Integrand size = 25, antiderivative size = 548 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{7 e^8 (d+e x)^7}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{6 e^8 (d+e x)^6}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{5 e^8 (d+e x)^5}+\frac {A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{4 e^8 (d+e x)^4}+\frac {B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{3 e^8 (d+e x)^3}+\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right )}{2 e^8 (d+e x)^2}+\frac {c^2 (7 B c d-3 b B e-A c e)}{e^8 (d+e x)}+\frac {B c^3 \log (d+e x)}{e^8} \]
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Time = 0.48 (sec) , antiderivative size = 546, 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} \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{4 e^8 (d+e x)^4}+\frac {3 c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{2 e^8 (d+e x)^2}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{5 e^8 (d+e x)^5}+\frac {B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^8 (d+e x)^3}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{6 e^8 (d+e x)^6}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{7 e^8 (d+e x)^7}+\frac {c^2 (-A c e-3 b B e+7 B c d)}{e^8 (d+e x)}+\frac {B c^3 \log (d+e x)}{e^8} \]
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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}{e^7 (d+e x)^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 )}{e^7 (d+e x)^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 )}{e^7 (d+e x)^6}+\frac {-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 )}{e^7 (d+e x)^5}+\frac {-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 )}{e^7 (d+e x)^4}+\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 )}{e^7 (d+e x)^3}+\frac {c^2 (-7 B c d+3 b B e+A c e)}{e^7 (d+e x)^2}+\frac {B c^3}{e^7 (d+e x)}\right ) \, dx \\ & = \frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{7 e^8 (d+e x)^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 )}{6 e^8 (d+e x)^6}+\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 )}{5 e^8 (d+e x)^5}+\frac {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 )}{4 e^8 (d+e x)^4}+\frac {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 )}{3 e^8 (d+e x)^3}+\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 )}{2 e^8 (d+e x)^2}+\frac {c^2 (7 B c d-3 b B e-A c e)}{e^8 (d+e x)}+\frac {B c^3 \log (d+e x)}{e^8} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 863, normalized size of antiderivative = 1.57 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {-3 A e \left (20 c^3 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )+e^3 \left (20 a^3 e^3+10 a^2 b e^2 (d+7 e x)+4 a b^2 e \left (d^2+7 d e x+21 e^2 x^2\right )+b^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 c e^2 \left (2 a^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 a b e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+2 b^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+2 c^2 e \left (2 a e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 b \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )\right )+B \left (c^3 d \left (1089 d^6+7203 d^5 e x+20139 d^4 e^2 x^2+30625 d^3 e^3 x^3+26950 d^2 e^4 x^4+13230 d e^5 x^5+2940 e^6 x^6\right )-e^3 \left (10 a^3 e^3 (d+7 e x)+12 a^2 b e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+9 a b^2 e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 b^3 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )-3 c e^2 \left (3 a^2 e^2 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+8 a b e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+10 b^2 \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )-30 c^2 e \left (a e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+6 b \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )\right )\right )+420 B c^3 (d+e x)^7 \log (d+e x)}{420 e^8 (d+e x)^7} \]
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Time = 0.39 (sec) , antiderivative size = 1047, normalized size of antiderivative = 1.91
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1047\) |
norman | \(\text {Expression too large to display}\) | \(1057\) |
default | \(\text {Expression too large to display}\) | \(1065\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1354\) |
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none
Time = 0.32 (sec) , antiderivative size = 1023, normalized size of antiderivative = 1.87 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {1089 \, B c^{3} d^{7} - 60 \, A a^{3} e^{7} - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} 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^{4} e^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3} e^{4} - 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} - 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6} + 420 \, {\left (7 \, B c^{3} d e^{6} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 630 \, {\left (21 \, B c^{3} d^{2} e^{5} - 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{6} - {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{7}\right )} x^{5} + 70 \, {\left (385 \, B c^{3} d^{3} e^{4} - 30 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} - 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{6} - 2 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{7}\right )} x^{4} + 35 \, {\left (875 \, B c^{3} d^{4} e^{3} - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{5} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{6} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{7}\right )} x^{3} + 21 \, {\left (959 \, B c^{3} d^{5} e^{2} - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{4} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{5} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{6} - 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 7 \, {\left (1029 \, B c^{3} d^{6} e - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} - 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^{4} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} e^{5} - 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{6} - 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x + 420 \, {\left (B c^{3} e^{7} x^{7} + 7 \, B c^{3} d e^{6} x^{6} + 21 \, B c^{3} d^{2} e^{5} x^{5} + 35 \, B c^{3} d^{3} e^{4} x^{4} + 35 \, B c^{3} d^{4} e^{3} x^{3} + 21 \, B c^{3} d^{5} e^{2} x^{2} + 7 \, B c^{3} d^{6} e x + B c^{3} d^{7}\right )} \log \left (e x + d\right )}{420 \, {\left (e^{15} x^{7} + 7 \, d e^{14} x^{6} + 21 \, d^{2} e^{13} x^{5} + 35 \, d^{3} e^{12} x^{4} + 35 \, d^{4} e^{11} x^{3} + 21 \, d^{5} e^{10} x^{2} + 7 \, d^{6} e^{9} x + d^{7} e^{8}\right )}} \]
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Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=\text {Timed out} \]
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none
Time = 0.23 (sec) , antiderivative size = 926, normalized size of antiderivative = 1.69 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {1089 \, B c^{3} d^{7} - 60 \, A a^{3} e^{7} - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} 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^{4} e^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3} e^{4} - 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} - 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6} + 420 \, {\left (7 \, B c^{3} d e^{6} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 630 \, {\left (21 \, B c^{3} d^{2} e^{5} - 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{6} - {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{7}\right )} x^{5} + 70 \, {\left (385 \, B c^{3} d^{3} e^{4} - 30 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} - 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{6} - 2 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{7}\right )} x^{4} + 35 \, {\left (875 \, B c^{3} d^{4} e^{3} - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{5} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{6} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{7}\right )} x^{3} + 21 \, {\left (959 \, B c^{3} d^{5} e^{2} - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{4} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{5} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{6} - 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 7 \, {\left (1029 \, B c^{3} d^{6} e - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} - 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} - 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^{4} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} e^{5} - 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{6} - 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x}{420 \, {\left (e^{15} x^{7} + 7 \, d e^{14} x^{6} + 21 \, d^{2} e^{13} x^{5} + 35 \, d^{3} e^{12} x^{4} + 35 \, d^{4} e^{11} x^{3} + 21 \, d^{5} e^{10} x^{2} + 7 \, d^{6} e^{9} x + d^{7} e^{8}\right )}} + \frac {B c^{3} \log \left (e x + d\right )}{e^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1079 vs. \(2 (537) = 1074\).
Time = 0.28 (sec) , antiderivative size = 1079, normalized size of antiderivative = 1.97 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {B c^{3} \log \left ({\left | e x + d \right |}\right )}{e^{8}} + \frac {420 \, {\left (7 \, B c^{3} d e^{5} - 3 \, B b c^{2} e^{6} - A c^{3} e^{6}\right )} x^{6} + 630 \, {\left (21 \, B c^{3} d^{2} e^{4} - 6 \, B b c^{2} d e^{5} - 2 \, A c^{3} d e^{5} - B b^{2} c e^{6} - B a c^{2} e^{6} - A b c^{2} e^{6}\right )} x^{5} + 70 \, {\left (385 \, B c^{3} d^{3} e^{3} - 90 \, B b c^{2} d^{2} e^{4} - 30 \, A c^{3} d^{2} e^{4} - 15 \, B b^{2} c d e^{5} - 15 \, B a c^{2} d e^{5} - 15 \, A b c^{2} d e^{5} - 2 \, B b^{3} e^{6} - 12 \, B a b c e^{6} - 6 \, A b^{2} c e^{6} - 6 \, A a c^{2} e^{6}\right )} x^{4} + 35 \, {\left (875 \, B c^{3} d^{4} e^{2} - 180 \, B b c^{2} d^{3} e^{3} - 60 \, A c^{3} d^{3} e^{3} - 30 \, B b^{2} c d^{2} e^{4} - 30 \, B a c^{2} d^{2} e^{4} - 30 \, A b c^{2} d^{2} e^{4} - 4 \, B b^{3} d e^{5} - 24 \, B a b c d e^{5} - 12 \, A b^{2} c d e^{5} - 12 \, A a c^{2} d e^{5} - 9 \, B a b^{2} e^{6} - 3 \, A b^{3} e^{6} - 9 \, B a^{2} c e^{6} - 18 \, A a b c e^{6}\right )} x^{3} + 21 \, {\left (959 \, B c^{3} d^{5} e - 180 \, B b c^{2} d^{4} e^{2} - 60 \, A c^{3} d^{4} e^{2} - 30 \, B b^{2} c d^{3} e^{3} - 30 \, B a c^{2} d^{3} e^{3} - 30 \, A b c^{2} d^{3} e^{3} - 4 \, B b^{3} d^{2} e^{4} - 24 \, B a b c d^{2} e^{4} - 12 \, A b^{2} c d^{2} e^{4} - 12 \, A a c^{2} d^{2} e^{4} - 9 \, B a b^{2} d e^{5} - 3 \, A b^{3} d e^{5} - 9 \, B a^{2} c d e^{5} - 18 \, A a b c d e^{5} - 12 \, B a^{2} b e^{6} - 12 \, A a b^{2} e^{6} - 12 \, A a^{2} c e^{6}\right )} x^{2} + 7 \, {\left (1029 \, B c^{3} d^{6} - 180 \, B b c^{2} d^{5} e - 60 \, A c^{3} d^{5} e - 30 \, B b^{2} c d^{4} e^{2} - 30 \, B a c^{2} d^{4} e^{2} - 30 \, A b c^{2} d^{4} e^{2} - 4 \, B b^{3} d^{3} e^{3} - 24 \, B a b c d^{3} e^{3} - 12 \, A b^{2} c d^{3} e^{3} - 12 \, A a c^{2} d^{3} e^{3} - 9 \, B a b^{2} d^{2} e^{4} - 3 \, A b^{3} d^{2} e^{4} - 9 \, B a^{2} c d^{2} e^{4} - 18 \, A a b c d^{2} e^{4} - 12 \, B a^{2} b d e^{5} - 12 \, A a b^{2} d e^{5} - 12 \, A a^{2} c d e^{5} - 10 \, B a^{3} e^{6} - 30 \, A a^{2} b e^{6}\right )} x + \frac {1089 \, B c^{3} d^{7} - 180 \, B b c^{2} d^{6} e - 60 \, A c^{3} d^{6} e - 30 \, B b^{2} c d^{5} e^{2} - 30 \, B a c^{2} d^{5} e^{2} - 30 \, A b c^{2} d^{5} e^{2} - 4 \, B b^{3} d^{4} e^{3} - 24 \, B a b c d^{4} e^{3} - 12 \, A b^{2} c d^{4} e^{3} - 12 \, A a c^{2} d^{4} e^{3} - 9 \, B a b^{2} d^{3} e^{4} - 3 \, A b^{3} d^{3} e^{4} - 9 \, B a^{2} c d^{3} e^{4} - 18 \, A a b c d^{3} e^{4} - 12 \, B a^{2} b d^{2} e^{5} - 12 \, A a b^{2} d^{2} e^{5} - 12 \, A a^{2} c d^{2} e^{5} - 10 \, B a^{3} d e^{6} - 30 \, A a^{2} b d e^{6} - 60 \, A a^{3} e^{7}}{e}}{420 \, {\left (e x + d\right )}^{7} e^{7}} \]
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Time = 11.42 (sec) , antiderivative size = 1353, normalized size of antiderivative = 2.47 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {60\,A\,a^3\,e^7-1089\,B\,c^3\,d^7+10\,B\,a^3\,d\,e^6+60\,A\,c^3\,d^6\,e-420\,B\,c^3\,d^7\,\ln \left (d+e\,x\right )+70\,B\,a^3\,e^7\,x+3\,A\,b^3\,d^3\,e^4+4\,B\,b^3\,d^4\,e^3+105\,A\,b^3\,e^7\,x^3+140\,B\,b^3\,e^7\,x^4+420\,A\,c^3\,e^7\,x^6-7203\,B\,c^3\,d^6\,e\,x+12\,A\,a\,b^2\,d^2\,e^5+12\,A\,a\,c^2\,d^4\,e^3+12\,A\,a^2\,c\,d^2\,e^5+9\,B\,a\,b^2\,d^3\,e^4+12\,B\,a^2\,b\,d^2\,e^5+30\,A\,b\,c^2\,d^5\,e^2+12\,A\,b^2\,c\,d^4\,e^3+30\,B\,a\,c^2\,d^5\,e^2+9\,B\,a^2\,c\,d^3\,e^4+30\,B\,b^2\,c\,d^5\,e^2+252\,A\,a\,b^2\,e^7\,x^2+252\,A\,a^2\,c\,e^7\,x^2+252\,B\,a^2\,b\,e^7\,x^2+315\,B\,a\,b^2\,e^7\,x^3+420\,A\,a\,c^2\,e^7\,x^4+315\,B\,a^2\,c\,e^7\,x^3+420\,A\,b^2\,c\,e^7\,x^4+630\,A\,b\,c^2\,e^7\,x^5+630\,B\,a\,c^2\,e^7\,x^5+21\,A\,b^3\,d^2\,e^5\,x+63\,A\,b^3\,d\,e^6\,x^2+630\,B\,b^2\,c\,e^7\,x^5+1260\,B\,b\,c^2\,e^7\,x^6+420\,A\,c^3\,d^5\,e^2\,x+28\,B\,b^3\,d^3\,e^4\,x+140\,B\,b^3\,d\,e^6\,x^3+1260\,A\,c^3\,d\,e^6\,x^5-2940\,B\,c^3\,d\,e^6\,x^6-420\,B\,c^3\,e^7\,x^7\,\ln \left (d+e\,x\right )+1260\,A\,c^3\,d^4\,e^3\,x^2+84\,B\,b^3\,d^2\,e^5\,x^2+2100\,A\,c^3\,d^3\,e^4\,x^3+2100\,A\,c^3\,d^2\,e^5\,x^4-20139\,B\,c^3\,d^5\,e^2\,x^2-30625\,B\,c^3\,d^4\,e^3\,x^3-26950\,B\,c^3\,d^3\,e^4\,x^4-13230\,B\,c^3\,d^2\,e^5\,x^5+30\,A\,a^2\,b\,d\,e^6+180\,B\,b\,c^2\,d^6\,e+210\,A\,a^2\,b\,e^7\,x+252\,A\,a\,c^2\,d^2\,e^5\,x^2+630\,A\,b\,c^2\,d^3\,e^4\,x^2+252\,A\,b^2\,c\,d^2\,e^5\,x^2+630\,B\,a\,c^2\,d^3\,e^4\,x^2+1050\,A\,b\,c^2\,d^2\,e^5\,x^3+1050\,B\,a\,c^2\,d^2\,e^5\,x^3+3780\,B\,b\,c^2\,d^4\,e^3\,x^2+630\,B\,b^2\,c\,d^3\,e^4\,x^2+6300\,B\,b\,c^2\,d^3\,e^4\,x^3+1050\,B\,b^2\,c\,d^2\,e^5\,x^3+6300\,B\,b\,c^2\,d^2\,e^5\,x^4-8820\,B\,c^3\,d^5\,e^2\,x^2\,\ln \left (d+e\,x\right )-14700\,B\,c^3\,d^4\,e^3\,x^3\,\ln \left (d+e\,x\right )-14700\,B\,c^3\,d^3\,e^4\,x^4\,\ln \left (d+e\,x\right )-8820\,B\,c^3\,d^2\,e^5\,x^5\,\ln \left (d+e\,x\right )+18\,A\,a\,b\,c\,d^3\,e^4+24\,B\,a\,b\,c\,d^4\,e^3+630\,A\,a\,b\,c\,e^7\,x^3+84\,A\,a\,b^2\,d\,e^6\,x+840\,B\,a\,b\,c\,e^7\,x^4+84\,A\,a^2\,c\,d\,e^6\,x+84\,B\,a^2\,b\,d\,e^6\,x-2940\,B\,c^3\,d^6\,e\,x\,\ln \left (d+e\,x\right )+84\,A\,a\,c^2\,d^3\,e^4\,x+63\,B\,a\,b^2\,d^2\,e^5\,x+189\,B\,a\,b^2\,d\,e^6\,x^2+420\,A\,a\,c^2\,d\,e^6\,x^3+210\,A\,b\,c^2\,d^4\,e^3\,x+84\,A\,b^2\,c\,d^3\,e^4\,x+210\,B\,a\,c^2\,d^4\,e^3\,x+63\,B\,a^2\,c\,d^2\,e^5\,x+189\,B\,a^2\,c\,d\,e^6\,x^2+420\,A\,b^2\,c\,d\,e^6\,x^3+1050\,A\,b\,c^2\,d\,e^6\,x^4+1050\,B\,a\,c^2\,d\,e^6\,x^4+1260\,B\,b\,c^2\,d^5\,e^2\,x+210\,B\,b^2\,c\,d^4\,e^3\,x+1050\,B\,b^2\,c\,d\,e^6\,x^4+3780\,B\,b\,c^2\,d\,e^6\,x^5-2940\,B\,c^3\,d\,e^6\,x^6\,\ln \left (d+e\,x\right )+504\,B\,a\,b\,c\,d^2\,e^5\,x^2+126\,A\,a\,b\,c\,d^2\,e^5\,x+378\,A\,a\,b\,c\,d\,e^6\,x^2+168\,B\,a\,b\,c\,d^3\,e^4\,x+840\,B\,a\,b\,c\,d\,e^6\,x^3}{420\,e^8\,{\left (d+e\,x\right )}^7} \]
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