Integrand size = 20, antiderivative size = 272 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}{4 e^7}-\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}{5 e^7}+\frac {\left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^6}{2 e^7}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^7}{7 e^7}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^8}{8 e^7}-\frac {c^2 (2 c d-b e) (d+e x)^9}{3 e^7}+\frac {c^3 (d+e x)^{10}}{10 e^7} \]
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Time = 0.23 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {3 c (d+e x)^8 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{8 e^7}-\frac {(d+e x)^7 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^7}+\frac {(d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7}-\frac {3 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7}+\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right )^3}{4 e^7}-\frac {c^2 (d+e x)^9 (2 c d-b e)}{3 e^7}+\frac {c^3 (d+e x)^{10}}{10 e^7} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}{e^6}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}{e^6}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^5}{e^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^6}{e^6}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^7}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^8}{e^6}+\frac {c^3 (d+e x)^9}{e^6}\right ) \, dx \\ & = \frac {\left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}{4 e^7}-\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}{5 e^7}+\frac {\left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^6}{2 e^7}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^7}{7 e^7}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^8}{8 e^7}-\frac {c^2 (2 c d-b e) (d+e x)^9}{3 e^7}+\frac {c^3 (d+e x)^{10}}{10 e^7} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.37 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=a^3 d^3 x+\frac {3}{2} a^2 d^2 (b d+a e) x^2+a d \left (b^2 d^2+3 a b d e+a \left (c d^2+a e^2\right )\right ) x^3+\frac {1}{4} \left (b^3 d^3+9 a b^2 d^2 e+a^2 e \left (9 c d^2+a e^2\right )+3 a b d \left (2 c d^2+3 a e^2\right )\right ) x^4+\frac {3}{5} \left (b^3 d^2 e+a b e \left (6 c d^2+a e^2\right )+a c d \left (c d^2+3 a e^2\right )+b^2 \left (c d^3+3 a d e^2\right )\right ) x^5+\frac {1}{2} \left (b^3 d e^2+a c e \left (3 c d^2+a e^2\right )+b c d \left (c d^2+6 a e^2\right )+b^2 \left (3 c d^2 e+a e^3\right )\right ) x^6+\frac {1}{7} \left (c^3 d^3+b^3 e^3+9 c^2 d e (b d+a e)+3 b c e^2 (3 b d+2 a e)\right ) x^7+\frac {3}{8} c e \left (c^2 d^2+b^2 e^2+c e (3 b d+a e)\right ) x^8+\frac {1}{3} c^2 e^2 (c d+b e) x^9+\frac {1}{10} c^3 e^3 x^{10} \]
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Time = 3.18 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.49
method | result | size |
norman | \(\frac {c^{3} e^{3} x^{10}}{10}+\left (\frac {1}{3} e^{3} b \,c^{2}+\frac {1}{3} d \,e^{2} c^{3}\right ) x^{9}+\left (\frac {3}{8} a \,c^{2} e^{3}+\frac {3}{8} b^{2} c \,e^{3}+\frac {9}{8} d \,e^{2} b \,c^{2}+\frac {3}{8} d^{2} e \,c^{3}\right ) x^{8}+\left (\frac {6}{7} a b c \,e^{3}+\frac {9}{7} c^{2} a d \,e^{2}+\frac {1}{7} b^{3} e^{3}+\frac {9}{7} b^{2} d \,e^{2} c +\frac {9}{7} b \,c^{2} d^{2} e +\frac {1}{7} c^{3} d^{3}\right ) x^{7}+\left (\frac {1}{2} a^{2} c \,e^{3}+\frac {1}{2} b^{2} e^{3} a +3 a b c d \,e^{2}+\frac {3}{2} a \,c^{2} d^{2} e +\frac {1}{2} b^{3} d \,e^{2}+\frac {3}{2} b^{2} c \,d^{2} e +\frac {1}{2} d^{3} b \,c^{2}\right ) x^{6}+\left (\frac {3}{5} a^{2} b \,e^{3}+\frac {9}{5} d \,e^{2} a^{2} c +\frac {9}{5} a \,b^{2} d \,e^{2}+\frac {18}{5} a b c \,d^{2} e +\frac {3}{5} d^{3} c^{2} a +\frac {3}{5} d^{2} e \,b^{3}+\frac {3}{5} b^{2} c \,d^{3}\right ) x^{5}+\left (\frac {1}{4} a^{3} e^{3}+\frac {9}{4} a^{2} b d \,e^{2}+\frac {9}{4} d^{2} e \,a^{2} c +\frac {9}{4} a \,b^{2} d^{2} e +\frac {3}{2} a b c \,d^{3}+\frac {1}{4} b^{3} d^{3}\right ) x^{4}+\left (d \,e^{2} a^{3}+3 a^{2} b \,d^{2} e +a^{2} c \,d^{3}+a \,b^{2} d^{3}\right ) x^{3}+\left (\frac {3}{2} a^{3} d^{2} e +\frac {3}{2} a^{2} b \,d^{3}\right ) x^{2}+a^{3} d^{3} x\) | \(406\) |
gosper | \(\frac {18}{5} x^{5} a b c \,d^{2} e +3 a^{2} b \,d^{2} e \,x^{3}+\frac {9}{4} x^{4} a^{2} b d \,e^{2}+\frac {1}{4} d^{3} x^{4} b^{3}+x^{3} a \,b^{2} d^{3}+\frac {3}{2} x^{2} a^{2} b \,d^{3}+3 x^{6} a b c d \,e^{2}+\frac {9}{7} x^{7} b \,c^{2} d^{2} e +\frac {9}{4} x^{4} a \,b^{2} d^{2} e +\frac {9}{8} x^{8} d \,e^{2} b \,c^{2}+\frac {6}{7} x^{7} a b c \,e^{3}+\frac {9}{7} x^{7} b^{2} d \,e^{2} c +\frac {3}{2} x^{6} a \,c^{2} d^{2} e +\frac {3}{2} x^{6} b^{2} c \,d^{2} e +\frac {9}{5} x^{5} d \,e^{2} a^{2} c +\frac {9}{5} x^{5} a \,b^{2} d \,e^{2}+a^{3} d^{3} x +\frac {9}{7} x^{7} c^{2} a d \,e^{2}+\frac {9}{4} x^{4} d^{2} e \,a^{2} c +\frac {3}{2} c a b \,x^{4} d^{3}+\frac {3}{2} d^{2} e \,a^{3} x^{2}+\frac {3}{8} x^{8} d^{2} e \,c^{3}+\frac {1}{2} x^{6} b^{2} e^{3} a +\frac {1}{7} x^{7} b^{3} e^{3}+\frac {1}{7} x^{7} c^{3} d^{3}+\frac {3}{5} b^{2} c \,d^{3} x^{5}+\frac {1}{10} c^{3} e^{3} x^{10}+a^{3} d \,e^{2} x^{3}+a^{2} c \,d^{3} x^{3}+\frac {1}{3} x^{9} e^{3} b \,c^{2}+\frac {1}{4} a^{3} e^{3} x^{4}+\frac {1}{2} x^{6} b^{3} d \,e^{2}+\frac {1}{2} x^{6} d^{3} b \,c^{2}+\frac {3}{5} x^{5} a^{2} b \,e^{3}+\frac {3}{5} x^{5} d^{3} c^{2} a +\frac {3}{5} x^{5} d^{2} e \,b^{3}+\frac {3}{8} x^{8} b^{2} c \,e^{3}+\frac {1}{2} a^{2} c \,e^{3} x^{6}+\frac {3}{8} a \,c^{2} e^{3} x^{8}+\frac {1}{3} c^{3} d \,e^{2} x^{9}\) | \(480\) |
risch | \(\frac {18}{5} x^{5} a b c \,d^{2} e +3 a^{2} b \,d^{2} e \,x^{3}+\frac {9}{4} x^{4} a^{2} b d \,e^{2}+\frac {1}{4} d^{3} x^{4} b^{3}+x^{3} a \,b^{2} d^{3}+\frac {3}{2} x^{2} a^{2} b \,d^{3}+3 x^{6} a b c d \,e^{2}+\frac {9}{7} x^{7} b \,c^{2} d^{2} e +\frac {9}{4} x^{4} a \,b^{2} d^{2} e +\frac {9}{8} x^{8} d \,e^{2} b \,c^{2}+\frac {6}{7} x^{7} a b c \,e^{3}+\frac {9}{7} x^{7} b^{2} d \,e^{2} c +\frac {3}{2} x^{6} a \,c^{2} d^{2} e +\frac {3}{2} x^{6} b^{2} c \,d^{2} e +\frac {9}{5} x^{5} d \,e^{2} a^{2} c +\frac {9}{5} x^{5} a \,b^{2} d \,e^{2}+a^{3} d^{3} x +\frac {9}{7} x^{7} c^{2} a d \,e^{2}+\frac {9}{4} x^{4} d^{2} e \,a^{2} c +\frac {3}{2} c a b \,x^{4} d^{3}+\frac {3}{2} d^{2} e \,a^{3} x^{2}+\frac {3}{8} x^{8} d^{2} e \,c^{3}+\frac {1}{2} x^{6} b^{2} e^{3} a +\frac {1}{7} x^{7} b^{3} e^{3}+\frac {1}{7} x^{7} c^{3} d^{3}+\frac {3}{5} b^{2} c \,d^{3} x^{5}+\frac {1}{10} c^{3} e^{3} x^{10}+a^{3} d \,e^{2} x^{3}+a^{2} c \,d^{3} x^{3}+\frac {1}{3} x^{9} e^{3} b \,c^{2}+\frac {1}{4} a^{3} e^{3} x^{4}+\frac {1}{2} x^{6} b^{3} d \,e^{2}+\frac {1}{2} x^{6} d^{3} b \,c^{2}+\frac {3}{5} x^{5} a^{2} b \,e^{3}+\frac {3}{5} x^{5} d^{3} c^{2} a +\frac {3}{5} x^{5} d^{2} e \,b^{3}+\frac {3}{8} x^{8} b^{2} c \,e^{3}+\frac {1}{2} a^{2} c \,e^{3} x^{6}+\frac {3}{8} a \,c^{2} e^{3} x^{8}+\frac {1}{3} c^{3} d \,e^{2} x^{9}\) | \(480\) |
parallelrisch | \(\frac {18}{5} x^{5} a b c \,d^{2} e +3 a^{2} b \,d^{2} e \,x^{3}+\frac {9}{4} x^{4} a^{2} b d \,e^{2}+\frac {1}{4} d^{3} x^{4} b^{3}+x^{3} a \,b^{2} d^{3}+\frac {3}{2} x^{2} a^{2} b \,d^{3}+3 x^{6} a b c d \,e^{2}+\frac {9}{7} x^{7} b \,c^{2} d^{2} e +\frac {9}{4} x^{4} a \,b^{2} d^{2} e +\frac {9}{8} x^{8} d \,e^{2} b \,c^{2}+\frac {6}{7} x^{7} a b c \,e^{3}+\frac {9}{7} x^{7} b^{2} d \,e^{2} c +\frac {3}{2} x^{6} a \,c^{2} d^{2} e +\frac {3}{2} x^{6} b^{2} c \,d^{2} e +\frac {9}{5} x^{5} d \,e^{2} a^{2} c +\frac {9}{5} x^{5} a \,b^{2} d \,e^{2}+a^{3} d^{3} x +\frac {9}{7} x^{7} c^{2} a d \,e^{2}+\frac {9}{4} x^{4} d^{2} e \,a^{2} c +\frac {3}{2} c a b \,x^{4} d^{3}+\frac {3}{2} d^{2} e \,a^{3} x^{2}+\frac {3}{8} x^{8} d^{2} e \,c^{3}+\frac {1}{2} x^{6} b^{2} e^{3} a +\frac {1}{7} x^{7} b^{3} e^{3}+\frac {1}{7} x^{7} c^{3} d^{3}+\frac {3}{5} b^{2} c \,d^{3} x^{5}+\frac {1}{10} c^{3} e^{3} x^{10}+a^{3} d \,e^{2} x^{3}+a^{2} c \,d^{3} x^{3}+\frac {1}{3} x^{9} e^{3} b \,c^{2}+\frac {1}{4} a^{3} e^{3} x^{4}+\frac {1}{2} x^{6} b^{3} d \,e^{2}+\frac {1}{2} x^{6} d^{3} b \,c^{2}+\frac {3}{5} x^{5} a^{2} b \,e^{3}+\frac {3}{5} x^{5} d^{3} c^{2} a +\frac {3}{5} x^{5} d^{2} e \,b^{3}+\frac {3}{8} x^{8} b^{2} c \,e^{3}+\frac {1}{2} a^{2} c \,e^{3} x^{6}+\frac {3}{8} a \,c^{2} e^{3} x^{8}+\frac {1}{3} c^{3} d \,e^{2} x^{9}\) | \(480\) |
default | \(\frac {c^{3} e^{3} x^{10}}{10}+\frac {\left (3 e^{3} b \,c^{2}+3 d \,e^{2} c^{3}\right ) x^{9}}{9}+\frac {\left (3 d^{2} e \,c^{3}+9 d \,e^{2} b \,c^{2}+e^{3} \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{8}}{8}+\frac {\left (c^{3} d^{3}+9 b \,c^{2} d^{2} e +3 d \,e^{2} \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+e^{3} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{7}}{7}+\frac {\left (3 d^{3} b \,c^{2}+3 d^{2} e \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+3 d \,e^{2} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+e^{3} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{6}}{6}+\frac {\left (d^{3} \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+3 d^{2} e \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+3 d \,e^{2} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+3 a^{2} b \,e^{3}\right ) x^{5}}{5}+\frac {\left (d^{3} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+3 d^{2} e \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+9 a^{2} b d \,e^{2}+a^{3} e^{3}\right ) x^{4}}{4}+\frac {\left (d^{3} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+9 a^{2} b \,d^{2} e +3 d \,e^{2} a^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{3} d^{2} e +3 a^{2} b \,d^{3}\right ) x^{2}}{2}+a^{3} d^{3} x\) | \(495\) |
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Time = 0.27 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.35 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{3} \, {\left (c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{9} + \frac {3}{8} \, {\left (c^{3} d^{2} e + 3 \, b c^{2} d e^{2} + {\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{3} + 9 \, b c^{2} d^{2} e + 9 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} + {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x^{7} + a^{3} d^{3} x + \frac {1}{2} \, {\left (b c^{2} d^{3} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e + {\left (b^{3} + 6 \, a b c\right )} d e^{2} + {\left (a b^{2} + a^{2} c\right )} e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (a^{2} b e^{3} + {\left (b^{2} c + a c^{2}\right )} d^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{2} e + 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (9 \, a^{2} b d e^{2} + a^{3} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{3} + 9 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e\right )} x^{4} + {\left (3 \, a^{2} b d^{2} e + a^{3} d e^{2} + {\left (a b^{2} + a^{2} c\right )} d^{3}\right )} x^{3} + \frac {3}{2} \, {\left (a^{2} b d^{3} + a^{3} d^{2} e\right )} x^{2} \]
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Time = 0.05 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.78 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=a^{3} d^{3} x + \frac {c^{3} e^{3} x^{10}}{10} + x^{9} \left (\frac {b c^{2} e^{3}}{3} + \frac {c^{3} d e^{2}}{3}\right ) + x^{8} \cdot \left (\frac {3 a c^{2} e^{3}}{8} + \frac {3 b^{2} c e^{3}}{8} + \frac {9 b c^{2} d e^{2}}{8} + \frac {3 c^{3} d^{2} e}{8}\right ) + x^{7} \cdot \left (\frac {6 a b c e^{3}}{7} + \frac {9 a c^{2} d e^{2}}{7} + \frac {b^{3} e^{3}}{7} + \frac {9 b^{2} c d e^{2}}{7} + \frac {9 b c^{2} d^{2} e}{7} + \frac {c^{3} d^{3}}{7}\right ) + x^{6} \left (\frac {a^{2} c e^{3}}{2} + \frac {a b^{2} e^{3}}{2} + 3 a b c d e^{2} + \frac {3 a c^{2} d^{2} e}{2} + \frac {b^{3} d e^{2}}{2} + \frac {3 b^{2} c d^{2} e}{2} + \frac {b c^{2} d^{3}}{2}\right ) + x^{5} \cdot \left (\frac {3 a^{2} b e^{3}}{5} + \frac {9 a^{2} c d e^{2}}{5} + \frac {9 a b^{2} d e^{2}}{5} + \frac {18 a b c d^{2} e}{5} + \frac {3 a c^{2} d^{3}}{5} + \frac {3 b^{3} d^{2} e}{5} + \frac {3 b^{2} c d^{3}}{5}\right ) + x^{4} \left (\frac {a^{3} e^{3}}{4} + \frac {9 a^{2} b d e^{2}}{4} + \frac {9 a^{2} c d^{2} e}{4} + \frac {9 a b^{2} d^{2} e}{4} + \frac {3 a b c d^{3}}{2} + \frac {b^{3} d^{3}}{4}\right ) + x^{3} \left (a^{3} d e^{2} + 3 a^{2} b d^{2} e + a^{2} c d^{3} + a b^{2} d^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{3} d^{2} e}{2} + \frac {3 a^{2} b d^{3}}{2}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.35 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{3} \, {\left (c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{9} + \frac {3}{8} \, {\left (c^{3} d^{2} e + 3 \, b c^{2} d e^{2} + {\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{3} + 9 \, b c^{2} d^{2} e + 9 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} + {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x^{7} + a^{3} d^{3} x + \frac {1}{2} \, {\left (b c^{2} d^{3} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e + {\left (b^{3} + 6 \, a b c\right )} d e^{2} + {\left (a b^{2} + a^{2} c\right )} e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (a^{2} b e^{3} + {\left (b^{2} c + a c^{2}\right )} d^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{2} e + 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (9 \, a^{2} b d e^{2} + a^{3} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{3} + 9 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e\right )} x^{4} + {\left (3 \, a^{2} b d^{2} e + a^{3} d e^{2} + {\left (a b^{2} + a^{2} c\right )} d^{3}\right )} x^{3} + \frac {3}{2} \, {\left (a^{2} b d^{3} + a^{3} d^{2} e\right )} x^{2} \]
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Time = 0.26 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.76 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{3} \, c^{3} d e^{2} x^{9} + \frac {1}{3} \, b c^{2} e^{3} x^{9} + \frac {3}{8} \, c^{3} d^{2} e x^{8} + \frac {9}{8} \, b c^{2} d e^{2} x^{8} + \frac {3}{8} \, b^{2} c e^{3} x^{8} + \frac {3}{8} \, a c^{2} e^{3} x^{8} + \frac {1}{7} \, c^{3} d^{3} x^{7} + \frac {9}{7} \, b c^{2} d^{2} e x^{7} + \frac {9}{7} \, b^{2} c d e^{2} x^{7} + \frac {9}{7} \, a c^{2} d e^{2} x^{7} + \frac {1}{7} \, b^{3} e^{3} x^{7} + \frac {6}{7} \, a b c e^{3} x^{7} + \frac {1}{2} \, b c^{2} d^{3} x^{6} + \frac {3}{2} \, b^{2} c d^{2} e x^{6} + \frac {3}{2} \, a c^{2} d^{2} e x^{6} + \frac {1}{2} \, b^{3} d e^{2} x^{6} + 3 \, a b c d e^{2} x^{6} + \frac {1}{2} \, a b^{2} e^{3} x^{6} + \frac {1}{2} \, a^{2} c e^{3} x^{6} + \frac {3}{5} \, b^{2} c d^{3} x^{5} + \frac {3}{5} \, a c^{2} d^{3} x^{5} + \frac {3}{5} \, b^{3} d^{2} e x^{5} + \frac {18}{5} \, a b c d^{2} e x^{5} + \frac {9}{5} \, a b^{2} d e^{2} x^{5} + \frac {9}{5} \, a^{2} c d e^{2} x^{5} + \frac {3}{5} \, a^{2} b e^{3} x^{5} + \frac {1}{4} \, b^{3} d^{3} x^{4} + \frac {3}{2} \, a b c d^{3} x^{4} + \frac {9}{4} \, a b^{2} d^{2} e x^{4} + \frac {9}{4} \, a^{2} c d^{2} e x^{4} + \frac {9}{4} \, a^{2} b d e^{2} x^{4} + \frac {1}{4} \, a^{3} e^{3} x^{4} + a b^{2} d^{3} x^{3} + a^{2} c d^{3} x^{3} + 3 \, a^{2} b d^{2} e x^{3} + a^{3} d e^{2} x^{3} + \frac {3}{2} \, a^{2} b d^{3} x^{2} + \frac {3}{2} \, a^{3} d^{2} e x^{2} + a^{3} d^{3} x \]
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Time = 0.14 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.40 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=x^4\,\left (\frac {a^3\,e^3}{4}+\frac {9\,a^2\,b\,d\,e^2}{4}+\frac {9\,c\,a^2\,d^2\,e}{4}+\frac {9\,a\,b^2\,d^2\,e}{4}+\frac {3\,c\,a\,b\,d^3}{2}+\frac {b^3\,d^3}{4}\right )+x^7\,\left (\frac {b^3\,e^3}{7}+\frac {9\,b^2\,c\,d\,e^2}{7}+\frac {9\,b\,c^2\,d^2\,e}{7}+\frac {6\,a\,b\,c\,e^3}{7}+\frac {c^3\,d^3}{7}+\frac {9\,a\,c^2\,d\,e^2}{7}\right )+x^5\,\left (\frac {3\,a^2\,b\,e^3}{5}+\frac {9\,a^2\,c\,d\,e^2}{5}+\frac {9\,a\,b^2\,d\,e^2}{5}+\frac {18\,a\,b\,c\,d^2\,e}{5}+\frac {3\,a\,c^2\,d^3}{5}+\frac {3\,b^3\,d^2\,e}{5}+\frac {3\,b^2\,c\,d^3}{5}\right )+x^6\,\left (\frac {a^2\,c\,e^3}{2}+\frac {a\,b^2\,e^3}{2}+3\,a\,b\,c\,d\,e^2+\frac {3\,a\,c^2\,d^2\,e}{2}+\frac {b^3\,d\,e^2}{2}+\frac {3\,b^2\,c\,d^2\,e}{2}+\frac {b\,c^2\,d^3}{2}\right )+a^3\,d^3\,x+\frac {c^3\,e^3\,x^{10}}{10}+\frac {3\,a^2\,d^2\,x^2\,\left (a\,e+b\,d\right )}{2}+\frac {c^2\,e^2\,x^9\,\left (b\,e+c\,d\right )}{3}+a\,d\,x^3\,\left (a^2\,e^2+3\,a\,b\,d\,e+c\,a\,d^2+b^2\,d^2\right )+\frac {3\,c\,e\,x^8\,\left (b^2\,e^2+3\,b\,c\,d\,e+c^2\,d^2+a\,c\,e^2\right )}{8} \]
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