Integrand size = 24, antiderivative size = 422 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {\left (A c e \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )-B \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )\right ) x}{e^7}-\frac {c \left (A c e (4 c d-3 b e)-B \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) x^2}{2 e^6}-\frac {c^2 (4 B c d-3 b B e-A c e) x^3}{3 e^5}+\frac {B c^3 x^4}{4 e^4}+\frac {d^3 (B d-A e) (c d-b e)^3}{3 e^8 (d+e x)^3}-\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{2 e^8 (d+e x)^2}-\frac {3 d (c d-b e) \left (A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right )}{e^8 (d+e x)}+\frac {\left (B d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right )-A e \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )\right ) \log (d+e x)}{e^8} \]
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Time = 0.40 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {785} \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=-\frac {3 d (c d-b e) \left (A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )\right )}{e^8 (d+e x)}-\frac {c x^2 \left (A c e (4 c d-3 b e)-B \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{2 e^6}+\frac {\log (d+e x) \left (B d \left (-4 b^3 e^3+30 b^2 c d e^2-60 b c^2 d^2 e+35 c^3 d^3\right )-A e \left (-b^3 e^3+12 b^2 c d e^2-30 b c^2 d^2 e+20 c^3 d^3\right )\right )}{e^8}+\frac {x \left (A c e \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )-B \left (-b^3 e^3+12 b^2 c d e^2-30 b c^2 d^2 e+20 c^3 d^3\right )\right )}{e^7}-\frac {c^2 x^3 (-A c e-3 b B e+4 B c d)}{3 e^5}+\frac {d^3 (B d-A e) (c d-b e)^3}{3 e^8 (d+e x)^3}-\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{2 e^8 (d+e x)^2}+\frac {B c^3 x^4}{4 e^4} \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A c e \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )-B \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )}{e^7}+\frac {c \left (-A c e (4 c d-3 b e)+B \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) x}{e^6}+\frac {c^2 (-4 B c d+3 b B e+A c e) x^2}{e^5}+\frac {B c^3 x^3}{e^4}-\frac {d^3 (B d-A e) (c d-b e)^3}{e^7 (d+e x)^4}+\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^7 (d+e x)^3}+\frac {3 d (c d-b e) \left (A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right )}{e^7 (d+e x)^2}+\frac {B d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right )-A e \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )}{e^7 (d+e x)}\right ) \, dx \\ & = \frac {\left (A c e \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )-B \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )\right ) x}{e^7}-\frac {c \left (A c e (4 c d-3 b e)-B \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) x^2}{2 e^6}-\frac {c^2 (4 B c d-3 b B e-A c e) x^3}{3 e^5}+\frac {B c^3 x^4}{4 e^4}+\frac {d^3 (B d-A e) (c d-b e)^3}{3 e^8 (d+e x)^3}-\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{2 e^8 (d+e x)^2}-\frac {3 d (c d-b e) \left (A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right )}{e^8 (d+e x)}+\frac {\left (B d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right )-A e \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )\right ) \log (d+e x)}{e^8} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {12 e \left (A c e \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )+B \left (-20 c^3 d^3+30 b c^2 d^2 e-12 b^2 c d e^2+b^3 e^3\right )\right ) x-6 c e^2 \left (A c e (4 c d-3 b e)+B \left (-10 c^2 d^2+12 b c d e-3 b^2 e^2\right )\right ) x^2+4 c^2 e^3 (-4 B c d+3 b B e+A c e) x^3+3 B c^3 e^4 x^4+\frac {4 d^3 (B d-A e) (c d-b e)^3}{(d+e x)^3}-\frac {6 d^2 (c d-b e)^2 (B d (7 c d-4 b e)+3 A e (-2 c d+b e))}{(d+e x)^2}+\frac {36 d (c d-b e) \left (-A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )+B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right )}{d+e x}+12 \left (B d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right )+A e \left (-20 c^3 d^3+30 b c^2 d^2 e-12 b^2 c d e^2+b^3 e^3\right )\right ) \log (d+e x)}{12 e^8} \]
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Time = 0.20 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.33
method | result | size |
norman | \(\frac {\frac {d^{3} \left (11 A \,b^{3} e^{4}-132 A \,b^{2} c d \,e^{3}+330 A b \,c^{2} d^{2} e^{2}-220 A \,c^{3} d^{3} e -44 B \,b^{3} d \,e^{3}+330 B \,b^{2} c \,d^{2} e^{2}-660 B b \,c^{2} d^{3} e +385 B \,c^{3} d^{4}\right )}{6 e^{8}}+\frac {\left (12 A \,b^{2} c \,e^{3}-30 A b \,c^{2} d \,e^{2}+20 A \,c^{3} d^{2} e +4 B \,b^{3} e^{3}-30 B \,b^{2} c d \,e^{2}+60 B b \,c^{2} d^{2} e -35 B \,c^{3} d^{3}\right ) x^{4}}{4 e^{4}}+\frac {B \,c^{3} x^{7}}{4 e}+\frac {c \left (6 A b c \,e^{2}-4 A \,c^{2} d e +6 B \,b^{2} e^{2}-12 B b c d e +7 B \,c^{2} d^{2}\right ) x^{5}}{4 e^{3}}+\frac {c^{2} \left (4 A c e +12 B b e -7 B c d \right ) x^{6}}{12 e^{2}}+\frac {3 d \left (A \,b^{3} e^{4}-12 A \,b^{2} c d \,e^{3}+30 A b \,c^{2} d^{2} e^{2}-20 A \,c^{3} d^{3} e -4 B \,b^{3} d \,e^{3}+30 B \,b^{2} c \,d^{2} e^{2}-60 B b \,c^{2} d^{3} e +35 B \,c^{3} d^{4}\right ) x^{2}}{e^{6}}+\frac {3 d^{2} \left (3 A \,b^{3} e^{4}-36 A \,b^{2} c d \,e^{3}+90 A b \,c^{2} d^{2} e^{2}-60 A \,c^{3} d^{3} e -12 B \,b^{3} d \,e^{3}+90 B \,b^{2} c \,d^{2} e^{2}-180 B b \,c^{2} d^{3} e +105 B \,c^{3} d^{4}\right ) x}{2 e^{7}}}{\left (e x +d \right )^{3}}+\frac {\left (A \,b^{3} e^{4}-12 A \,b^{2} c d \,e^{3}+30 A b \,c^{2} d^{2} e^{2}-20 A \,c^{3} d^{3} e -4 B \,b^{3} d \,e^{3}+30 B \,b^{2} c \,d^{2} e^{2}-60 B b \,c^{2} d^{3} e +35 B \,c^{3} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(563\) |
default | \(\frac {\frac {1}{4} B \,c^{3} x^{4} e^{3}+\frac {1}{3} A \,c^{3} e^{3} x^{3}+B b \,c^{2} e^{3} x^{3}-\frac {4}{3} B \,c^{3} d \,e^{2} x^{3}+\frac {3}{2} A b \,c^{2} e^{3} x^{2}-2 A \,c^{3} d \,e^{2} x^{2}+\frac {3}{2} B \,b^{2} c \,e^{3} x^{2}-6 B b \,c^{2} d \,e^{2} x^{2}+5 B \,c^{3} d^{2} e \,x^{2}+3 A \,b^{2} c \,e^{3} x -12 A b \,c^{2} d \,e^{2} x +10 A \,c^{3} d^{2} e x +B \,b^{3} e^{3} x -12 B \,b^{2} c d \,e^{2} x +30 B b \,c^{2} d^{2} e x -20 B \,c^{3} d^{3} x}{e^{7}}+\frac {d^{3} \left (A \,b^{3} e^{4}-3 A \,b^{2} c d \,e^{3}+3 A b \,c^{2} d^{2} e^{2}-A \,c^{3} d^{3} e -B \,b^{3} d \,e^{3}+3 B \,b^{2} c \,d^{2} e^{2}-3 B b \,c^{2} d^{3} e +B \,c^{3} d^{4}\right )}{3 e^{8} \left (e x +d \right )^{3}}+\frac {3 d \left (A \,b^{3} e^{4}-6 A \,b^{2} c d \,e^{3}+10 A b \,c^{2} d^{2} e^{2}-5 A \,c^{3} d^{3} e -2 B \,b^{3} d \,e^{3}+10 B \,b^{2} c \,d^{2} e^{2}-15 B b \,c^{2} d^{3} e +7 B \,c^{3} d^{4}\right )}{e^{8} \left (e x +d \right )}+\frac {\left (A \,b^{3} e^{4}-12 A \,b^{2} c d \,e^{3}+30 A b \,c^{2} d^{2} e^{2}-20 A \,c^{3} d^{3} e -4 B \,b^{3} d \,e^{3}+30 B \,b^{2} c \,d^{2} e^{2}-60 B b \,c^{2} d^{3} e +35 B \,c^{3} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}-\frac {d^{2} \left (3 A \,b^{3} e^{4}-12 A \,b^{2} c d \,e^{3}+15 A b \,c^{2} d^{2} e^{2}-6 A \,c^{3} d^{3} e -4 B \,b^{3} d \,e^{3}+15 B \,b^{2} c \,d^{2} e^{2}-18 B b \,c^{2} d^{3} e +7 B \,c^{3} d^{4}\right )}{2 e^{8} \left (e x +d \right )^{2}}\) | \(596\) |
risch | \(\frac {B \,c^{3} x^{4}}{4 e^{4}}-\frac {12 B \,b^{2} c d x}{e^{5}}+\frac {30 B b \,c^{2} d^{2} x}{e^{6}}-\frac {12 \ln \left (e x +d \right ) A \,b^{2} c d}{e^{5}}+\frac {30 \ln \left (e x +d \right ) A b \,c^{2} d^{2}}{e^{6}}+\frac {30 \ln \left (e x +d \right ) B \,b^{2} c \,d^{2}}{e^{6}}-\frac {60 \ln \left (e x +d \right ) B b \,c^{2} d^{3}}{e^{7}}-\frac {6 B b \,c^{2} d \,x^{2}}{e^{5}}-\frac {12 A b \,c^{2} d x}{e^{5}}+\frac {A \,c^{3} x^{3}}{3 e^{4}}+\frac {B \,b^{3} x}{e^{4}}+\frac {\ln \left (e x +d \right ) A \,b^{3}}{e^{4}}+\frac {B b \,c^{2} x^{3}}{e^{4}}-\frac {4 B \,c^{3} d \,x^{3}}{3 e^{5}}+\frac {3 A b \,c^{2} x^{2}}{2 e^{4}}-\frac {2 A \,c^{3} d \,x^{2}}{e^{5}}+\frac {3 B \,b^{2} c \,x^{2}}{2 e^{4}}+\frac {5 B \,c^{3} d^{2} x^{2}}{e^{6}}+\frac {3 A \,b^{2} c x}{e^{4}}+\frac {10 A \,c^{3} d^{2} x}{e^{6}}-\frac {20 B \,c^{3} d^{3} x}{e^{7}}-\frac {20 \ln \left (e x +d \right ) A \,c^{3} d^{3}}{e^{7}}-\frac {4 \ln \left (e x +d \right ) B \,b^{3} d}{e^{5}}+\frac {35 \ln \left (e x +d \right ) B \,c^{3} d^{4}}{e^{8}}+\frac {\left (3 A \,b^{3} d \,e^{5}-18 A \,b^{2} c \,d^{2} e^{4}+30 A b \,c^{2} d^{3} e^{3}-15 A \,c^{3} d^{4} e^{2}-6 B \,b^{3} d^{2} e^{4}+30 B \,b^{2} c \,d^{3} e^{3}-45 B b \,c^{2} d^{4} e^{2}+21 B \,c^{3} d^{5} e \right ) x^{2}+\frac {d^{2} \left (9 A \,b^{3} e^{4}-60 A \,b^{2} c d \,e^{3}+105 A b \,c^{2} d^{2} e^{2}-54 A \,c^{3} d^{3} e -20 B \,b^{3} d \,e^{3}+105 B \,b^{2} c \,d^{2} e^{2}-162 B b \,c^{2} d^{3} e +77 B \,c^{3} d^{4}\right ) x}{2}+\frac {d^{3} \left (11 A \,b^{3} e^{4}-78 A \,b^{2} c d \,e^{3}+141 A b \,c^{2} d^{2} e^{2}-74 A \,c^{3} d^{3} e -26 B \,b^{3} d \,e^{3}+141 B \,b^{2} c \,d^{2} e^{2}-222 B b \,c^{2} d^{3} e +107 B \,c^{3} d^{4}\right )}{6 e}}{e^{7} \left (e x +d \right )^{3}}\) | \(644\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1182\) |
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Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (412) = 824\).
Time = 0.29 (sec) , antiderivative size = 910, normalized size of antiderivative = 2.16 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {3 \, B c^{3} e^{7} x^{7} + 214 \, B c^{3} d^{7} + 22 \, A b^{3} d^{3} e^{4} - 148 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 282 \, {\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - 52 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3} - {\left (7 \, B c^{3} d e^{6} - 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 3 \, {\left (7 \, B c^{3} d^{2} e^{5} - 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{6} + 6 \, {\left (B b^{2} c + A b c^{2}\right )} e^{7}\right )} x^{5} - 3 \, {\left (35 \, B c^{3} d^{3} e^{4} - 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 30 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{6} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{7}\right )} x^{4} - 2 \, {\left (278 \, B c^{3} d^{4} e^{3} - 146 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 189 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{5} - 18 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{6}\right )} x^{3} - 6 \, {\left (68 \, B c^{3} d^{5} e^{2} - 6 \, A b^{3} d e^{6} - 26 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 9 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{4} + 6 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{5}\right )} x^{2} + 6 \, {\left (37 \, B c^{3} d^{6} e + 9 \, A b^{3} d^{2} e^{5} - 34 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 81 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{3} - 18 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{4}\right )} x + 12 \, {\left (35 \, B c^{3} d^{7} + A b^{3} d^{3} e^{4} - 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 30 \, {\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3} + {\left (35 \, B c^{3} d^{4} e^{3} + A b^{3} e^{7} - 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 30 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{5} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{6}\right )} x^{3} + 3 \, {\left (35 \, B c^{3} d^{5} e^{2} + A b^{3} d e^{6} - 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 30 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{4} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{5}\right )} x^{2} + 3 \, {\left (35 \, B c^{3} d^{6} e + A b^{3} d^{2} e^{5} - 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 30 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{3} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{4}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} \]
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Time = 63.64 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.66 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {B c^{3} x^{4}}{4 e^{4}} + x^{3} \left (\frac {A c^{3}}{3 e^{4}} + \frac {B b c^{2}}{e^{4}} - \frac {4 B c^{3} d}{3 e^{5}}\right ) + x^{2} \cdot \left (\frac {3 A b c^{2}}{2 e^{4}} - \frac {2 A c^{3} d}{e^{5}} + \frac {3 B b^{2} c}{2 e^{4}} - \frac {6 B b c^{2} d}{e^{5}} + \frac {5 B c^{3} d^{2}}{e^{6}}\right ) + x \left (\frac {3 A b^{2} c}{e^{4}} - \frac {12 A b c^{2} d}{e^{5}} + \frac {10 A c^{3} d^{2}}{e^{6}} + \frac {B b^{3}}{e^{4}} - \frac {12 B b^{2} c d}{e^{5}} + \frac {30 B b c^{2} d^{2}}{e^{6}} - \frac {20 B c^{3} d^{3}}{e^{7}}\right ) + \frac {11 A b^{3} d^{3} e^{4} - 78 A b^{2} c d^{4} e^{3} + 141 A b c^{2} d^{5} e^{2} - 74 A c^{3} d^{6} e - 26 B b^{3} d^{4} e^{3} + 141 B b^{2} c d^{5} e^{2} - 222 B b c^{2} d^{6} e + 107 B c^{3} d^{7} + x^{2} \cdot \left (18 A b^{3} d e^{6} - 108 A b^{2} c d^{2} e^{5} + 180 A b c^{2} d^{3} e^{4} - 90 A c^{3} d^{4} e^{3} - 36 B b^{3} d^{2} e^{5} + 180 B b^{2} c d^{3} e^{4} - 270 B b c^{2} d^{4} e^{3} + 126 B c^{3} d^{5} e^{2}\right ) + x \left (27 A b^{3} d^{2} e^{5} - 180 A b^{2} c d^{3} e^{4} + 315 A b c^{2} d^{4} e^{3} - 162 A c^{3} d^{5} e^{2} - 60 B b^{3} d^{3} e^{4} + 315 B b^{2} c d^{4} e^{3} - 486 B b c^{2} d^{5} e^{2} + 231 B c^{3} d^{6} e\right )}{6 d^{3} e^{8} + 18 d^{2} e^{9} x + 18 d e^{10} x^{2} + 6 e^{11} x^{3}} - \frac {\left (- A b^{3} e^{4} + 12 A b^{2} c d e^{3} - 30 A b c^{2} d^{2} e^{2} + 20 A c^{3} d^{3} e + 4 B b^{3} d e^{3} - 30 B b^{2} c d^{2} e^{2} + 60 B b c^{2} d^{3} e - 35 B c^{3} d^{4}\right ) \log {\left (d + e x \right )}}{e^{8}} \]
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Time = 0.20 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.33 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {107 \, B c^{3} d^{7} + 11 \, A b^{3} d^{3} e^{4} - 74 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 141 \, {\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - 26 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3} + 18 \, {\left (7 \, B c^{3} d^{5} e^{2} + A b^{3} d e^{6} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 10 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{4} - 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{5}\right )} x^{2} + 3 \, {\left (77 \, B c^{3} d^{6} e + 9 \, A b^{3} d^{2} e^{5} - 54 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 105 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{3} - 20 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{4}\right )} x}{6 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} + \frac {3 \, B c^{3} e^{3} x^{4} - 4 \, {\left (4 \, B c^{3} d e^{2} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{3}\right )} x^{3} + 6 \, {\left (10 \, B c^{3} d^{2} e - 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{2} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e^{3}\right )} x^{2} - 12 \, {\left (20 \, B c^{3} d^{3} - 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e + 12 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{2} - {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{3}\right )} x}{12 \, e^{7}} + \frac {{\left (35 \, B c^{3} d^{4} + A b^{3} e^{4} - 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 30 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{2} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{3}\right )} \log \left (e x + d\right )}{e^{8}} \]
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Time = 0.26 (sec) , antiderivative size = 618, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {{\left (35 \, B c^{3} d^{4} - 60 \, B b c^{2} d^{3} e - 20 \, A c^{3} d^{3} e + 30 \, B b^{2} c d^{2} e^{2} + 30 \, A b c^{2} d^{2} e^{2} - 4 \, B b^{3} d e^{3} - 12 \, A b^{2} c d e^{3} + A b^{3} e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} + \frac {107 \, B c^{3} d^{7} - 222 \, B b c^{2} d^{6} e - 74 \, A c^{3} d^{6} e + 141 \, B b^{2} c d^{5} e^{2} + 141 \, A b c^{2} d^{5} e^{2} - 26 \, B b^{3} d^{4} e^{3} - 78 \, A b^{2} c d^{4} e^{3} + 11 \, A b^{3} d^{3} e^{4} + 18 \, {\left (7 \, B c^{3} d^{5} e^{2} - 15 \, B b c^{2} d^{4} e^{3} - 5 \, A c^{3} d^{4} e^{3} + 10 \, B b^{2} c d^{3} e^{4} + 10 \, A b c^{2} d^{3} e^{4} - 2 \, B b^{3} d^{2} e^{5} - 6 \, A b^{2} c d^{2} e^{5} + A b^{3} d e^{6}\right )} x^{2} + 3 \, {\left (77 \, B c^{3} d^{6} e - 162 \, B b c^{2} d^{5} e^{2} - 54 \, A c^{3} d^{5} e^{2} + 105 \, B b^{2} c d^{4} e^{3} + 105 \, A b c^{2} d^{4} e^{3} - 20 \, B b^{3} d^{3} e^{4} - 60 \, A b^{2} c d^{3} e^{4} + 9 \, A b^{3} d^{2} e^{5}\right )} x}{6 \, {\left (e x + d\right )}^{3} e^{8}} + \frac {3 \, B c^{3} e^{12} x^{4} - 16 \, B c^{3} d e^{11} x^{3} + 12 \, B b c^{2} e^{12} x^{3} + 4 \, A c^{3} e^{12} x^{3} + 60 \, B c^{3} d^{2} e^{10} x^{2} - 72 \, B b c^{2} d e^{11} x^{2} - 24 \, A c^{3} d e^{11} x^{2} + 18 \, B b^{2} c e^{12} x^{2} + 18 \, A b c^{2} e^{12} x^{2} - 240 \, B c^{3} d^{3} e^{9} x + 360 \, B b c^{2} d^{2} e^{10} x + 120 \, A c^{3} d^{2} e^{10} x - 144 \, B b^{2} c d e^{11} x - 144 \, A b c^{2} d e^{11} x + 12 \, B b^{3} e^{12} x + 36 \, A b^{2} c e^{12} x}{12 \, e^{16}} \]
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Time = 0.21 (sec) , antiderivative size = 676, normalized size of antiderivative = 1.60 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {x\,\left (-10\,B\,b^3\,d^3\,e^3+\frac {9\,A\,b^3\,d^2\,e^4}{2}+\frac {105\,B\,b^2\,c\,d^4\,e^2}{2}-30\,A\,b^2\,c\,d^3\,e^3-81\,B\,b\,c^2\,d^5\,e+\frac {105\,A\,b\,c^2\,d^4\,e^2}{2}+\frac {77\,B\,c^3\,d^6}{2}-27\,A\,c^3\,d^5\,e\right )+x^2\,\left (-6\,B\,b^3\,d^2\,e^4+3\,A\,b^3\,d\,e^5+30\,B\,b^2\,c\,d^3\,e^3-18\,A\,b^2\,c\,d^2\,e^4-45\,B\,b\,c^2\,d^4\,e^2+30\,A\,b\,c^2\,d^3\,e^3+21\,B\,c^3\,d^5\,e-15\,A\,c^3\,d^4\,e^2\right )+\frac {-26\,B\,b^3\,d^4\,e^3+11\,A\,b^3\,d^3\,e^4+141\,B\,b^2\,c\,d^5\,e^2-78\,A\,b^2\,c\,d^4\,e^3-222\,B\,b\,c^2\,d^6\,e+141\,A\,b\,c^2\,d^5\,e^2+107\,B\,c^3\,d^7-74\,A\,c^3\,d^6\,e}{6\,e}}{d^3\,e^7+3\,d^2\,e^8\,x+3\,d\,e^9\,x^2+e^{10}\,x^3}+x\,\left (\frac {B\,b^3+3\,A\,c\,b^2}{e^4}+\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{e^4}+\frac {6\,B\,c^3\,d^2}{e^6}\right )}{e}-\frac {6\,d^2\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e^2}-\frac {4\,B\,c^3\,d^3}{e^7}\right )+x^3\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{3\,e^4}-\frac {4\,B\,c^3\,d}{3\,e^5}\right )-x^2\,\left (\frac {2\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{2\,e^4}+\frac {3\,B\,c^3\,d^2}{e^6}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-4\,B\,b^3\,d\,e^3+A\,b^3\,e^4+30\,B\,b^2\,c\,d^2\,e^2-12\,A\,b^2\,c\,d\,e^3-60\,B\,b\,c^2\,d^3\,e+30\,A\,b\,c^2\,d^2\,e^2+35\,B\,c^3\,d^4-20\,A\,c^3\,d^3\,e\right )}{e^8}+\frac {B\,c^3\,x^4}{4\,e^4} \]
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