d}{2} + a b^{3} c\right ) + x^{3} \left (\frac {4 a^{3} b d}{3} + 2 a^{2} b^{2} c\right ) + x^{2} \left (\frac {a^{4} d}{2} + 2 a^{3} b c\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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