Integrand size = 25, antiderivative size = 544 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right ) x}{e^7}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^2}{2 e^8}-\frac {\left (A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) (d+e x)^3}{3 e^8}-\frac {\left (B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^4}{4 e^8}-\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right ) (d+e x)^5}{5 e^8}-\frac {c^2 (7 B c d-3 b B e-A c e) (d+e x)^6}{6 e^8}+\frac {B c^3 (d+e x)^7}{7 e^8}-\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3 \log (d+e x)}{e^8} \]
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Time = 0.75 (sec) , antiderivative size = 541, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {785} \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx=-\frac {(d+e x)^3 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{3 e^8}-\frac {3 c (d+e x)^5 \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{5 e^8}-\frac {3 (d+e x)^2 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{2 e^8}-\frac {(d+e x)^4 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{4 e^8}-\frac {(B d-A e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^8}+\frac {x \left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^7}-\frac {c^2 (d+e x)^6 (-A c e-3 b B e+7 B c d)}{6 e^8}+\frac {B c^3 (d+e x)^7}{7 e^8} \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (\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}+\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)}+\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)}{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)^2}{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)^3}{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)^4}{e^7}+\frac {c^2 (-7 B c d+3 b B e+A c e) (d+e x)^5}{e^7}+\frac {B c^3 (d+e x)^6}{e^7}\right ) \, dx \\ & = \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 ) x}{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)^2}{2 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)^3}{3 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)^4}{4 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)^5}{5 e^8}-\frac {c^2 (7 B c d-3 b B e-A c e) (d+e x)^6}{6 e^8}+\frac {B c^3 (d+e x)^7}{7 e^8}-\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3 \log (d+e x)}{e^8} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.29 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {e x \left (7 A e \left (c^3 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+10 b e^3 \left (18 a^2 e^2+9 a b e (-2 d+e x)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+15 c e^2 \left (6 a^2 e^2 (-2 d+e x)+4 a b e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+3 c^2 e \left (5 a e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+b \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )\right )+B \left (c^3 \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )+35 e^3 \left (12 a^3 e^3+18 a^2 b e^2 (-2 d+e x)+6 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^3 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+21 c e^2 \left (10 a^2 e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+10 a b e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+b^2 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+21 c^2 e \left (a e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+b \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )\right )\right )-420 (B d-A e) \left (c d^2+e (-b d+a e)\right )^3 \log (d+e x)}{420 e^8} \]
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Time = 0.37 (sec) , antiderivative size = 1030, normalized size of antiderivative = 1.89
method | result | size |
norman | \(\text {Expression too large to display}\) | \(1030\) |
default | \(\text {Expression too large to display}\) | \(1199\) |
risch | \(\text {Expression too large to display}\) | \(1319\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1326\) |
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Time = 0.31 (sec) , antiderivative size = 843, normalized size of antiderivative = 1.55 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {60 \, B c^{3} e^{7} x^{7} - 70 \, {\left (B c^{3} d e^{6} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 84 \, {\left (B c^{3} d^{2} e^{5} - {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{6} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{7}\right )} x^{5} - 105 \, {\left (B c^{3} d^{3} e^{4} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{6} - {\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} + 140 \, {\left (B c^{3} d^{4} e^{3} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{5} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{6} + {\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} - 210 \, {\left (B c^{3} d^{5} e^{2} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{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^{2} e^{5} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{6} - 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 420 \, {\left (B c^{3} d^{6} e - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} - {\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} + {\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} - 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{6} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x - 420 \, {\left (B c^{3} d^{7} - A a^{3} e^{7} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} - {\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} + {\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} - 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6}\right )} \log \left (e x + d\right )}{420 \, e^{8}} \]
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Time = 1.11 (sec) , antiderivative size = 979, normalized size of antiderivative = 1.80 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {B c^{3} x^{7}}{7 e} + x^{6} \left (\frac {A c^{3}}{6 e} + \frac {B b c^{2}}{2 e} - \frac {B c^{3} d}{6 e^{2}}\right ) + x^{5} \cdot \left (\frac {3 A b c^{2}}{5 e} - \frac {A c^{3} d}{5 e^{2}} + \frac {3 B a c^{2}}{5 e} + \frac {3 B b^{2} c}{5 e} - \frac {3 B b c^{2} d}{5 e^{2}} + \frac {B c^{3} d^{2}}{5 e^{3}}\right ) + x^{4} \cdot \left (\frac {3 A a c^{2}}{4 e} + \frac {3 A b^{2} c}{4 e} - \frac {3 A b c^{2} d}{4 e^{2}} + \frac {A c^{3} d^{2}}{4 e^{3}} + \frac {3 B a b c}{2 e} - \frac {3 B a c^{2} d}{4 e^{2}} + \frac {B b^{3}}{4 e} - \frac {3 B b^{2} c d}{4 e^{2}} + \frac {3 B b c^{2} d^{2}}{4 e^{3}} - \frac {B c^{3} d^{3}}{4 e^{4}}\right ) + x^{3} \cdot \left (\frac {2 A a b c}{e} - \frac {A a c^{2} d}{e^{2}} + \frac {A b^{3}}{3 e} - \frac {A b^{2} c d}{e^{2}} + \frac {A b c^{2} d^{2}}{e^{3}} - \frac {A c^{3} d^{3}}{3 e^{4}} + \frac {B a^{2} c}{e} + \frac {B a b^{2}}{e} - \frac {2 B a b c d}{e^{2}} + \frac {B a c^{2} d^{2}}{e^{3}} - \frac {B b^{3} d}{3 e^{2}} + \frac {B b^{2} c d^{2}}{e^{3}} - \frac {B b c^{2} d^{3}}{e^{4}} + \frac {B c^{3} d^{4}}{3 e^{5}}\right ) + x^{2} \cdot \left (\frac {3 A a^{2} c}{2 e} + \frac {3 A a b^{2}}{2 e} - \frac {3 A a b c d}{e^{2}} + \frac {3 A a c^{2} d^{2}}{2 e^{3}} - \frac {A b^{3} d}{2 e^{2}} + \frac {3 A b^{2} c d^{2}}{2 e^{3}} - \frac {3 A b c^{2} d^{3}}{2 e^{4}} + \frac {A c^{3} d^{4}}{2 e^{5}} + \frac {3 B a^{2} b}{2 e} - \frac {3 B a^{2} c d}{2 e^{2}} - \frac {3 B a b^{2} d}{2 e^{2}} + \frac {3 B a b c d^{2}}{e^{3}} - \frac {3 B a c^{2} d^{3}}{2 e^{4}} + \frac {B b^{3} d^{2}}{2 e^{3}} - \frac {3 B b^{2} c d^{3}}{2 e^{4}} + \frac {3 B b c^{2} d^{4}}{2 e^{5}} - \frac {B c^{3} d^{5}}{2 e^{6}}\right ) + x \left (\frac {3 A a^{2} b}{e} - \frac {3 A a^{2} c d}{e^{2}} - \frac {3 A a b^{2} d}{e^{2}} + \frac {6 A a b c d^{2}}{e^{3}} - \frac {3 A a c^{2} d^{3}}{e^{4}} + \frac {A b^{3} d^{2}}{e^{3}} - \frac {3 A b^{2} c d^{3}}{e^{4}} + \frac {3 A b c^{2} d^{4}}{e^{5}} - \frac {A c^{3} d^{5}}{e^{6}} + \frac {B a^{3}}{e} - \frac {3 B a^{2} b d}{e^{2}} + \frac {3 B a^{2} c d^{2}}{e^{3}} + \frac {3 B a b^{2} d^{2}}{e^{3}} - \frac {6 B a b c d^{3}}{e^{4}} + \frac {3 B a c^{2} d^{4}}{e^{5}} - \frac {B b^{3} d^{3}}{e^{4}} + \frac {3 B b^{2} c d^{4}}{e^{5}} - \frac {3 B b c^{2} d^{5}}{e^{6}} + \frac {B c^{3} d^{6}}{e^{7}}\right ) - \frac {\left (- A e + B d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{8}} \]
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Time = 0.20 (sec) , antiderivative size = 842, normalized size of antiderivative = 1.55 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {60 \, B c^{3} e^{6} x^{7} - 70 \, {\left (B c^{3} d e^{5} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{6}\right )} x^{6} + 84 \, {\left (B c^{3} d^{2} e^{4} - {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{5} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{6}\right )} x^{5} - 105 \, {\left (B c^{3} d^{3} e^{3} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{4} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{5} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{6}\right )} x^{4} + 140 \, {\left (B c^{3} d^{4} e^{2} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{3} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{4} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{5} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{6}\right )} x^{3} - 210 \, {\left (B c^{3} d^{5} e - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{2} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{3} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{4} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{5} - 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{6}\right )} x^{2} + 420 \, {\left (B c^{3} d^{6} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{2} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} e^{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^{4} - 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{5} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{6}\right )} x}{420 \, e^{7}} - \frac {{\left (B c^{3} d^{7} - A a^{3} e^{7} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} - {\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} + {\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} - 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6}\right )} \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. 1209 vs. \(2 (533) = 1066\).
Time = 0.29 (sec) , antiderivative size = 1209, normalized size of antiderivative = 2.22 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {60 \, B c^{3} e^{6} x^{7} - 70 \, B c^{3} d e^{5} x^{6} + 210 \, B b c^{2} e^{6} x^{6} + 70 \, A c^{3} e^{6} x^{6} + 84 \, B c^{3} d^{2} e^{4} x^{5} - 252 \, B b c^{2} d e^{5} x^{5} - 84 \, A c^{3} d e^{5} x^{5} + 252 \, B b^{2} c e^{6} x^{5} + 252 \, B a c^{2} e^{6} x^{5} + 252 \, A b c^{2} e^{6} x^{5} - 105 \, B c^{3} d^{3} e^{3} x^{4} + 315 \, B b c^{2} d^{2} e^{4} x^{4} + 105 \, A c^{3} d^{2} e^{4} x^{4} - 315 \, B b^{2} c d e^{5} x^{4} - 315 \, B a c^{2} d e^{5} x^{4} - 315 \, A b c^{2} d e^{5} x^{4} + 105 \, B b^{3} e^{6} x^{4} + 630 \, B a b c e^{6} x^{4} + 315 \, A b^{2} c e^{6} x^{4} + 315 \, A a c^{2} e^{6} x^{4} + 140 \, B c^{3} d^{4} e^{2} x^{3} - 420 \, B b c^{2} d^{3} e^{3} x^{3} - 140 \, A c^{3} d^{3} e^{3} x^{3} + 420 \, B b^{2} c d^{2} e^{4} x^{3} + 420 \, B a c^{2} d^{2} e^{4} x^{3} + 420 \, A b c^{2} d^{2} e^{4} x^{3} - 140 \, B b^{3} d e^{5} x^{3} - 840 \, B a b c d e^{5} x^{3} - 420 \, A b^{2} c d e^{5} x^{3} - 420 \, A a c^{2} d e^{5} x^{3} + 420 \, B a b^{2} e^{6} x^{3} + 140 \, A b^{3} e^{6} x^{3} + 420 \, B a^{2} c e^{6} x^{3} + 840 \, A a b c e^{6} x^{3} - 210 \, B c^{3} d^{5} e x^{2} + 630 \, B b c^{2} d^{4} e^{2} x^{2} + 210 \, A c^{3} d^{4} e^{2} x^{2} - 630 \, B b^{2} c d^{3} e^{3} x^{2} - 630 \, B a c^{2} d^{3} e^{3} x^{2} - 630 \, A b c^{2} d^{3} e^{3} x^{2} + 210 \, B b^{3} d^{2} e^{4} x^{2} + 1260 \, B a b c d^{2} e^{4} x^{2} + 630 \, A b^{2} c d^{2} e^{4} x^{2} + 630 \, A a c^{2} d^{2} e^{4} x^{2} - 630 \, B a b^{2} d e^{5} x^{2} - 210 \, A b^{3} d e^{5} x^{2} - 630 \, B a^{2} c d e^{5} x^{2} - 1260 \, A a b c d e^{5} x^{2} + 630 \, B a^{2} b e^{6} x^{2} + 630 \, A a b^{2} e^{6} x^{2} + 630 \, A a^{2} c e^{6} x^{2} + 420 \, B c^{3} d^{6} x - 1260 \, B b c^{2} d^{5} e x - 420 \, A c^{3} d^{5} e x + 1260 \, B b^{2} c d^{4} e^{2} x + 1260 \, B a c^{2} d^{4} e^{2} x + 1260 \, A b c^{2} d^{4} e^{2} x - 420 \, B b^{3} d^{3} e^{3} x - 2520 \, B a b c d^{3} e^{3} x - 1260 \, A b^{2} c d^{3} e^{3} x - 1260 \, A a c^{2} d^{3} e^{3} x + 1260 \, B a b^{2} d^{2} e^{4} x + 420 \, A b^{3} d^{2} e^{4} x + 1260 \, B a^{2} c d^{2} e^{4} x + 2520 \, A a b c d^{2} e^{4} x - 1260 \, B a^{2} b d e^{5} x - 1260 \, A a b^{2} d e^{5} x - 1260 \, A a^{2} c d e^{5} x + 420 \, B a^{3} e^{6} x + 1260 \, A a^{2} b e^{6} x}{420 \, e^{7}} - \frac {{\left (B c^{3} d^{7} - 3 \, B b c^{2} d^{6} e - A c^{3} d^{6} e + 3 \, B b^{2} c d^{5} e^{2} + 3 \, B a c^{2} d^{5} e^{2} + 3 \, A b c^{2} d^{5} e^{2} - B b^{3} d^{4} e^{3} - 6 \, B a b c d^{4} e^{3} - 3 \, A b^{2} c d^{4} e^{3} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a b^{2} d^{3} e^{4} + A b^{3} d^{3} e^{4} + 3 \, B a^{2} c d^{3} e^{4} + 6 \, A a b c d^{3} e^{4} - 3 \, B a^{2} b d^{2} e^{5} - 3 \, A a b^{2} d^{2} e^{5} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} + 3 \, A a^{2} b d e^{6} - A a^{3} e^{7}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} \]
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Time = 11.15 (sec) , antiderivative size = 968, normalized size of antiderivative = 1.78 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx=x^2\,\left (\frac {3\,B\,a^2\,b+3\,A\,c\,a^2+3\,A\,a\,b^2}{2\,e}-\frac {d\,\left (\frac {3\,B\,c\,a^2+3\,B\,a\,b^2+6\,A\,c\,a\,b+A\,b^3}{e}-\frac {d\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{e}-\frac {d\,\left (\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e}-\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}\right )}{e}\right )}{e}\right )}{2\,e}\right )+x^5\,\left (\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{5\,e}-\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{5\,e}\right )+x^4\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{4\,e}-\frac {d\,\left (\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e}-\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}\right )}{4\,e}\right )+x\,\left (\frac {B\,a^3+3\,A\,b\,a^2}{e}-\frac {d\,\left (\frac {3\,B\,a^2\,b+3\,A\,c\,a^2+3\,A\,a\,b^2}{e}-\frac {d\,\left (\frac {3\,B\,c\,a^2+3\,B\,a\,b^2+6\,A\,c\,a\,b+A\,b^3}{e}-\frac {d\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{e}-\frac {d\,\left (\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e}-\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )+x^3\,\left (\frac {3\,B\,c\,a^2+3\,B\,a\,b^2+6\,A\,c\,a\,b+A\,b^3}{3\,e}-\frac {d\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{e}-\frac {d\,\left (\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e}-\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}\right )}{e}\right )}{3\,e}\right )+x^6\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{6\,e}-\frac {B\,c^3\,d}{6\,e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-B\,a^3\,d\,e^6+A\,a^3\,e^7+3\,B\,a^2\,b\,d^2\,e^5-3\,A\,a^2\,b\,d\,e^6-3\,B\,a^2\,c\,d^3\,e^4+3\,A\,a^2\,c\,d^2\,e^5-3\,B\,a\,b^2\,d^3\,e^4+3\,A\,a\,b^2\,d^2\,e^5+6\,B\,a\,b\,c\,d^4\,e^3-6\,A\,a\,b\,c\,d^3\,e^4-3\,B\,a\,c^2\,d^5\,e^2+3\,A\,a\,c^2\,d^4\,e^3+B\,b^3\,d^4\,e^3-A\,b^3\,d^3\,e^4-3\,B\,b^2\,c\,d^5\,e^2+3\,A\,b^2\,c\,d^4\,e^3+3\,B\,b\,c^2\,d^6\,e-3\,A\,b\,c^2\,d^5\,e^2-B\,c^3\,d^7+A\,c^3\,d^6\,e\right )}{e^8}+\frac {B\,c^3\,x^7}{7\,e} \]
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