Integrand size = 20, antiderivative size = 159 \[ \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx=\frac {(A b-a B) (b d-a e)^3 (a+b x)^4}{4 b^5}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^5}{5 b^5}+\frac {e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^6}{2 b^5}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)^7}{7 b^5}+\frac {B e^3 (a+b x)^8}{8 b^5} \]
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Time = 0.17 (sec) , antiderivative size = 159, 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 = {78} \[ \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx=\frac {e^2 (a+b x)^7 (-4 a B e+A b e+3 b B d)}{7 b^5}+\frac {e (a+b x)^6 (b d-a e) (-2 a B e+A b e+b B d)}{2 b^5}+\frac {(a+b x)^5 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{5 b^5}+\frac {(a+b x)^4 (A b-a B) (b d-a e)^3}{4 b^5}+\frac {B e^3 (a+b x)^8}{8 b^5} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(A b-a B) (b d-a e)^3 (a+b x)^3}{b^4}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^4}{b^4}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^5}{b^4}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)^6}{b^4}+\frac {B e^3 (a+b x)^7}{b^4}\right ) \, dx \\ & = \frac {(A b-a B) (b d-a e)^3 (a+b x)^4}{4 b^5}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^5}{5 b^5}+\frac {e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^6}{2 b^5}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)^7}{7 b^5}+\frac {B e^3 (a+b x)^8}{8 b^5} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.87 \[ \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx=a^3 A d^3 x+\frac {1}{2} a^2 d^2 (a B d+3 A (b d+a e)) x^2+a d \left (a B d (b d+a e)+A \left (b^2 d^2+3 a b d e+a^2 e^2\right )\right ) x^3+\frac {1}{4} \left (3 a B d \left (b^2 d^2+3 a b d e+a^2 e^2\right )+A \left (b^3 d^3+9 a b^2 d^2 e+9 a^2 b d e^2+a^3 e^3\right )\right ) x^4+\frac {1}{5} \left (a^3 B e^3+9 a b^2 d e (B d+A e)+3 a^2 b e^2 (3 B d+A e)+b^3 d^2 (B d+3 A e)\right ) x^5+\frac {1}{2} b e \left (a^2 B e^2+b^2 d (B d+A e)+a b e (3 B d+A e)\right ) x^6+\frac {1}{7} b^2 e^2 (3 b B d+A b e+3 a B e) x^7+\frac {1}{8} b^3 B e^3 x^8 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(338\) vs. \(2(149)=298\).
Time = 0.71 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.13
method | result | size |
default | \(\frac {b^{3} B \,e^{3} x^{8}}{8}+\frac {\left (\left (b^{3} A +3 a \,b^{2} B \right ) e^{3}+3 b^{3} B d \,e^{2}\right ) x^{7}}{7}+\frac {\left (\left (3 a \,b^{2} A +3 a^{2} b B \right ) e^{3}+3 \left (b^{3} A +3 a \,b^{2} B \right ) d \,e^{2}+3 b^{3} B \,d^{2} e \right ) x^{6}}{6}+\frac {\left (\left (3 a^{2} b A +a^{3} B \right ) e^{3}+3 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d \,e^{2}+3 \left (b^{3} A +3 a \,b^{2} B \right ) d^{2} e +b^{3} B \,d^{3}\right ) x^{5}}{5}+\frac {\left (a^{3} A \,e^{3}+3 \left (3 a^{2} b A +a^{3} B \right ) d \,e^{2}+3 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{2} e +\left (b^{3} A +3 a \,b^{2} B \right ) d^{3}\right ) x^{4}}{4}+\frac {\left (3 a^{3} A d \,e^{2}+3 \left (3 a^{2} b A +a^{3} B \right ) d^{2} e +\left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{3} A \,d^{2} e +\left (3 a^{2} b A +a^{3} B \right ) d^{3}\right ) x^{2}}{2}+a^{3} A \,d^{3} x\) | \(339\) |
norman | \(\frac {b^{3} B \,e^{3} x^{8}}{8}+\left (\frac {1}{7} A \,b^{3} e^{3}+\frac {3}{7} B a \,b^{2} e^{3}+\frac {3}{7} b^{3} B d \,e^{2}\right ) x^{7}+\left (\frac {1}{2} A a \,b^{2} e^{3}+\frac {1}{2} A \,b^{3} d \,e^{2}+\frac {1}{2} B \,a^{2} b \,e^{3}+\frac {3}{2} B a \,b^{2} d \,e^{2}+\frac {1}{2} b^{3} B \,d^{2} e \right ) x^{6}+\left (\frac {3}{5} A \,a^{2} b \,e^{3}+\frac {9}{5} A a \,b^{2} d \,e^{2}+\frac {3}{5} A \,b^{3} d^{2} e +\frac {1}{5} B \,a^{3} e^{3}+\frac {9}{5} B \,a^{2} b d \,e^{2}+\frac {9}{5} B a \,b^{2} d^{2} e +\frac {1}{5} b^{3} B \,d^{3}\right ) x^{5}+\left (\frac {1}{4} a^{3} A \,e^{3}+\frac {9}{4} A \,a^{2} b d \,e^{2}+\frac {9}{4} A a \,b^{2} d^{2} e +\frac {1}{4} A \,b^{3} d^{3}+\frac {3}{4} B \,a^{3} d \,e^{2}+\frac {9}{4} B \,a^{2} b \,d^{2} e +\frac {3}{4} B a \,b^{2} d^{3}\right ) x^{4}+\left (a^{3} A d \,e^{2}+3 A \,a^{2} b \,d^{2} e +A a \,b^{2} d^{3}+B \,a^{3} d^{2} e +B \,a^{2} b \,d^{3}\right ) x^{3}+\left (\frac {3}{2} a^{3} A \,d^{2} e +\frac {3}{2} A \,a^{2} b \,d^{3}+\frac {1}{2} B \,a^{3} d^{3}\right ) x^{2}+a^{3} A \,d^{3} x\) | \(351\) |
gosper | \(\frac {3}{2} x^{6} B a \,b^{2} d \,e^{2}+\frac {9}{5} x^{5} A a \,b^{2} d \,e^{2}+\frac {9}{5} x^{5} B \,a^{2} b d \,e^{2}+\frac {9}{5} x^{5} B a \,b^{2} d^{2} e +\frac {9}{4} x^{4} A \,a^{2} b d \,e^{2}+\frac {9}{4} x^{4} A a \,b^{2} d^{2} e +\frac {1}{2} x^{2} B \,a^{3} d^{3}+\frac {1}{8} b^{3} B \,e^{3} x^{8}+a^{3} A \,d^{3} x +\frac {1}{7} x^{7} A \,b^{3} e^{3}+\frac {1}{5} x^{5} B \,a^{3} e^{3}+\frac {1}{5} x^{5} b^{3} B \,d^{3}+\frac {1}{4} x^{4} a^{3} A \,e^{3}+\frac {1}{4} x^{4} A \,b^{3} d^{3}+3 A \,a^{2} b \,d^{2} e \,x^{3}+\frac {9}{4} x^{4} B \,a^{2} b \,d^{2} e +\frac {3}{7} x^{7} B a \,b^{2} e^{3}+\frac {3}{7} x^{7} b^{3} B d \,e^{2}+\frac {1}{2} x^{6} A a \,b^{2} e^{3}+\frac {1}{2} x^{6} A \,b^{3} d \,e^{2}+A a \,b^{2} d^{3} x^{3}+B \,a^{3} d^{2} e \,x^{3}+B \,a^{2} b \,d^{3} x^{3}+\frac {1}{2} x^{6} B \,a^{2} b \,e^{3}+\frac {1}{2} x^{6} b^{3} B \,d^{2} e +\frac {3}{5} x^{5} A \,a^{2} b \,e^{3}+\frac {3}{5} x^{5} A \,b^{3} d^{2} e +\frac {3}{4} x^{4} B \,a^{3} d \,e^{2}+\frac {3}{4} x^{4} B a \,b^{2} d^{3}+\frac {3}{2} x^{2} a^{3} A \,d^{2} e +\frac {3}{2} x^{2} A \,a^{2} b \,d^{3}+A \,a^{3} d \,e^{2} x^{3}\) | \(411\) |
risch | \(\frac {3}{2} x^{6} B a \,b^{2} d \,e^{2}+\frac {9}{5} x^{5} A a \,b^{2} d \,e^{2}+\frac {9}{5} x^{5} B \,a^{2} b d \,e^{2}+\frac {9}{5} x^{5} B a \,b^{2} d^{2} e +\frac {9}{4} x^{4} A \,a^{2} b d \,e^{2}+\frac {9}{4} x^{4} A a \,b^{2} d^{2} e +\frac {1}{2} x^{2} B \,a^{3} d^{3}+\frac {1}{8} b^{3} B \,e^{3} x^{8}+a^{3} A \,d^{3} x +\frac {1}{7} x^{7} A \,b^{3} e^{3}+\frac {1}{5} x^{5} B \,a^{3} e^{3}+\frac {1}{5} x^{5} b^{3} B \,d^{3}+\frac {1}{4} x^{4} a^{3} A \,e^{3}+\frac {1}{4} x^{4} A \,b^{3} d^{3}+3 A \,a^{2} b \,d^{2} e \,x^{3}+\frac {9}{4} x^{4} B \,a^{2} b \,d^{2} e +\frac {3}{7} x^{7} B a \,b^{2} e^{3}+\frac {3}{7} x^{7} b^{3} B d \,e^{2}+\frac {1}{2} x^{6} A a \,b^{2} e^{3}+\frac {1}{2} x^{6} A \,b^{3} d \,e^{2}+A a \,b^{2} d^{3} x^{3}+B \,a^{3} d^{2} e \,x^{3}+B \,a^{2} b \,d^{3} x^{3}+\frac {1}{2} x^{6} B \,a^{2} b \,e^{3}+\frac {1}{2} x^{6} b^{3} B \,d^{2} e +\frac {3}{5} x^{5} A \,a^{2} b \,e^{3}+\frac {3}{5} x^{5} A \,b^{3} d^{2} e +\frac {3}{4} x^{4} B \,a^{3} d \,e^{2}+\frac {3}{4} x^{4} B a \,b^{2} d^{3}+\frac {3}{2} x^{2} a^{3} A \,d^{2} e +\frac {3}{2} x^{2} A \,a^{2} b \,d^{3}+A \,a^{3} d \,e^{2} x^{3}\) | \(411\) |
parallelrisch | \(\frac {3}{2} x^{6} B a \,b^{2} d \,e^{2}+\frac {9}{5} x^{5} A a \,b^{2} d \,e^{2}+\frac {9}{5} x^{5} B \,a^{2} b d \,e^{2}+\frac {9}{5} x^{5} B a \,b^{2} d^{2} e +\frac {9}{4} x^{4} A \,a^{2} b d \,e^{2}+\frac {9}{4} x^{4} A a \,b^{2} d^{2} e +\frac {1}{2} x^{2} B \,a^{3} d^{3}+\frac {1}{8} b^{3} B \,e^{3} x^{8}+a^{3} A \,d^{3} x +\frac {1}{7} x^{7} A \,b^{3} e^{3}+\frac {1}{5} x^{5} B \,a^{3} e^{3}+\frac {1}{5} x^{5} b^{3} B \,d^{3}+\frac {1}{4} x^{4} a^{3} A \,e^{3}+\frac {1}{4} x^{4} A \,b^{3} d^{3}+3 A \,a^{2} b \,d^{2} e \,x^{3}+\frac {9}{4} x^{4} B \,a^{2} b \,d^{2} e +\frac {3}{7} x^{7} B a \,b^{2} e^{3}+\frac {3}{7} x^{7} b^{3} B d \,e^{2}+\frac {1}{2} x^{6} A a \,b^{2} e^{3}+\frac {1}{2} x^{6} A \,b^{3} d \,e^{2}+A a \,b^{2} d^{3} x^{3}+B \,a^{3} d^{2} e \,x^{3}+B \,a^{2} b \,d^{3} x^{3}+\frac {1}{2} x^{6} B \,a^{2} b \,e^{3}+\frac {1}{2} x^{6} b^{3} B \,d^{2} e +\frac {3}{5} x^{5} A \,a^{2} b \,e^{3}+\frac {3}{5} x^{5} A \,b^{3} d^{2} e +\frac {3}{4} x^{4} B \,a^{3} d \,e^{2}+\frac {3}{4} x^{4} B a \,b^{2} d^{3}+\frac {3}{2} x^{2} a^{3} A \,d^{2} e +\frac {3}{2} x^{2} A \,a^{2} b \,d^{3}+A \,a^{3} d \,e^{2} x^{3}\) | \(411\) |
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Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (149) = 298\).
Time = 0.21 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.04 \[ \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx=\frac {1}{8} \, B b^{3} e^{3} x^{8} + A a^{3} d^{3} x + \frac {1}{7} \, {\left (3 \, B b^{3} d e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (B b^{3} d^{2} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (A a^{3} e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{2}\right )} x^{4} + {\left (A a^{3} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} d^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{3} d^{2} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3}\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (155) = 310\).
Time = 0.04 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.65 \[ \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx=A a^{3} d^{3} x + \frac {B b^{3} e^{3} x^{8}}{8} + x^{7} \left (\frac {A b^{3} e^{3}}{7} + \frac {3 B a b^{2} e^{3}}{7} + \frac {3 B b^{3} d e^{2}}{7}\right ) + x^{6} \left (\frac {A a b^{2} e^{3}}{2} + \frac {A b^{3} d e^{2}}{2} + \frac {B a^{2} b e^{3}}{2} + \frac {3 B a b^{2} d e^{2}}{2} + \frac {B b^{3} d^{2} e}{2}\right ) + x^{5} \cdot \left (\frac {3 A a^{2} b e^{3}}{5} + \frac {9 A a b^{2} d e^{2}}{5} + \frac {3 A b^{3} d^{2} e}{5} + \frac {B a^{3} e^{3}}{5} + \frac {9 B a^{2} b d e^{2}}{5} + \frac {9 B a b^{2} d^{2} e}{5} + \frac {B b^{3} d^{3}}{5}\right ) + x^{4} \left (\frac {A a^{3} e^{3}}{4} + \frac {9 A a^{2} b d e^{2}}{4} + \frac {9 A a b^{2} d^{2} e}{4} + \frac {A b^{3} d^{3}}{4} + \frac {3 B a^{3} d e^{2}}{4} + \frac {9 B a^{2} b d^{2} e}{4} + \frac {3 B a b^{2} d^{3}}{4}\right ) + x^{3} \left (A a^{3} d e^{2} + 3 A a^{2} b d^{2} e + A a b^{2} d^{3} + B a^{3} d^{2} e + B a^{2} b d^{3}\right ) + x^{2} \cdot \left (\frac {3 A a^{3} d^{2} e}{2} + \frac {3 A a^{2} b d^{3}}{2} + \frac {B a^{3} d^{3}}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (149) = 298\).
Time = 0.20 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.04 \[ \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx=\frac {1}{8} \, B b^{3} e^{3} x^{8} + A a^{3} d^{3} x + \frac {1}{7} \, {\left (3 \, B b^{3} d e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (B b^{3} d^{2} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (A a^{3} e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{2}\right )} x^{4} + {\left (A a^{3} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} d^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{3} d^{2} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3}\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (149) = 298\).
Time = 0.29 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.58 \[ \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx=\frac {1}{8} \, B b^{3} e^{3} x^{8} + \frac {3}{7} \, B b^{3} d e^{2} x^{7} + \frac {3}{7} \, B a b^{2} e^{3} x^{7} + \frac {1}{7} \, A b^{3} e^{3} x^{7} + \frac {1}{2} \, B b^{3} d^{2} e x^{6} + \frac {3}{2} \, B a b^{2} d e^{2} x^{6} + \frac {1}{2} \, A b^{3} d e^{2} x^{6} + \frac {1}{2} \, B a^{2} b e^{3} x^{6} + \frac {1}{2} \, A a b^{2} e^{3} x^{6} + \frac {1}{5} \, B b^{3} d^{3} x^{5} + \frac {9}{5} \, B a b^{2} d^{2} e x^{5} + \frac {3}{5} \, A b^{3} d^{2} e x^{5} + \frac {9}{5} \, B a^{2} b d e^{2} x^{5} + \frac {9}{5} \, A a b^{2} d e^{2} x^{5} + \frac {1}{5} \, B a^{3} e^{3} x^{5} + \frac {3}{5} \, A a^{2} b e^{3} x^{5} + \frac {3}{4} \, B a b^{2} d^{3} x^{4} + \frac {1}{4} \, A b^{3} d^{3} x^{4} + \frac {9}{4} \, B a^{2} b d^{2} e x^{4} + \frac {9}{4} \, A a b^{2} d^{2} e x^{4} + \frac {3}{4} \, B a^{3} d e^{2} x^{4} + \frac {9}{4} \, A a^{2} b d e^{2} x^{4} + \frac {1}{4} \, A a^{3} e^{3} x^{4} + B a^{2} b d^{3} x^{3} + A a b^{2} d^{3} x^{3} + B a^{3} d^{2} e x^{3} + 3 \, A a^{2} b d^{2} e x^{3} + A a^{3} d e^{2} x^{3} + \frac {1}{2} \, B a^{3} d^{3} x^{2} + \frac {3}{2} \, A a^{2} b d^{3} x^{2} + \frac {3}{2} \, A a^{3} d^{2} e x^{2} + A a^{3} d^{3} x \]
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Time = 0.12 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.10 \[ \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx=x^3\,\left (B\,a^3\,d^2\,e+A\,a^3\,d\,e^2+B\,a^2\,b\,d^3+3\,A\,a^2\,b\,d^2\,e+A\,a\,b^2\,d^3\right )+x^6\,\left (\frac {B\,a^2\,b\,e^3}{2}+\frac {3\,B\,a\,b^2\,d\,e^2}{2}+\frac {A\,a\,b^2\,e^3}{2}+\frac {B\,b^3\,d^2\,e}{2}+\frac {A\,b^3\,d\,e^2}{2}\right )+x^4\,\left (\frac {3\,B\,a^3\,d\,e^2}{4}+\frac {A\,a^3\,e^3}{4}+\frac {9\,B\,a^2\,b\,d^2\,e}{4}+\frac {9\,A\,a^2\,b\,d\,e^2}{4}+\frac {3\,B\,a\,b^2\,d^3}{4}+\frac {9\,A\,a\,b^2\,d^2\,e}{4}+\frac {A\,b^3\,d^3}{4}\right )+x^5\,\left (\frac {B\,a^3\,e^3}{5}+\frac {9\,B\,a^2\,b\,d\,e^2}{5}+\frac {3\,A\,a^2\,b\,e^3}{5}+\frac {9\,B\,a\,b^2\,d^2\,e}{5}+\frac {9\,A\,a\,b^2\,d\,e^2}{5}+\frac {B\,b^3\,d^3}{5}+\frac {3\,A\,b^3\,d^2\,e}{5}\right )+\frac {a^2\,d^2\,x^2\,\left (3\,A\,a\,e+3\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^2\,e^2\,x^7\,\left (A\,b\,e+3\,B\,a\,e+3\,B\,b\,d\right )}{7}+A\,a^3\,d^3\,x+\frac {B\,b^3\,e^3\,x^8}{8} \]
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