Optimal. Leaf size=397 \[ -\frac{3 a^3 x^2 e^{-a-b x}}{b^2}-\frac{36 a^2 x^2 e^{-a-b x}}{b^2}-\frac{6 a^3 x e^{-a-b x}}{b^3}-\frac{72 a^2 x e^{-a-b x}}{b^3}-\frac{6 a^3 e^{-a-b x}}{b^4}-\frac{72 a^2 e^{-a-b x}}{b^4}-3 a^2 x^4 e^{-a-b x}-\frac{a^3 x^3 e^{-a-b x}}{b}-\frac{12 a^2 x^3 e^{-a-b x}}{b}-b^2 x^6 e^{-a-b x}-\frac{180 a x^2 e^{-a-b x}}{b^2}-\frac{360 x^2 e^{-a-b x}}{b^2}-\frac{360 a x e^{-a-b x}}{b^3}-\frac{720 x e^{-a-b x}}{b^3}-\frac{360 a e^{-a-b x}}{b^4}-\frac{720 e^{-a-b x}}{b^4}-3 a b x^5 e^{-a-b x}-6 b x^5 e^{-a-b x}-15 a x^4 e^{-a-b x}-30 x^4 e^{-a-b x}-\frac{60 a x^3 e^{-a-b x}}{b}-\frac{120 x^3 e^{-a-b x}}{b} \]
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Rubi [A] time = 0.516056, antiderivative size = 397, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2196, 2176, 2194} \[ -\frac{3 a^3 x^2 e^{-a-b x}}{b^2}-\frac{36 a^2 x^2 e^{-a-b x}}{b^2}-\frac{6 a^3 x e^{-a-b x}}{b^3}-\frac{72 a^2 x e^{-a-b x}}{b^3}-\frac{6 a^3 e^{-a-b x}}{b^4}-\frac{72 a^2 e^{-a-b x}}{b^4}-3 a^2 x^4 e^{-a-b x}-\frac{a^3 x^3 e^{-a-b x}}{b}-\frac{12 a^2 x^3 e^{-a-b x}}{b}-b^2 x^6 e^{-a-b x}-\frac{180 a x^2 e^{-a-b x}}{b^2}-\frac{360 x^2 e^{-a-b x}}{b^2}-\frac{360 a x e^{-a-b x}}{b^3}-\frac{720 x e^{-a-b x}}{b^3}-\frac{360 a e^{-a-b x}}{b^4}-\frac{720 e^{-a-b x}}{b^4}-3 a b x^5 e^{-a-b x}-6 b x^5 e^{-a-b x}-15 a x^4 e^{-a-b x}-30 x^4 e^{-a-b x}-\frac{60 a x^3 e^{-a-b x}}{b}-\frac{120 x^3 e^{-a-b x}}{b} \]
Antiderivative was successfully verified.
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Rule 2196
Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{-a-b x} x^3 (a+b x)^3 \, dx &=\int \left (a^3 e^{-a-b x} x^3+3 a^2 b e^{-a-b x} x^4+3 a b^2 e^{-a-b x} x^5+b^3 e^{-a-b x} x^6\right ) \, dx\\ &=a^3 \int e^{-a-b x} x^3 \, dx+\left (3 a^2 b\right ) \int e^{-a-b x} x^4 \, dx+\left (3 a b^2\right ) \int e^{-a-b x} x^5 \, dx+b^3 \int e^{-a-b x} x^6 \, dx\\ &=-\frac{a^3 e^{-a-b x} x^3}{b}-3 a^2 e^{-a-b x} x^4-3 a b e^{-a-b x} x^5-b^2 e^{-a-b x} x^6+\left (12 a^2\right ) \int e^{-a-b x} x^3 \, dx+\frac{\left (3 a^3\right ) \int e^{-a-b x} x^2 \, dx}{b}+(15 a b) \int e^{-a-b x} x^4 \, dx+\left (6 b^2\right ) \int e^{-a-b x} x^5 \, dx\\ &=-\frac{3 a^3 e^{-a-b x} x^2}{b^2}-\frac{12 a^2 e^{-a-b x} x^3}{b}-\frac{a^3 e^{-a-b x} x^3}{b}-15 a e^{-a-b x} x^4-3 a^2 e^{-a-b x} x^4-6 b e^{-a-b x} x^5-3 a b e^{-a-b x} x^5-b^2 e^{-a-b x} x^6+(60 a) \int e^{-a-b x} x^3 \, dx+\frac{\left (6 a^3\right ) \int e^{-a-b x} x \, dx}{b^2}+\frac{\left (36 a^2\right ) \int e^{-a-b x} x^2 \, dx}{b}+(30 b) \int e^{-a-b x} x^4 \, dx\\ &=-\frac{6 a^3 e^{-a-b x} x}{b^3}-\frac{36 a^2 e^{-a-b x} x^2}{b^2}-\frac{3 a^3 e^{-a-b x} x^2}{b^2}-\frac{60 a e^{-a-b x} x^3}{b}-\frac{12 a^2 e^{-a-b x} x^3}{b}-\frac{a^3 e^{-a-b x} x^3}{b}-30 e^{-a-b x} x^4-15 a e^{-a-b x} x^4-3 a^2 e^{-a-b x} x^4-6 b e^{-a-b x} x^5-3 a b e^{-a-b x} x^5-b^2 e^{-a-b x} x^6+120 \int e^{-a-b x} x^3 \, dx+\frac{\left (6 a^3\right ) \int e^{-a-b x} \, dx}{b^3}+\frac{\left (72 a^2\right ) \int e^{-a-b x} x \, dx}{b^2}+\frac{(180 a) \int e^{-a-b x} x^2 \, dx}{b}\\ &=-\frac{6 a^3 e^{-a-b x}}{b^4}-\frac{72 a^2 e^{-a-b x} x}{b^3}-\frac{6 a^3 e^{-a-b x} x}{b^3}-\frac{180 a e^{-a-b x} x^2}{b^2}-\frac{36 a^2 e^{-a-b x} x^2}{b^2}-\frac{3 a^3 e^{-a-b x} x^2}{b^2}-\frac{120 e^{-a-b x} x^3}{b}-\frac{60 a e^{-a-b x} x^3}{b}-\frac{12 a^2 e^{-a-b x} x^3}{b}-\frac{a^3 e^{-a-b x} x^3}{b}-30 e^{-a-b x} x^4-15 a e^{-a-b x} x^4-3 a^2 e^{-a-b x} x^4-6 b e^{-a-b x} x^5-3 a b e^{-a-b x} x^5-b^2 e^{-a-b x} x^6+\frac{\left (72 a^2\right ) \int e^{-a-b x} \, dx}{b^3}+\frac{(360 a) \int e^{-a-b x} x \, dx}{b^2}+\frac{360 \int e^{-a-b x} x^2 \, dx}{b}\\ &=-\frac{72 a^2 e^{-a-b x}}{b^4}-\frac{6 a^3 e^{-a-b x}}{b^4}-\frac{360 a e^{-a-b x} x}{b^3}-\frac{72 a^2 e^{-a-b x} x}{b^3}-\frac{6 a^3 e^{-a-b x} x}{b^3}-\frac{360 e^{-a-b x} x^2}{b^2}-\frac{180 a e^{-a-b x} x^2}{b^2}-\frac{36 a^2 e^{-a-b x} x^2}{b^2}-\frac{3 a^3 e^{-a-b x} x^2}{b^2}-\frac{120 e^{-a-b x} x^3}{b}-\frac{60 a e^{-a-b x} x^3}{b}-\frac{12 a^2 e^{-a-b x} x^3}{b}-\frac{a^3 e^{-a-b x} x^3}{b}-30 e^{-a-b x} x^4-15 a e^{-a-b x} x^4-3 a^2 e^{-a-b x} x^4-6 b e^{-a-b x} x^5-3 a b e^{-a-b x} x^5-b^2 e^{-a-b x} x^6+\frac{(360 a) \int e^{-a-b x} \, dx}{b^3}+\frac{720 \int e^{-a-b x} x \, dx}{b^2}\\ &=-\frac{360 a e^{-a-b x}}{b^4}-\frac{72 a^2 e^{-a-b x}}{b^4}-\frac{6 a^3 e^{-a-b x}}{b^4}-\frac{720 e^{-a-b x} x}{b^3}-\frac{360 a e^{-a-b x} x}{b^3}-\frac{72 a^2 e^{-a-b x} x}{b^3}-\frac{6 a^3 e^{-a-b x} x}{b^3}-\frac{360 e^{-a-b x} x^2}{b^2}-\frac{180 a e^{-a-b x} x^2}{b^2}-\frac{36 a^2 e^{-a-b x} x^2}{b^2}-\frac{3 a^3 e^{-a-b x} x^2}{b^2}-\frac{120 e^{-a-b x} x^3}{b}-\frac{60 a e^{-a-b x} x^3}{b}-\frac{12 a^2 e^{-a-b x} x^3}{b}-\frac{a^3 e^{-a-b x} x^3}{b}-30 e^{-a-b x} x^4-15 a e^{-a-b x} x^4-3 a^2 e^{-a-b x} x^4-6 b e^{-a-b x} x^5-3 a b e^{-a-b x} x^5-b^2 e^{-a-b x} x^6+\frac{720 \int e^{-a-b x} \, dx}{b^3}\\ &=-\frac{720 e^{-a-b x}}{b^4}-\frac{360 a e^{-a-b x}}{b^4}-\frac{72 a^2 e^{-a-b x}}{b^4}-\frac{6 a^3 e^{-a-b x}}{b^4}-\frac{720 e^{-a-b x} x}{b^3}-\frac{360 a e^{-a-b x} x}{b^3}-\frac{72 a^2 e^{-a-b x} x}{b^3}-\frac{6 a^3 e^{-a-b x} x}{b^3}-\frac{360 e^{-a-b x} x^2}{b^2}-\frac{180 a e^{-a-b x} x^2}{b^2}-\frac{36 a^2 e^{-a-b x} x^2}{b^2}-\frac{3 a^3 e^{-a-b x} x^2}{b^2}-\frac{120 e^{-a-b x} x^3}{b}-\frac{60 a e^{-a-b x} x^3}{b}-\frac{12 a^2 e^{-a-b x} x^3}{b}-\frac{a^3 e^{-a-b x} x^3}{b}-30 e^{-a-b x} x^4-15 a e^{-a-b x} x^4-3 a^2 e^{-a-b x} x^4-6 b e^{-a-b x} x^5-3 a b e^{-a-b x} x^5-b^2 e^{-a-b x} x^6\\ \end{align*}
Mathematica [A] time = 0.323201, size = 121, normalized size = 0.3 \[ e^{-a-b x} \left (-\frac{3 \left (a^3+12 a^2+60 a+120\right ) x^2}{b^2}-\frac{6 \left (a^3+12 a^2+60 a+120\right ) x}{b^3}-\frac{6 \left (a^3+12 a^2+60 a+120\right )}{b^4}-\frac{\left (a^3+12 a^2+60 a+120\right ) x^3}{b}-3 \left (a^2+5 a+10\right ) x^4-3 (a+2) b x^5-b^2 x^6\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 182, normalized size = 0.5 \begin{align*} -{\frac{ \left ({b}^{6}{x}^{6}+3\,{b}^{5}{x}^{5}a+3\,{a}^{2}{b}^{4}{x}^{4}+6\,{b}^{5}{x}^{5}+{a}^{3}{b}^{3}{x}^{3}+15\,a{b}^{4}{x}^{4}+12\,{a}^{2}{b}^{3}{x}^{3}+30\,{x}^{4}{b}^{4}+3\,{a}^{3}{b}^{2}{x}^{2}+60\,a{b}^{3}{x}^{3}+36\,{a}^{2}{b}^{2}{x}^{2}+120\,{x}^{3}{b}^{3}+6\,{a}^{3}bx+180\,a{b}^{2}{x}^{2}+72\,{a}^{2}bx+360\,{b}^{2}{x}^{2}+6\,{a}^{3}+360\,abx+72\,{a}^{2}+720\,bx+360\,a+720 \right ){{\rm e}^{-bx-a}}}{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06296, size = 265, normalized size = 0.67 \begin{align*} -\frac{{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{3} e^{\left (-b x - a\right )}}{b^{4}} - \frac{3 \,{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a^{2} e^{\left (-b x - a\right )}}{b^{4}} - \frac{3 \,{\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 )} a e^{\left (-b x - a\right )}}{b^{4}} - \frac{{\left (b^{6} x^{6} + 6 \, b^{5} x^{5} + 30 \, b^{4} x^{4} + 120 \, b^{3} x^{3} + 360 \, b^{2} x^{2} + 720 \, b x + 720\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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Fricas [A] time = 1.47151, size = 301, normalized size = 0.76 \begin{align*} -\frac{{\left (b^{6} x^{6} + 3 \,{\left (a + 2\right )} b^{5} x^{5} + 3 \,{\left (a^{2} + 5 \, a + 10\right )} b^{4} x^{4} +{\left (a^{3} + 12 \, a^{2} + 60 \, a + 120\right )} b^{3} x^{3} + 3 \,{\left (a^{3} + 12 \, a^{2} + 60 \, a + 120\right )} b^{2} x^{2} + 6 \, a^{3} + 6 \,{\left (a^{3} + 12 \, a^{2} + 60 \, a + 120\right )} b x + 72 \, a^{2} + 360 \, a + 720\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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Sympy [A] time = 0.192338, size = 236, normalized size = 0.59 \begin{align*} \begin{cases} \frac{\left (- a^{3} b^{3} x^{3} - 3 a^{3} b^{2} x^{2} - 6 a^{3} b x - 6 a^{3} - 3 a^{2} b^{4} x^{4} - 12 a^{2} b^{3} x^{3} - 36 a^{2} b^{2} x^{2} - 72 a^{2} b x - 72 a^{2} - 3 a b^{5} x^{5} - 15 a b^{4} x^{4} - 60 a b^{3} x^{3} - 180 a b^{2} x^{2} - 360 a b x - 360 a - b^{6} x^{6} - 6 b^{5} x^{5} - 30 b^{4} x^{4} - 120 b^{3} x^{3} - 360 b^{2} x^{2} - 720 b x - 720\right ) e^{- a - b x}}{b^{4}} & \text{for}\: b^{4} \neq 0 \\\frac{a^{3} x^{4}}{4} + \frac{3 a^{2} b x^{5}}{5} + \frac{a b^{2} x^{6}}{2} + \frac{b^{3} x^{7}}{7} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25142, size = 273, normalized size = 0.69 \begin{align*} -\frac{{\left (b^{9} x^{6} + 3 \, a b^{8} x^{5} + 3 \, a^{2} b^{7} x^{4} + 6 \, b^{8} x^{5} + a^{3} b^{6} x^{3} + 15 \, a b^{7} x^{4} + 12 \, a^{2} b^{6} x^{3} + 30 \, b^{7} x^{4} + 3 \, a^{3} b^{5} x^{2} + 60 \, a b^{6} x^{3} + 36 \, a^{2} b^{5} x^{2} + 120 \, b^{6} x^{3} + 6 \, a^{3} b^{4} x + 180 \, a b^{5} x^{2} + 72 \, a^{2} b^{4} x + 360 \, b^{5} x^{2} + 6 \, a^{3} b^{3} + 360 \, a b^{4} x + 72 \, a^{2} b^{3} + 720 \, b^{4} x + 360 \, a b^{3} + 720 \, b^{3}\right )} e^{\left (-b x - a\right )}}{b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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