Integrand size = 25, antiderivative size = 555 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=-\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}{5 e^8}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right ) (d+e x)^6}{6 e^8}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^7}{7 e^8}-\frac {\left (A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) (d+e x)^8}{8 e^8}-\frac {\left (B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^9}{9 e^8}-\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right ) (d+e x)^{10}}{10 e^8}-\frac {c^2 (7 B c d-3 b B e-A c e) (d+e x)^{11}}{11 e^8}+\frac {B c^3 (d+e x)^{12}}{12 e^8} \]
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Time = 0.81 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {785} \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=-\frac {(d+e x)^8 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{8 e^8}-\frac {3 c (d+e x)^{10} \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{10 e^8}-\frac {3 (d+e x)^7 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{7 e^8}-\frac {(d+e x)^9 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{9 e^8}+\frac {(d+e x)^6 \left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{6 e^8}-\frac {(d+e x)^5 (B d-A e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8}-\frac {c^2 (d+e x)^{11} (-A c e-3 b B e+7 B c d)}{11 e^8}+\frac {B c^3 (d+e x)^{12}}{12 e^8} \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}{e^7}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right ) (d+e x)^5}{e^7}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (-B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )+A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^6}{e^7}+\frac {\left (-A e (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 )+B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) (d+e x)^7}{e^7}+\frac {\left (-B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )+3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^8}{e^7}+\frac {3 c \left (-A c e (2 c d-b e)+B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right ) (d+e x)^9}{e^7}+\frac {c^2 (-7 B c d+3 b B e+A c e) (d+e x)^{10}}{e^7}+\frac {B c^3 (d+e x)^{11}}{e^7}\right ) \, dx \\ & = -\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}{5 e^8}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right ) (d+e x)^6}{6 e^8}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^7}{7 e^8}-\frac {\left (A e (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 )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) (d+e x)^8}{8 e^8}-\frac {\left (B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^9}{9 e^8}-\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right ) (d+e x)^{10}}{10 e^8}-\frac {c^2 (7 B c d-3 b B e-A c e) (d+e x)^{11}}{11 e^8}+\frac {B c^3 (d+e x)^{12}}{12 e^8} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 957, normalized size of antiderivative = 1.72 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=a^3 A d^4 x+\frac {1}{2} a^2 d^3 (3 A b d+a B d+4 a A e) x^2+\frac {1}{3} a d^2 \left (a B d (3 b d+4 a e)+3 A \left (b^2 d^2+4 a b d e+a \left (c d^2+2 a e^2\right )\right )\right ) x^3+\frac {1}{4} d \left (3 a B d \left (b^2 d^2+4 a b d e+a \left (c d^2+2 a e^2\right )\right )+A \left (b^3 d^3+12 a b^2 d^2 e+4 a^2 e \left (3 c d^2+a e^2\right )+6 a b d \left (c d^2+3 a e^2\right )\right )\right ) x^4+\frac {1}{5} \left (b^3 d^3 (B d+4 A e)+3 b^2 d^2 \left (A c d^2+4 a B d e+6 a A e^2\right )+6 a b d \left (B c d^3+4 A c d^2 e+3 a B d e^2+2 a A e^3\right )+a \left (4 a B d e \left (3 c d^2+a e^2\right )+A \left (3 c^2 d^4+18 a c d^2 e^2+a^2 e^4\right )\right )\right ) x^5+\frac {1}{6} \left (2 b^3 d^2 e (2 B d+3 A e)+12 a A c d e \left (c d^2+a e^2\right )+12 a b B d e \left (2 c d^2+a e^2\right )+3 b^2 d \left (B c d^3+4 A c d^2 e+6 a B d e^2+4 a A e^3\right )+3 A b \left (c^2 d^4+12 a c d^2 e^2+a^2 e^4\right )+a B \left (3 c^2 d^4+18 a c d^2 e^2+a^2 e^4\right )\right ) x^6+\frac {1}{7} \left (2 b^3 d e^2 (3 B d+2 A e)+12 a B c d e \left (c d^2+a e^2\right )+12 A b c d e \left (c d^2+2 a e^2\right )+3 b^2 e \left (4 B c d^3+6 A c d^2 e+4 a B d e^2+a A e^3\right )+3 b B \left (c^2 d^4+12 a c d^2 e^2+a^2 e^4\right )+A c \left (c^2 d^4+18 a c d^2 e^2+3 a^2 e^4\right )\right ) x^7+\frac {1}{8} \left (A e \left (4 c^3 d^3+b^3 e^3+6 b c e^2 (2 b d+a e)+6 c^2 d e (3 b d+2 a e)\right )+B \left (c^3 d^4+6 c^2 d^2 e (2 b d+3 a e)+b^2 e^3 (4 b d+3 a e)+3 c e^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right )\right ) x^8+\frac {1}{9} e \left (3 A c e \left (2 c^2 d^2+b^2 e^2+c e (4 b d+a e)\right )+B \left (4 c^3 d^3+b^3 e^3+6 b c e^2 (2 b d+a e)+6 c^2 d e (3 b d+2 a e)\right )\right ) x^9+\frac {1}{10} c e^2 \left (A c e (4 c d+3 b e)+3 B \left (2 c^2 d^2+b^2 e^2+c e (4 b d+a e)\right )\right ) x^{10}+\frac {1}{11} c^2 e^3 (4 B c d+3 b B e+A c e) x^{11}+\frac {1}{12} B c^3 e^4 x^{12} \]
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Time = 0.38 (sec) , antiderivative size = 1041, normalized size of antiderivative = 1.88
method | result | size |
default | \(\text {Expression too large to display}\) | \(1041\) |
norman | \(\text {Expression too large to display}\) | \(1110\) |
gosper | \(\text {Expression too large to display}\) | \(1354\) |
risch | \(\text {Expression too large to display}\) | \(1354\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1354\) |
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none
Time = 0.31 (sec) , antiderivative size = 919, normalized size of antiderivative = 1.66 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{12} \, B c^{3} e^{4} x^{12} + \frac {1}{11} \, {\left (4 \, B c^{3} d e^{3} + {\left (3 \, B b c^{2} + A c^{3}\right )} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (6 \, B c^{3} d^{2} e^{2} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{3} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{3} d^{3} e + 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{2} + 12 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{3} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{4}\right )} x^{9} + A a^{3} d^{4} x + \frac {1}{8} \, {\left (B c^{3} d^{4} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 18 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{2} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} + 12 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e + 6 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{2} + 4 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} e + 6 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A a^{3} e^{4} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4} + 4 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3} e + 18 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{2} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{4} + 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{3} e + 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{3} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{4} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{3} d^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4}\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1401 vs. \(2 (542) = 1084\).
Time = 0.09 (sec) , antiderivative size = 1401, normalized size of antiderivative = 2.52 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
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Time = 0.20 (sec) , antiderivative size = 919, normalized size of antiderivative = 1.66 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{12} \, B c^{3} e^{4} x^{12} + \frac {1}{11} \, {\left (4 \, B c^{3} d e^{3} + {\left (3 \, B b c^{2} + A c^{3}\right )} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (6 \, B c^{3} d^{2} e^{2} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{3} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{3} d^{3} e + 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{2} + 12 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{3} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{4}\right )} x^{9} + A a^{3} d^{4} x + \frac {1}{8} \, {\left (B c^{3} d^{4} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 18 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{2} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} + 12 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e + 6 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{2} + 4 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} e + 6 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A a^{3} e^{4} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4} + 4 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3} e + 18 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{2} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{4} + 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{3} e + 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{3} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{4} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{3} d^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4}\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1353 vs. \(2 (540) = 1080\).
Time = 0.31 (sec) , antiderivative size = 1353, normalized size of antiderivative = 2.44 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
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Time = 0.32 (sec) , antiderivative size = 1093, normalized size of antiderivative = 1.97 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=x^5\,\left (\frac {4\,B\,a^3\,d\,e^3}{5}+\frac {A\,a^3\,e^4}{5}+\frac {18\,B\,a^2\,b\,d^2\,e^2}{5}+\frac {12\,A\,a^2\,b\,d\,e^3}{5}+\frac {12\,B\,a^2\,c\,d^3\,e}{5}+\frac {18\,A\,a^2\,c\,d^2\,e^2}{5}+\frac {12\,B\,a\,b^2\,d^3\,e}{5}+\frac {18\,A\,a\,b^2\,d^2\,e^2}{5}+\frac {6\,B\,a\,b\,c\,d^4}{5}+\frac {24\,A\,a\,b\,c\,d^3\,e}{5}+\frac {3\,A\,a\,c^2\,d^4}{5}+\frac {B\,b^3\,d^4}{5}+\frac {4\,A\,b^3\,d^3\,e}{5}+\frac {3\,A\,b^2\,c\,d^4}{5}\right )+x^8\,\left (\frac {3\,B\,a^2\,c\,e^4}{8}+\frac {3\,B\,a\,b^2\,e^4}{8}+3\,B\,a\,b\,c\,d\,e^3+\frac {3\,A\,a\,b\,c\,e^4}{4}+\frac {9\,B\,a\,c^2\,d^2\,e^2}{4}+\frac {3\,A\,a\,c^2\,d\,e^3}{2}+\frac {B\,b^3\,d\,e^3}{2}+\frac {A\,b^3\,e^4}{8}+\frac {9\,B\,b^2\,c\,d^2\,e^2}{4}+\frac {3\,A\,b^2\,c\,d\,e^3}{2}+\frac {3\,B\,b\,c^2\,d^3\,e}{2}+\frac {9\,A\,b\,c^2\,d^2\,e^2}{4}+\frac {B\,c^3\,d^4}{8}+\frac {A\,c^3\,d^3\,e}{2}\right )+x^3\,\left (\frac {4\,B\,a^3\,d^3\,e}{3}+2\,A\,a^3\,d^2\,e^2+B\,a^2\,b\,d^4+4\,A\,a^2\,b\,d^3\,e+A\,c\,a^2\,d^4+A\,a\,b^2\,d^4\right )+x^{10}\,\left (\frac {3\,B\,b^2\,c\,e^4}{10}+\frac {6\,B\,b\,c^2\,d\,e^3}{5}+\frac {3\,A\,b\,c^2\,e^4}{10}+\frac {3\,B\,c^3\,d^2\,e^2}{5}+\frac {2\,A\,c^3\,d\,e^3}{5}+\frac {3\,B\,a\,c^2\,e^4}{10}\right )+x^6\,\left (\frac {B\,a^3\,e^4}{6}+2\,B\,a^2\,b\,d\,e^3+\frac {A\,a^2\,b\,e^4}{2}+3\,B\,a^2\,c\,d^2\,e^2+2\,A\,a^2\,c\,d\,e^3+3\,B\,a\,b^2\,d^2\,e^2+2\,A\,a\,b^2\,d\,e^3+4\,B\,a\,b\,c\,d^3\,e+6\,A\,a\,b\,c\,d^2\,e^2+\frac {B\,a\,c^2\,d^4}{2}+2\,A\,a\,c^2\,d^3\,e+\frac {2\,B\,b^3\,d^3\,e}{3}+A\,b^3\,d^2\,e^2+\frac {B\,b^2\,c\,d^4}{2}+2\,A\,b^2\,c\,d^3\,e+\frac {A\,b\,c^2\,d^4}{2}\right )+x^7\,\left (\frac {3\,B\,a^2\,b\,e^4}{7}+\frac {12\,B\,a^2\,c\,d\,e^3}{7}+\frac {3\,A\,a^2\,c\,e^4}{7}+\frac {12\,B\,a\,b^2\,d\,e^3}{7}+\frac {3\,A\,a\,b^2\,e^4}{7}+\frac {36\,B\,a\,b\,c\,d^2\,e^2}{7}+\frac {24\,A\,a\,b\,c\,d\,e^3}{7}+\frac {12\,B\,a\,c^2\,d^3\,e}{7}+\frac {18\,A\,a\,c^2\,d^2\,e^2}{7}+\frac {6\,B\,b^3\,d^2\,e^2}{7}+\frac {4\,A\,b^3\,d\,e^3}{7}+\frac {12\,B\,b^2\,c\,d^3\,e}{7}+\frac {18\,A\,b^2\,c\,d^2\,e^2}{7}+\frac {3\,B\,b\,c^2\,d^4}{7}+\frac {12\,A\,b\,c^2\,d^3\,e}{7}+\frac {A\,c^3\,d^4}{7}\right )+x^4\,\left (\frac {3\,B\,a^3\,d^2\,e^2}{2}+A\,a^3\,d\,e^3+3\,B\,a^2\,b\,d^3\,e+\frac {9\,A\,a^2\,b\,d^2\,e^2}{2}+\frac {3\,B\,c\,a^2\,d^4}{4}+3\,A\,c\,a^2\,d^3\,e+\frac {3\,B\,a\,b^2\,d^4}{4}+3\,A\,a\,b^2\,d^3\,e+\frac {3\,A\,c\,a\,b\,d^4}{2}+\frac {A\,b^3\,d^4}{4}\right )+x^9\,\left (\frac {B\,b^3\,e^4}{9}+\frac {4\,B\,b^2\,c\,d\,e^3}{3}+\frac {A\,b^2\,c\,e^4}{3}+2\,B\,b\,c^2\,d^2\,e^2+\frac {4\,A\,b\,c^2\,d\,e^3}{3}+\frac {2\,B\,a\,b\,c\,e^4}{3}+\frac {4\,B\,c^3\,d^3\,e}{9}+\frac {2\,A\,c^3\,d^2\,e^2}{3}+\frac {4\,B\,a\,c^2\,d\,e^3}{3}+\frac {A\,a\,c^2\,e^4}{3}\right )+\frac {a^2\,d^3\,x^2\,\left (4\,A\,a\,e+3\,A\,b\,d+B\,a\,d\right )}{2}+\frac {c^2\,e^3\,x^{11}\,\left (A\,c\,e+3\,B\,b\,e+4\,B\,c\,d\right )}{11}+A\,a^3\,d^4\,x+\frac {B\,c^3\,e^4\,x^{12}}{12} \]
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