Integrand size = 20, antiderivative size = 272 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx=-\frac {b^5 (6 b B d-A b e-6 a B e) x}{e^7}+\frac {b^6 B x^2}{2 e^6}+\frac {(b d-a e)^6 (B d-A e)}{5 e^8 (d+e x)^5}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{4 e^8 (d+e x)^4}+\frac {b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{e^8 (d+e x)^3}-\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{2 e^8 (d+e x)^2}+\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{e^8 (d+e x)}+\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) \log (d+e x)}{e^8} \]
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Time = 0.25 (sec) , antiderivative size = 272, 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} \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx=-\frac {b^5 x (-6 a B e-A b e+6 b B d)}{e^7}+\frac {3 b^4 (b d-a e) \log (d+e x) (-5 a B e-2 A b e+7 b B d)}{e^8}+\frac {5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{e^8 (d+e x)}-\frac {5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{2 e^8 (d+e x)^2}+\frac {b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8 (d+e x)^3}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{4 e^8 (d+e x)^4}+\frac {(b d-a e)^6 (B d-A e)}{5 e^8 (d+e x)^5}+\frac {b^6 B x^2}{2 e^6} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b^5 (-6 b B d+A b e+6 a B e)}{e^7}+\frac {b^6 B x}{e^6}+\frac {(-b d+a e)^6 (-B d+A e)}{e^7 (d+e x)^6}+\frac {(-b d+a e)^5 (-7 b B d+6 A b e+a B e)}{e^7 (d+e x)^5}+\frac {3 b (b d-a e)^4 (-7 b B d+5 A b e+2 a B e)}{e^7 (d+e x)^4}-\frac {5 b^2 (b d-a e)^3 (-7 b B d+4 A b e+3 a B e)}{e^7 (d+e x)^3}+\frac {5 b^3 (b d-a e)^2 (-7 b B d+3 A b e+4 a B e)}{e^7 (d+e x)^2}-\frac {3 b^4 (b d-a e) (-7 b B d+2 A b e+5 a B e)}{e^7 (d+e x)}\right ) \, dx \\ & = -\frac {b^5 (6 b B d-A b e-6 a B e) x}{e^7}+\frac {b^6 B x^2}{2 e^6}+\frac {(b d-a e)^6 (B d-A e)}{5 e^8 (d+e x)^5}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{4 e^8 (d+e x)^4}+\frac {b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{e^8 (d+e x)^3}-\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{2 e^8 (d+e x)^2}+\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{e^8 (d+e x)}+\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) \log (d+e x)}{e^8} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(633\) vs. \(2(272)=544\).
Time = 0.21 (sec) , antiderivative size = 633, normalized size of antiderivative = 2.33 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx=\frac {-a^6 e^6 (4 A e+B (d+5 e x))-2 a^5 b e^5 \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )-5 a^4 b^2 e^4 \left (2 A e \left (d^2+5 d e x+10 e^2 x^2\right )+3 B \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )-20 a^3 b^3 e^3 \left (A e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 B \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+5 a^2 b^4 e^2 \left (-12 A e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )+B d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+2 a b^5 e \left (A d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )-6 B \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )\right )+b^6 \left (-2 A e \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )+B \left (459 d^7+1875 d^6 e x+2700 d^5 e^2 x^2+1300 d^4 e^3 x^3-400 d^3 e^4 x^4-500 d^2 e^5 x^5-70 d e^6 x^6+10 e^7 x^7\right )\right )+60 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^5 \log (d+e x)}{20 e^8 (d+e x)^5} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(799\) vs. \(2(264)=528\).
Time = 0.70 (sec) , antiderivative size = 800, normalized size of antiderivative = 2.94
method | result | size |
default | \(\frac {b^{5} \left (\frac {1}{2} B b e \,x^{2}+A b e x +6 B a e x -6 B b d x \right )}{e^{7}}-\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right )}{e^{8} \left (e x +d \right )}-\frac {A \,a^{6} e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{5 e^{8} \left (e x +d \right )^{5}}-\frac {b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{e^{8} \left (e x +d \right )^{3}}-\frac {5 b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{2 e^{8} \left (e x +d \right )^{2}}+\frac {3 b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right ) \ln \left (e x +d \right )}{e^{8}}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{4 e^{8} \left (e x +d \right )^{4}}\) | \(800\) |
norman | \(\frac {-\frac {4 A \,a^{6} e^{7}+6 A \,a^{5} b d \,e^{6}+10 A \,a^{4} b^{2} d^{2} e^{5}+20 A \,a^{3} b^{3} d^{3} e^{4}+60 A \,a^{2} b^{4} d^{4} e^{3}-274 A a \,b^{5} d^{5} e^{2}+274 A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+4 B \,a^{5} b \,d^{2} e^{5}+15 B \,a^{4} b^{2} d^{3} e^{4}+80 B \,a^{3} b^{3} d^{4} e^{3}-685 B \,a^{2} b^{4} d^{5} e^{2}+1644 B a \,b^{5} d^{6} e -959 b^{6} B \,d^{7}}{20 e^{8}}-\frac {5 \left (3 A \,a^{2} b^{4} e^{3}-6 A a \,b^{5} d \,e^{2}+6 A \,b^{6} d^{2} e +4 B \,a^{3} b^{3} e^{3}-15 B \,a^{2} b^{4} d \,e^{2}+36 B a \,b^{5} d^{2} e -21 b^{6} B \,d^{3}\right ) x^{4}}{e^{4}}-\frac {5 \left (4 A \,a^{3} b^{3} e^{4}+12 A \,a^{2} b^{4} d \,e^{3}-36 A a \,b^{5} d^{2} e^{2}+36 A \,b^{6} d^{3} e +3 B \,a^{4} b^{2} e^{4}+16 B \,a^{3} b^{3} d \,e^{3}-90 B \,a^{2} b^{4} d^{2} e^{2}+216 B a \,b^{5} d^{3} e -126 b^{6} B \,d^{4}\right ) x^{3}}{2 e^{5}}-\frac {\left (10 A \,a^{4} b^{2} e^{5}+20 A \,a^{3} b^{3} d \,e^{4}+60 A \,a^{2} b^{4} d^{2} e^{3}-220 A a \,b^{5} d^{3} e^{2}+220 A \,b^{6} d^{4} e +4 B \,a^{5} b \,e^{5}+15 B \,a^{4} b^{2} d \,e^{4}+80 B \,a^{3} b^{3} d^{2} e^{3}-550 B \,a^{2} b^{4} d^{3} e^{2}+1320 B a \,b^{5} d^{4} e -770 b^{6} B \,d^{5}\right ) x^{2}}{2 e^{6}}-\frac {\left (6 A \,a^{5} b \,e^{6}+10 A \,a^{4} b^{2} d \,e^{5}+20 A \,a^{3} b^{3} d^{2} e^{4}+60 A \,a^{2} b^{4} d^{3} e^{3}-250 A a \,b^{5} d^{4} e^{2}+250 A \,b^{6} d^{5} e +B \,a^{6} e^{6}+4 B \,a^{5} b d \,e^{5}+15 B \,a^{4} b^{2} d^{2} e^{4}+80 B \,a^{3} b^{3} d^{3} e^{3}-625 B \,a^{2} b^{4} d^{4} e^{2}+1500 B a \,b^{5} d^{5} e -875 b^{6} B \,d^{6}\right ) x}{4 e^{7}}+\frac {b^{5} \left (2 A b e +12 B a e -7 B b d \right ) x^{6}}{2 e^{2}}+\frac {b^{6} B \,x^{7}}{2 e}}{\left (e x +d \right )^{5}}+\frac {3 b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(803\) |
risch | \(\frac {b^{6} B \,x^{2}}{2 e^{6}}+\frac {b^{6} A x}{e^{6}}+\frac {6 b^{5} B a x}{e^{6}}-\frac {6 b^{6} B d x}{e^{7}}+\frac {\left (-15 A \,a^{2} b^{4} e^{6}+30 A a \,b^{5} d \,e^{5}-15 A \,b^{6} d^{2} e^{4}-20 B \,a^{3} b^{3} e^{6}+75 B \,a^{2} b^{4} d \,e^{5}-90 B a \,b^{5} d^{2} e^{4}+35 b^{6} B \,d^{3} e^{3}\right ) x^{4}-\frac {5 b^{2} e^{2} \left (4 A \,a^{3} b \,e^{4}+12 A \,a^{2} b^{2} d \,e^{3}-36 A a \,b^{3} d^{2} e^{2}+20 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}+16 B \,a^{3} b d \,e^{3}-90 B \,a^{2} b^{2} d^{2} e^{2}+120 B a \,b^{3} d^{3} e -49 B \,b^{4} d^{4}\right ) x^{3}}{2}-\frac {b e \left (10 A \,a^{4} b \,e^{5}+20 A \,a^{3} b^{2} d \,e^{4}+60 A \,a^{2} b^{3} d^{2} e^{3}-220 A a \,b^{4} d^{3} e^{2}+130 A \,b^{5} d^{4} e +4 B \,a^{5} e^{5}+15 B \,a^{4} b d \,e^{4}+80 B \,a^{3} b^{2} d^{2} e^{3}-550 B \,a^{2} b^{3} d^{3} e^{2}+780 B a \,b^{4} d^{4} e -329 B \,b^{5} d^{5}\right ) x^{2}}{2}+\left (-\frac {3}{2} A \,a^{5} b \,e^{6}-\frac {5}{2} A \,a^{4} b^{2} d \,e^{5}-5 A \,a^{3} b^{3} d^{2} e^{4}-15 A \,a^{2} b^{4} d^{3} e^{3}+\frac {125}{2} A a \,b^{5} d^{4} e^{2}-\frac {77}{2} A \,b^{6} d^{5} e -\frac {1}{4} B \,a^{6} e^{6}-B \,a^{5} b d \,e^{5}-\frac {15}{4} B \,a^{4} b^{2} d^{2} e^{4}-20 B \,a^{3} b^{3} d^{3} e^{3}+\frac {625}{4} B \,a^{2} b^{4} d^{4} e^{2}-231 B a \,b^{5} d^{5} e +\frac {399}{4} b^{6} B \,d^{6}\right ) x -\frac {4 A \,a^{6} e^{7}+6 A \,a^{5} b d \,e^{6}+10 A \,a^{4} b^{2} d^{2} e^{5}+20 A \,a^{3} b^{3} d^{3} e^{4}+60 A \,a^{2} b^{4} d^{4} e^{3}-274 A a \,b^{5} d^{5} e^{2}+174 A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+4 B \,a^{5} b \,d^{2} e^{5}+15 B \,a^{4} b^{2} d^{3} e^{4}+80 B \,a^{3} b^{3} d^{4} e^{3}-685 B \,a^{2} b^{4} d^{5} e^{2}+1044 B a \,b^{5} d^{6} e -459 b^{6} B \,d^{7}}{20 e}}{e^{7} \left (e x +d \right )^{5}}+\frac {6 b^{5} \ln \left (e x +d \right ) A a}{e^{6}}-\frac {6 b^{6} \ln \left (e x +d \right ) A d}{e^{7}}+\frac {15 b^{4} \ln \left (e x +d \right ) B \,a^{2}}{e^{6}}-\frac {36 b^{5} \ln \left (e x +d \right ) B a d}{e^{7}}+\frac {21 b^{6} \ln \left (e x +d \right ) B \,d^{2}}{e^{8}}\) | \(829\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1455\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1157 vs. \(2 (264) = 528\).
Time = 0.25 (sec) , antiderivative size = 1157, normalized size of antiderivative = 4.25 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 814 vs. \(2 (264) = 528\).
Time = 0.24 (sec) , antiderivative size = 814, normalized size of antiderivative = 2.99 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx=\frac {459 \, B b^{6} d^{7} - 4 \, A a^{6} e^{7} - 174 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 137 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} - 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 2 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} - {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 100 \, {\left (7 \, B b^{6} d^{3} e^{4} - 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} - {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 50 \, {\left (49 \, B b^{6} d^{4} e^{3} - 20 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 18 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} - 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} - {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 10 \, {\left (329 \, B b^{6} d^{5} e^{2} - 130 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 110 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} - 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} - 2 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 5 \, {\left (399 \, B b^{6} d^{6} e - 154 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 125 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} - 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} - 2 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} - {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{20 \, {\left (e^{13} x^{5} + 5 \, d e^{12} x^{4} + 10 \, d^{2} e^{11} x^{3} + 10 \, d^{3} e^{10} x^{2} + 5 \, d^{4} e^{9} x + d^{5} e^{8}\right )}} + \frac {B b^{6} e x^{2} - 2 \, {\left (6 \, B b^{6} d - {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} x}{2 \, e^{7}} + \frac {3 \, {\left (7 \, B b^{6} d^{2} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{2}\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. 831 vs. \(2 (264) = 528\).
Time = 0.29 (sec) , antiderivative size = 831, normalized size of antiderivative = 3.06 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx=\frac {3 \, {\left (7 \, B b^{6} d^{2} - 12 \, B a b^{5} d e - 2 \, A b^{6} d e + 5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} + \frac {B b^{6} e^{6} x^{2} - 12 \, B b^{6} d e^{5} x + 12 \, B a b^{5} e^{6} x + 2 \, A b^{6} e^{6} x}{2 \, e^{12}} + \frac {459 \, B b^{6} d^{7} - 1044 \, B a b^{5} d^{6} e - 174 \, A b^{6} d^{6} e + 685 \, B a^{2} b^{4} d^{5} e^{2} + 274 \, A a b^{5} d^{5} e^{2} - 80 \, B a^{3} b^{3} d^{4} e^{3} - 60 \, A a^{2} b^{4} d^{4} e^{3} - 15 \, B a^{4} b^{2} d^{3} e^{4} - 20 \, A a^{3} b^{3} d^{3} e^{4} - 4 \, B a^{5} b d^{2} e^{5} - 10 \, A a^{4} b^{2} d^{2} e^{5} - B a^{6} d e^{6} - 6 \, A a^{5} b d e^{6} - 4 \, A a^{6} e^{7} + 100 \, {\left (7 \, B b^{6} d^{3} e^{4} - 18 \, B a b^{5} d^{2} e^{5} - 3 \, A b^{6} d^{2} e^{5} + 15 \, B a^{2} b^{4} d e^{6} + 6 \, A a b^{5} d e^{6} - 4 \, B a^{3} b^{3} e^{7} - 3 \, A a^{2} b^{4} e^{7}\right )} x^{4} + 50 \, {\left (49 \, B b^{6} d^{4} e^{3} - 120 \, B a b^{5} d^{3} e^{4} - 20 \, A b^{6} d^{3} e^{4} + 90 \, B a^{2} b^{4} d^{2} e^{5} + 36 \, A a b^{5} d^{2} e^{5} - 16 \, B a^{3} b^{3} d e^{6} - 12 \, A a^{2} b^{4} d e^{6} - 3 \, B a^{4} b^{2} e^{7} - 4 \, A a^{3} b^{3} e^{7}\right )} x^{3} + 10 \, {\left (329 \, B b^{6} d^{5} e^{2} - 780 \, B a b^{5} d^{4} e^{3} - 130 \, A b^{6} d^{4} e^{3} + 550 \, B a^{2} b^{4} d^{3} e^{4} + 220 \, A a b^{5} d^{3} e^{4} - 80 \, B a^{3} b^{3} d^{2} e^{5} - 60 \, A a^{2} b^{4} d^{2} e^{5} - 15 \, B a^{4} b^{2} d e^{6} - 20 \, A a^{3} b^{3} d e^{6} - 4 \, B a^{5} b e^{7} - 10 \, A a^{4} b^{2} e^{7}\right )} x^{2} + 5 \, {\left (399 \, B b^{6} d^{6} e - 924 \, B a b^{5} d^{5} e^{2} - 154 \, A b^{6} d^{5} e^{2} + 625 \, B a^{2} b^{4} d^{4} e^{3} + 250 \, A a b^{5} d^{4} e^{3} - 80 \, B a^{3} b^{3} d^{3} e^{4} - 60 \, A a^{2} b^{4} d^{3} e^{4} - 15 \, B a^{4} b^{2} d^{2} e^{5} - 20 \, A a^{3} b^{3} d^{2} e^{5} - 4 \, B a^{5} b d e^{6} - 10 \, A a^{4} b^{2} d e^{6} - B a^{6} e^{7} - 6 \, A a^{5} b e^{7}\right )} x}{20 \, {\left (e x + d\right )}^{5} e^{8}} \]
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Time = 0.40 (sec) , antiderivative size = 862, normalized size of antiderivative = 3.17 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^6} \, dx=x\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^6}-\frac {6\,B\,b^6\,d}{e^7}\right )-\frac {x^3\,\left (\frac {15\,B\,a^4\,b^2\,e^6}{2}+40\,B\,a^3\,b^3\,d\,e^5+10\,A\,a^3\,b^3\,e^6-225\,B\,a^2\,b^4\,d^2\,e^4+30\,A\,a^2\,b^4\,d\,e^5+300\,B\,a\,b^5\,d^3\,e^3-90\,A\,a\,b^5\,d^2\,e^4-\frac {245\,B\,b^6\,d^4\,e^2}{2}+50\,A\,b^6\,d^3\,e^3\right )+\frac {B\,a^6\,d\,e^6+4\,A\,a^6\,e^7+4\,B\,a^5\,b\,d^2\,e^5+6\,A\,a^5\,b\,d\,e^6+15\,B\,a^4\,b^2\,d^3\,e^4+10\,A\,a^4\,b^2\,d^2\,e^5+80\,B\,a^3\,b^3\,d^4\,e^3+20\,A\,a^3\,b^3\,d^3\,e^4-685\,B\,a^2\,b^4\,d^5\,e^2+60\,A\,a^2\,b^4\,d^4\,e^3+1044\,B\,a\,b^5\,d^6\,e-274\,A\,a\,b^5\,d^5\,e^2-459\,B\,b^6\,d^7+174\,A\,b^6\,d^6\,e}{20\,e}+x\,\left (\frac {B\,a^6\,e^6}{4}+B\,a^5\,b\,d\,e^5+\frac {3\,A\,a^5\,b\,e^6}{2}+\frac {15\,B\,a^4\,b^2\,d^2\,e^4}{4}+\frac {5\,A\,a^4\,b^2\,d\,e^5}{2}+20\,B\,a^3\,b^3\,d^3\,e^3+5\,A\,a^3\,b^3\,d^2\,e^4-\frac {625\,B\,a^2\,b^4\,d^4\,e^2}{4}+15\,A\,a^2\,b^4\,d^3\,e^3+231\,B\,a\,b^5\,d^5\,e-\frac {125\,A\,a\,b^5\,d^4\,e^2}{2}-\frac {399\,B\,b^6\,d^6}{4}+\frac {77\,A\,b^6\,d^5\,e}{2}\right )+x^2\,\left (2\,B\,a^5\,b\,e^6+\frac {15\,B\,a^4\,b^2\,d\,e^5}{2}+5\,A\,a^4\,b^2\,e^6+40\,B\,a^3\,b^3\,d^2\,e^4+10\,A\,a^3\,b^3\,d\,e^5-275\,B\,a^2\,b^4\,d^3\,e^3+30\,A\,a^2\,b^4\,d^2\,e^4+390\,B\,a\,b^5\,d^4\,e^2-110\,A\,a\,b^5\,d^3\,e^3-\frac {329\,B\,b^6\,d^5\,e}{2}+65\,A\,b^6\,d^4\,e^2\right )+x^4\,\left (20\,B\,a^3\,b^3\,e^6-75\,B\,a^2\,b^4\,d\,e^5+15\,A\,a^2\,b^4\,e^6+90\,B\,a\,b^5\,d^2\,e^4-30\,A\,a\,b^5\,d\,e^5-35\,B\,b^6\,d^3\,e^3+15\,A\,b^6\,d^2\,e^4\right )}{d^5\,e^7+5\,d^4\,e^8\,x+10\,d^3\,e^9\,x^2+10\,d^2\,e^{10}\,x^3+5\,d\,e^{11}\,x^4+e^{12}\,x^5}+\frac {\ln \left (d+e\,x\right )\,\left (15\,B\,a^2\,b^4\,e^2-36\,B\,a\,b^5\,d\,e+6\,A\,a\,b^5\,e^2+21\,B\,b^6\,d^2-6\,A\,b^6\,d\,e\right )}{e^8}+\frac {B\,b^6\,x^2}{2\,e^6} \]
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