Integrand size = 21, antiderivative size = 397 \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=-\frac {720 e^{-a-b x}}{b^4}-\frac {360 a e^{-a-b x}}{b^4}-\frac {72 a^2 e^{-a-b x}}{b^4}-\frac {6 a^3 e^{-a-b x}}{b^4}-\frac {720 e^{-a-b x} x}{b^3}-\frac {360 a e^{-a-b x} x}{b^3}-\frac {72 a^2 e^{-a-b x} x}{b^3}-\frac {6 a^3 e^{-a-b x} x}{b^3}-\frac {360 e^{-a-b x} x^2}{b^2}-\frac {180 a e^{-a-b x} x^2}{b^2}-\frac {36 a^2 e^{-a-b x} x^2}{b^2}-\frac {3 a^3 e^{-a-b x} x^2}{b^2}-\frac {120 e^{-a-b x} x^3}{b}-\frac {60 a e^{-a-b x} x^3}{b}-\frac {12 a^2 e^{-a-b x} x^3}{b}-\frac {a^3 e^{-a-b x} x^3}{b}-30 e^{-a-b x} x^4-15 a e^{-a-b x} x^4-3 a^2 e^{-a-b x} x^4-6 b e^{-a-b x} x^5-3 a b e^{-a-b x} x^5-b^2 e^{-a-b x} x^6 \]
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Time = 0.34 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2227, 2207, 2225} \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=-\frac {6 a^3 e^{-a-b x}}{b^4}-\frac {6 a^3 x e^{-a-b x}}{b^3}-\frac {3 a^3 x^2 e^{-a-b x}}{b^2}-\frac {a^3 x^3 e^{-a-b x}}{b}-\frac {72 a^2 e^{-a-b x}}{b^4}-\frac {72 a^2 x e^{-a-b x}}{b^3}-\frac {36 a^2 x^2 e^{-a-b x}}{b^2}-3 a^2 x^4 e^{-a-b x}-\frac {12 a^2 x^3 e^{-a-b x}}{b}-\frac {360 a e^{-a-b x}}{b^4}-\frac {720 e^{-a-b x}}{b^4}-\frac {360 a x e^{-a-b x}}{b^3}-\frac {720 x e^{-a-b x}}{b^3}-b^2 x^6 e^{-a-b x}-\frac {180 a x^2 e^{-a-b x}}{b^2}-\frac {360 x^2 e^{-a-b x}}{b^2}-3 a b x^5 e^{-a-b x}-6 b x^5 e^{-a-b x}-15 a x^4 e^{-a-b x}-30 x^4 e^{-a-b x}-\frac {60 a x^3 e^{-a-b x}}{b}-\frac {120 x^3 e^{-a-b x}}{b} \]
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Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.30 \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=e^{-a-b x} \left (-\frac {6 \left (120+60 a+12 a^2+a^3\right )}{b^4}-\frac {6 \left (120+60 a+12 a^2+a^3\right ) x}{b^3}-\frac {3 \left (120+60 a+12 a^2+a^3\right ) x^2}{b^2}-\frac {\left (120+60 a+12 a^2+a^3\right ) x^3}{b}-3 \left (10+5 a+a^2\right ) x^4-3 (2+a) b x^5-b^2 x^6\right ) \]
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Time = 0.21 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.45
method | result | size |
norman | \(\left (-3 a b -6 b \right ) x^{5} {\mathrm e}^{-b x -a}+\left (-3 a^{2}-15 a -30\right ) x^{4} {\mathrm e}^{-b x -a}-b^{2} {\mathrm e}^{-b x -a} x^{6}-\frac {6 \left (a^{3}+12 a^{2}+60 a +120\right ) {\mathrm e}^{-b x -a}}{b^{4}}-\frac {6 \left (a^{3}+12 a^{2}+60 a +120\right ) x \,{\mathrm e}^{-b x -a}}{b^{3}}-\frac {3 \left (a^{3}+12 a^{2}+60 a +120\right ) x^{2} {\mathrm e}^{-b x -a}}{b^{2}}-\frac {\left (a^{3}+12 a^{2}+60 a +120\right ) x^{3} {\mathrm e}^{-b x -a}}{b}\) | \(178\) |
gosper | \(-\frac {\left (b^{6} x^{6}+3 b^{5} x^{5} a +3 a^{2} b^{4} x^{4}+6 b^{5} x^{5}+a^{3} b^{3} x^{3}+15 a \,b^{4} x^{4}+12 a^{2} b^{3} x^{3}+30 b^{4} x^{4}+3 a^{3} b^{2} x^{2}+60 a \,b^{3} x^{3}+36 a^{2} b^{2} x^{2}+120 b^{3} x^{3}+6 a^{3} b x +180 a \,b^{2} x^{2}+72 a^{2} b x +360 b^{2} x^{2}+6 a^{3}+360 a b x +72 a^{2}+720 b x +360 a +720\right ) {\mathrm e}^{-b x -a}}{b^{4}}\) | \(182\) |
risch | \(-\frac {\left (b^{6} x^{6}+3 b^{5} x^{5} a +3 a^{2} b^{4} x^{4}+6 b^{5} x^{5}+a^{3} b^{3} x^{3}+15 a \,b^{4} x^{4}+12 a^{2} b^{3} x^{3}+30 b^{4} x^{4}+3 a^{3} b^{2} x^{2}+60 a \,b^{3} x^{3}+36 a^{2} b^{2} x^{2}+120 b^{3} x^{3}+6 a^{3} b x +180 a \,b^{2} x^{2}+72 a^{2} b x +360 b^{2} x^{2}+6 a^{3}+360 a b x +72 a^{2}+720 b x +360 a +720\right ) {\mathrm e}^{-b x -a}}{b^{4}}\) | \(182\) |
meijerg | \(\frac {{\mathrm e}^{-a} \left (720-\frac {\left (7 b^{6} x^{6}+42 b^{5} x^{5}+210 b^{4} x^{4}+840 b^{3} x^{3}+2520 b^{2} x^{2}+5040 b x +5040\right ) {\mathrm e}^{-b x}}{7}\right )}{b^{4}}+\frac {3 \,{\mathrm e}^{-a} a \left (120-\frac {\left (6 b^{5} x^{5}+30 b^{4} x^{4}+120 b^{3} x^{3}+360 b^{2} x^{2}+720 b x +720\right ) {\mathrm e}^{-b x}}{6}\right )}{b^{4}}+\frac {3 \,{\mathrm e}^{-a} a^{2} \left (24-\frac {\left (5 b^{4} x^{4}+20 b^{3} x^{3}+60 b^{2} x^{2}+120 b x +120\right ) {\mathrm e}^{-b x}}{5}\right )}{b^{4}}+\frac {{\mathrm e}^{-a} a^{3} \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b^{4}}\) | \(215\) |
parallelrisch | \(-\frac {b^{6} {\mathrm e}^{-b x -a} x^{6}+3 b^{5} {\mathrm e}^{-b x -a} x^{5} a +6 b^{5} {\mathrm e}^{-b x -a} x^{5}+3 x^{4} {\mathrm e}^{-b x -a} a^{2} b^{4}+15 x^{4} {\mathrm e}^{-b x -a} a \,b^{4}+x^{3} {\mathrm e}^{-b x -a} a^{3} b^{3}+30 \,{\mathrm e}^{-b x -a} b^{4} x^{4}+12 x^{3} {\mathrm e}^{-b x -a} a^{2} b^{3}+60 \,{\mathrm e}^{-b x -a} a \,b^{3} x^{3}+3 x^{2} {\mathrm e}^{-b x -a} a^{3} b^{2}+120 \,{\mathrm e}^{-b x -a} x^{3} b^{3}+36 \,{\mathrm e}^{-b x -a} a^{2} b^{2} x^{2}+180 x^{2} {\mathrm e}^{-b x -a} a \,b^{2}+6 \,{\mathrm e}^{-b x -a} a^{3} b x +360 b^{2} {\mathrm e}^{-b x -a} x^{2}+72 x \,{\mathrm e}^{-b x -a} a^{2} b +360 a b \,{\mathrm e}^{-b x -a} x +6 \,{\mathrm e}^{-b x -a} a^{3}+720 b \,{\mathrm e}^{-b x -a} x +72 a^{2} {\mathrm e}^{-b x -a}+360 a \,{\mathrm e}^{-b x -a}+720 \,{\mathrm e}^{-b x -a}}{b^{4}}\) | \(372\) |
derivativedivides | \(-\frac {{\mathrm e}^{-b x -a} \left (-b x -a \right )^{6}-6 \left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}+30 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+360 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-720 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+720 \,{\mathrm e}^{-b x -a}+a^{3} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )+3 a^{2} \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )+3 a \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b^{4}}\) | \(432\) |
default | \(-\frac {{\mathrm e}^{-b x -a} \left (-b x -a \right )^{6}-6 \left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}+30 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+360 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-720 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+720 \,{\mathrm e}^{-b x -a}+a^{3} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )+3 a^{2} \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )+3 a \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b^{4}}\) | \(432\) |
parts | \(-b^{2} {\mathrm e}^{-b x -a} x^{6}-3 a b \,{\mathrm e}^{-b x -a} x^{5}-3 a^{2} {\mathrm e}^{-b x -a} x^{4}-\frac {a^{3} {\mathrm e}^{-b x -a} x^{3}}{b}-\frac {3 \left (-\frac {2 \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {a^{3} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {4 a^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {5 a \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}\right )}{b^{2}}\) | \(435\) |
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Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.30 \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=-\frac {{\left (b^{6} x^{6} + 3 \, {\left (a + 2\right )} b^{5} x^{5} + 3 \, {\left (a^{2} + 5 \, a + 10\right )} b^{4} x^{4} + {\left (a^{3} + 12 \, a^{2} + 60 \, a + 120\right )} b^{3} x^{3} + 3 \, {\left (a^{3} + 12 \, a^{2} + 60 \, a + 120\right )} b^{2} x^{2} + 6 \, a^{3} + 6 \, {\left (a^{3} + 12 \, a^{2} + 60 \, a + 120\right )} b x + 72 \, a^{2} + 360 \, a + 720\right )} e^{\left (-b x - a\right )}}{b^{4}} \]
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Time = 0.10 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.59 \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=\begin {cases} \frac {\left (- a^{3} b^{3} x^{3} - 3 a^{3} b^{2} x^{2} - 6 a^{3} b x - 6 a^{3} - 3 a^{2} b^{4} x^{4} - 12 a^{2} b^{3} x^{3} - 36 a^{2} b^{2} x^{2} - 72 a^{2} b x - 72 a^{2} - 3 a b^{5} x^{5} - 15 a b^{4} x^{4} - 60 a b^{3} x^{3} - 180 a b^{2} x^{2} - 360 a b x - 360 a - b^{6} x^{6} - 6 b^{5} x^{5} - 30 b^{4} x^{4} - 120 b^{3} x^{3} - 360 b^{2} x^{2} - 720 b x - 720\right ) e^{- a - b x}}{b^{4}} & \text {for}\: b^{4} \neq 0 \\\frac {a^{3} x^{4}}{4} + \frac {3 a^{2} b x^{5}}{5} + \frac {a b^{2} x^{6}}{2} + \frac {b^{3} x^{7}}{7} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.49 \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=-\frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{3} e^{\left (-b x - a\right )}}{b^{4}} - \frac {3 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a^{2} e^{\left (-b x - a\right )}}{b^{4}} - \frac {3 \, {\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} a e^{\left (-b x - a\right )}}{b^{4}} - \frac {{\left (b^{6} x^{6} + 6 \, b^{5} x^{5} + 30 \, b^{4} x^{4} + 120 \, b^{3} x^{3} + 360 \, b^{2} x^{2} + 720 \, b x + 720\right )} e^{\left (-b x - a\right )}}{b^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.51 \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=-\frac {{\left (b^{9} x^{6} + 3 \, a b^{8} x^{5} + 3 \, a^{2} b^{7} x^{4} + 6 \, b^{8} x^{5} + a^{3} b^{6} x^{3} + 15 \, a b^{7} x^{4} + 12 \, a^{2} b^{6} x^{3} + 30 \, b^{7} x^{4} + 3 \, a^{3} b^{5} x^{2} + 60 \, a b^{6} x^{3} + 36 \, a^{2} b^{5} x^{2} + 120 \, b^{6} x^{3} + 6 \, a^{3} b^{4} x + 180 \, a b^{5} x^{2} + 72 \, a^{2} b^{4} x + 360 \, b^{5} x^{2} + 6 \, a^{3} b^{3} + 360 \, a b^{4} x + 72 \, a^{2} b^{3} + 720 \, b^{4} x + 360 \, a b^{3} + 720 \, b^{3}\right )} e^{\left (-b x - a\right )}}{b^{7}} \]
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Time = 0.20 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.44 \[ \int e^{-a-b x} x^3 (a+b x)^3 \, dx=-x^4\,{\mathrm {e}}^{-a-b\,x}\,\left (3\,a^2+15\,a+30\right )-b^2\,x^6\,{\mathrm {e}}^{-a-b\,x}-\frac {6\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+12\,a^2+60\,a+120\right )}{b^4}-3\,b\,x^5\,{\mathrm {e}}^{-a-b\,x}\,\left (a+2\right )-\frac {6\,x\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+12\,a^2+60\,a+120\right )}{b^3}-\frac {x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+12\,a^2+60\,a+120\right )}{b}-\frac {3\,x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+12\,a^2+60\,a+120\right )}{b^2} \]
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