Optimal. Leaf size=318 \[ -\frac{2 a^3 x e^{-a-b x}}{b^2}-\frac{18 a^2 x e^{-a-b x}}{b^2}-\frac{2 a^3 e^{-a-b x}}{b^3}-\frac{18 a^2 e^{-a-b x}}{b^3}-3 a^2 x^3 e^{-a-b x}-\frac{a^3 x^2 e^{-a-b x}}{b}-\frac{9 a^2 x^2 e^{-a-b x}}{b}-b^2 x^5 e^{-a-b x}-\frac{72 a x e^{-a-b x}}{b^2}-\frac{120 x e^{-a-b x}}{b^2}-\frac{72 a e^{-a-b x}}{b^3}-\frac{120 e^{-a-b x}}{b^3}-3 a b x^4 e^{-a-b x}-5 b x^4 e^{-a-b x}-12 a x^3 e^{-a-b x}-20 x^3 e^{-a-b x}-\frac{36 a x^2 e^{-a-b x}}{b}-\frac{60 x^2 e^{-a-b x}}{b} \]
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Rubi [A] time = 0.412468, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2196, 2176, 2194} \[ -\frac{2 a^3 x e^{-a-b x}}{b^2}-\frac{18 a^2 x e^{-a-b x}}{b^2}-\frac{2 a^3 e^{-a-b x}}{b^3}-\frac{18 a^2 e^{-a-b x}}{b^3}-3 a^2 x^3 e^{-a-b x}-\frac{a^3 x^2 e^{-a-b x}}{b}-\frac{9 a^2 x^2 e^{-a-b x}}{b}-b^2 x^5 e^{-a-b x}-\frac{72 a x e^{-a-b x}}{b^2}-\frac{120 x e^{-a-b x}}{b^2}-\frac{72 a e^{-a-b x}}{b^3}-\frac{120 e^{-a-b x}}{b^3}-3 a b x^4 e^{-a-b x}-5 b x^4 e^{-a-b x}-12 a x^3 e^{-a-b x}-20 x^3 e^{-a-b x}-\frac{36 a x^2 e^{-a-b x}}{b}-\frac{60 x^2 e^{-a-b x}}{b} \]
Antiderivative was successfully verified.
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Rule 2196
Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{-a-b x} x^2 (a+b x)^3 \, dx &=\int \left (a^3 e^{-a-b x} x^2+3 a^2 b e^{-a-b x} x^3+3 a b^2 e^{-a-b x} x^4+b^3 e^{-a-b x} x^5\right ) \, dx\\ &=a^3 \int e^{-a-b x} x^2 \, dx+\left (3 a^2 b\right ) \int e^{-a-b x} x^3 \, dx+\left (3 a b^2\right ) \int e^{-a-b x} x^4 \, dx+b^3 \int e^{-a-b x} x^5 \, dx\\ &=-\frac{a^3 e^{-a-b x} x^2}{b}-3 a^2 e^{-a-b x} x^3-3 a b e^{-a-b x} x^4-b^2 e^{-a-b x} x^5+\left (9 a^2\right ) \int e^{-a-b x} x^2 \, dx+\frac{\left (2 a^3\right ) \int e^{-a-b x} x \, dx}{b}+(12 a b) \int e^{-a-b x} x^3 \, dx+\left (5 b^2\right ) \int e^{-a-b x} x^4 \, dx\\ &=-\frac{2 a^3 e^{-a-b x} x}{b^2}-\frac{9 a^2 e^{-a-b x} x^2}{b}-\frac{a^3 e^{-a-b x} x^2}{b}-12 a e^{-a-b x} x^3-3 a^2 e^{-a-b x} x^3-5 b e^{-a-b x} x^4-3 a b e^{-a-b x} x^4-b^2 e^{-a-b x} x^5+(36 a) \int e^{-a-b x} x^2 \, dx+\frac{\left (2 a^3\right ) \int e^{-a-b x} \, dx}{b^2}+\frac{\left (18 a^2\right ) \int e^{-a-b x} x \, dx}{b}+(20 b) \int e^{-a-b x} x^3 \, dx\\ &=-\frac{2 a^3 e^{-a-b x}}{b^3}-\frac{18 a^2 e^{-a-b x} x}{b^2}-\frac{2 a^3 e^{-a-b x} x}{b^2}-\frac{36 a e^{-a-b x} x^2}{b}-\frac{9 a^2 e^{-a-b x} x^2}{b}-\frac{a^3 e^{-a-b x} x^2}{b}-20 e^{-a-b x} x^3-12 a e^{-a-b x} x^3-3 a^2 e^{-a-b x} x^3-5 b e^{-a-b x} x^4-3 a b e^{-a-b x} x^4-b^2 e^{-a-b x} x^5+60 \int e^{-a-b x} x^2 \, dx+\frac{\left (18 a^2\right ) \int e^{-a-b x} \, dx}{b^2}+\frac{(72 a) \int e^{-a-b x} x \, dx}{b}\\ &=-\frac{18 a^2 e^{-a-b x}}{b^3}-\frac{2 a^3 e^{-a-b x}}{b^3}-\frac{72 a e^{-a-b x} x}{b^2}-\frac{18 a^2 e^{-a-b x} x}{b^2}-\frac{2 a^3 e^{-a-b x} x}{b^2}-\frac{60 e^{-a-b x} x^2}{b}-\frac{36 a e^{-a-b x} x^2}{b}-\frac{9 a^2 e^{-a-b x} x^2}{b}-\frac{a^3 e^{-a-b x} x^2}{b}-20 e^{-a-b x} x^3-12 a e^{-a-b x} x^3-3 a^2 e^{-a-b x} x^3-5 b e^{-a-b x} x^4-3 a b e^{-a-b x} x^4-b^2 e^{-a-b x} x^5+\frac{(72 a) \int e^{-a-b x} \, dx}{b^2}+\frac{120 \int e^{-a-b x} x \, dx}{b}\\ &=-\frac{72 a e^{-a-b x}}{b^3}-\frac{18 a^2 e^{-a-b x}}{b^3}-\frac{2 a^3 e^{-a-b x}}{b^3}-\frac{120 e^{-a-b x} x}{b^2}-\frac{72 a e^{-a-b x} x}{b^2}-\frac{18 a^2 e^{-a-b x} x}{b^2}-\frac{2 a^3 e^{-a-b x} x}{b^2}-\frac{60 e^{-a-b x} x^2}{b}-\frac{36 a e^{-a-b x} x^2}{b}-\frac{9 a^2 e^{-a-b x} x^2}{b}-\frac{a^3 e^{-a-b x} x^2}{b}-20 e^{-a-b x} x^3-12 a e^{-a-b x} x^3-3 a^2 e^{-a-b x} x^3-5 b e^{-a-b x} x^4-3 a b e^{-a-b x} x^4-b^2 e^{-a-b x} x^5+\frac{120 \int e^{-a-b x} \, dx}{b^2}\\ &=-\frac{120 e^{-a-b x}}{b^3}-\frac{72 a e^{-a-b x}}{b^3}-\frac{18 a^2 e^{-a-b x}}{b^3}-\frac{2 a^3 e^{-a-b x}}{b^3}-\frac{120 e^{-a-b x} x}{b^2}-\frac{72 a e^{-a-b x} x}{b^2}-\frac{18 a^2 e^{-a-b x} x}{b^2}-\frac{2 a^3 e^{-a-b x} x}{b^2}-\frac{60 e^{-a-b x} x^2}{b}-\frac{36 a e^{-a-b x} x^2}{b}-\frac{9 a^2 e^{-a-b x} x^2}{b}-\frac{a^3 e^{-a-b x} x^2}{b}-20 e^{-a-b x} x^3-12 a e^{-a-b x} x^3-3 a^2 e^{-a-b x} x^3-5 b e^{-a-b x} x^4-3 a b e^{-a-b x} x^4-b^2 e^{-a-b x} x^5\\ \end{align*}
Mathematica [A] time = 0.198432, size = 130, normalized size = 0.41 \[ e^{-b x} \left (-\frac{2 \left (a^3+9 a^2+36 a+60\right ) e^{-a} x}{b^2}-\frac{2 \left (a^3+9 a^2+36 a+60\right ) e^{-a}}{b^3}-\frac{\left (a^3+9 a^2+36 a+60\right ) e^{-a} x^2}{b}-\left (3 a^2+12 a+20\right ) e^{-a} x^3+e^{-a} \left (-b^2\right ) x^5-(3 a+5) e^{-a} b x^4\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 143, normalized size = 0.5 \begin{align*} -{\frac{ \left ({b}^{5}{x}^{5}+3\,{b}^{4}{x}^{4}a+3\,{a}^{2}{b}^{3}{x}^{3}+5\,{b}^{4}{x}^{4}+{a}^{3}{b}^{2}{x}^{2}+12\,a{b}^{3}{x}^{3}+9\,{a}^{2}{b}^{2}{x}^{2}+20\,{x}^{3}{b}^{3}+2\,{a}^{3}bx+36\,a{b}^{2}{x}^{2}+18\,{a}^{2}bx+60\,{b}^{2}{x}^{2}+2\,{a}^{3}+72\,abx+18\,{a}^{2}+120\,bx+72\,a+120 \right ){{\rm e}^{-bx-a}}}{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05629, size = 221, normalized size = 0.69 \begin{align*} -\frac{{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{3} e^{\left (-b x - a\right )}}{b^{3}} - \frac{3 \,{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac{3 \,{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a e^{\left (-b x - a\right )}}{b^{3}} - \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 )} e^{\left (-b x - a\right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52069, size = 242, normalized size = 0.76 \begin{align*} -\frac{{\left (b^{5} x^{5} +{\left (3 \, a + 5\right )} b^{4} x^{4} +{\left (3 \, a^{2} + 12 \, a + 20\right )} b^{3} x^{3} +{\left (a^{3} + 9 \, a^{2} + 36 \, a + 60\right )} b^{2} x^{2} + 2 \, a^{3} + 2 \,{\left (a^{3} + 9 \, a^{2} + 36 \, a + 60\right )} b x + 18 \, a^{2} + 72 \, a + 120\right )} e^{\left (-b x - a\right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.175937, size = 196, normalized size = 0.62 \begin{align*} \begin{cases} \frac{\left (- a^{3} b^{2} x^{2} - 2 a^{3} b x - 2 a^{3} - 3 a^{2} b^{3} x^{3} - 9 a^{2} b^{2} x^{2} - 18 a^{2} b x - 18 a^{2} - 3 a b^{4} x^{4} - 12 a b^{3} x^{3} - 36 a b^{2} x^{2} - 72 a b x - 72 a - b^{5} x^{5} - 5 b^{4} x^{4} - 20 b^{3} x^{3} - 60 b^{2} x^{2} - 120 b x - 120\right ) e^{- a - b x}}{b^{3}} & \text{for}\: b^{3} \neq 0 \\\frac{a^{3} x^{3}}{3} + \frac{3 a^{2} b x^{4}}{4} + \frac{3 a b^{2} x^{5}}{5} + \frac{b^{3} x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36125, size = 220, normalized size = 0.69 \begin{align*} -\frac{{\left (b^{8} x^{5} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{3} + 5 \, b^{7} x^{4} + a^{3} b^{5} x^{2} + 12 \, a b^{6} x^{3} + 9 \, a^{2} b^{5} x^{2} + 20 \, b^{6} x^{3} + 2 \, a^{3} b^{4} x + 36 \, a b^{5} x^{2} + 18 \, a^{2} b^{4} x + 60 \, b^{5} x^{2} + 2 \, a^{3} b^{3} + 72 \, a b^{4} x + 18 \, a^{2} b^{3} + 120 \, b^{4} x + 72 \, a b^{3} + 120 \, b^{3}\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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