Integrand size = 20, antiderivative size = 276 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx=\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) x}{e^7}+\frac {(b d-a e)^6 (B d-A e)}{2 e^8 (d+e x)^2}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{e^8 (d+e x)}-\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^2}{2 e^8}+\frac {b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^3}{e^8}-\frac {b^5 (7 b B d-A b e-6 a B e) (d+e x)^4}{4 e^8}+\frac {b^6 B (d+e x)^5}{5 e^8}-\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) \log (d+e x)}{e^8} \]
[Out]
Time = 0.34 (sec) , antiderivative size = 276, 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} \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx=-\frac {b^5 (d+e x)^4 (-6 a B e-A b e+7 b B d)}{4 e^8}+\frac {b^4 (d+e x)^3 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^8}-\frac {5 b^3 (d+e x)^2 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{2 e^8}+\frac {5 b^2 x (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^7}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8 (d+e x)}+\frac {(b d-a e)^6 (B d-A e)}{2 e^8 (d+e x)^2}-\frac {3 b (b d-a e)^4 \log (d+e x) (-2 a B e-5 A b e+7 b B d)}{e^8}+\frac {b^6 B (d+e x)^5}{5 e^8} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {5 b^2 (b d-a e)^3 (-7 b B d+4 A b e+3 a B e)}{e^7}+\frac {(-b d+a e)^6 (-B d+A e)}{e^7 (d+e x)^3}+\frac {(-b d+a e)^5 (-7 b B d+6 A b e+a B e)}{e^7 (d+e x)^2}+\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)}+\frac {5 b^3 (b d-a e)^2 (-7 b B d+3 A b e+4 a B e) (d+e x)}{e^7}-\frac {3 b^4 (b d-a e) (-7 b B d+2 A b e+5 a B e) (d+e x)^2}{e^7}+\frac {b^5 (-7 b B d+A b e+6 a B e) (d+e x)^3}{e^7}+\frac {b^6 B (d+e x)^4}{e^7}\right ) \, dx \\ & = \frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) x}{e^7}+\frac {(b d-a e)^6 (B d-A e)}{2 e^8 (d+e x)^2}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{e^8 (d+e x)}-\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^2}{2 e^8}+\frac {b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^3}{e^8}-\frac {b^5 (7 b B d-A b e-6 a B e) (d+e x)^4}{4 e^8}+\frac {b^6 B (d+e x)^5}{5 e^8}-\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) \log (d+e x)}{e^8} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.28 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx=\frac {-20 b^2 e \left (-15 a^4 B e^4+12 a b^3 d^2 e (5 B d-3 A e)-5 b^4 d^3 (3 B d-2 A e)-20 a^3 b e^3 (-3 B d+A e)+45 a^2 b^2 d e^2 (-2 B d+A e)\right ) x+10 b^3 e^2 \left (20 a^3 B e^3+18 a b^2 d e (2 B d-A e)+15 a^2 b e^2 (-3 B d+A e)+2 b^3 d^2 (-5 B d+3 A e)\right ) x^2-20 b^4 e^3 \left (-5 a^2 B e^2-2 a b e (-3 B d+A e)+b^2 d (-2 B d+A e)\right ) x^3+5 b^5 e^4 (-3 b B d+A b e+6 a B e) x^4+4 b^6 B e^5 x^5+\frac {10 (b d-a e)^6 (B d-A e)}{(d+e x)^2}-\frac {20 (b d-a e)^5 (7 b B d-6 A b e-a B e)}{d+e x}-60 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) \log (d+e x)}{20 e^8} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(788\) vs. \(2(268)=536\).
Time = 0.70 (sec) , antiderivative size = 789, normalized size of antiderivative = 2.86
method | result | size |
norman | \(\frac {\frac {b^{2} \left (20 A \,a^{3} b \,e^{4}-30 A \,a^{2} b^{2} d \,e^{3}+20 A a \,b^{3} d^{2} e^{2}-5 A \,b^{4} d^{3} e +15 B \,a^{4} e^{4}-40 B \,a^{3} b d \,e^{3}+50 B \,a^{2} b^{2} d^{2} e^{2}-30 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) x^{3}}{e^{5}}-\frac {A \,a^{6} e^{7}+6 A \,a^{5} b d \,e^{6}-45 A \,a^{4} b^{2} d^{2} e^{5}+180 A \,a^{3} b^{3} d^{3} e^{4}-270 A \,a^{2} b^{4} d^{4} e^{3}+180 A a \,b^{5} d^{5} e^{2}-45 A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}-18 B \,a^{5} b \,d^{2} e^{5}+135 B \,a^{4} b^{2} d^{3} e^{4}-360 B \,a^{3} b^{3} d^{4} e^{3}+450 B \,a^{2} b^{4} d^{5} e^{2}-270 B a \,b^{5} d^{6} e +63 b^{6} B \,d^{7}}{2 e^{8}}-\frac {\left (6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+120 A \,a^{3} b^{3} d^{2} e^{4}-180 A \,a^{2} b^{4} d^{3} e^{3}+120 A a \,b^{5} d^{4} e^{2}-30 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+90 B \,a^{4} b^{2} d^{2} e^{4}-240 B \,a^{3} b^{3} d^{3} e^{3}+300 B \,a^{2} b^{4} d^{4} e^{2}-180 B a \,b^{5} d^{5} e +42 b^{6} B \,d^{6}\right ) x}{e^{7}}+\frac {b^{3} \left (30 A \,a^{2} b \,e^{3}-20 A a \,b^{2} d \,e^{2}+5 A \,b^{3} d^{2} e +40 B \,a^{3} e^{3}-50 B \,a^{2} b d \,e^{2}+30 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right ) x^{4}}{4 e^{4}}+\frac {b^{4} \left (20 A a b \,e^{2}-5 A \,b^{2} d e +50 B \,a^{2} e^{2}-30 B a b d e +7 b^{2} B \,d^{2}\right ) x^{5}}{10 e^{3}}+\frac {b^{5} \left (5 A b e +30 B a e -7 B b d \right ) x^{6}}{20 e^{2}}+\frac {b^{6} B \,x^{7}}{5 e}}{\left (e x +d \right )^{2}}+\frac {3 b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(789\) |
default | \(\frac {b^{2} \left (18 B a \,b^{3} d^{2} e^{2} x^{2}-45 A \,a^{2} b^{2} d \,e^{3} x +36 A a \,b^{3} d^{2} e^{2} x -9 A a \,b^{3} d \,e^{3} x^{2}-\frac {45}{2} B \,a^{2} b^{2} d \,e^{3} x^{2}+\frac {1}{4} A \,b^{4} e^{4} x^{4}+\frac {15}{2} A \,a^{2} b^{2} e^{4} x^{2}+3 A \,b^{4} d^{2} e^{2} x^{2}+10 B \,a^{3} b \,e^{4} x^{2}-5 B \,b^{4} d^{3} e \,x^{2}+20 A \,a^{3} b \,e^{4} x -10 A \,b^{4} d^{3} e x +2 A a \,b^{3} e^{4} x^{3}-A \,b^{4} d \,e^{3} x^{3}+5 B \,a^{2} b^{2} e^{4} x^{3}+2 B \,b^{4} d^{2} e^{2} x^{3}+\frac {3}{2} B a \,b^{3} e^{4} x^{4}-\frac {3}{4} B \,b^{4} d \,e^{3} x^{4}+\frac {1}{5} b^{4} B \,x^{5} e^{4}+15 B \,a^{4} e^{4} x +15 B \,b^{4} d^{4} x -6 B a \,b^{3} d \,e^{3} x^{3}-60 B \,a^{3} b d \,e^{3} x +90 B \,a^{2} b^{2} d^{2} e^{2} x -60 B a \,b^{3} d^{3} e x \right )}{e^{7}}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{e^{8} \left (e x +d \right )}-\frac {A \,a^{6} e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{2 e^{8} \left (e x +d \right )^{2}}+\frac {3 b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(846\) |
risch | \(\frac {b^{6} A \,x^{4}}{4 e^{3}}+\frac {b^{6} B \,x^{5}}{5 e^{3}}-\frac {45 b^{2} \ln \left (e x +d \right ) B \,a^{4} d}{e^{4}}+\frac {90 b^{4} B \,a^{2} d^{2} x}{e^{5}}-\frac {60 b^{5} B a \,d^{3} x}{e^{6}}+\frac {120 b^{3} \ln \left (e x +d \right ) B \,a^{3} d^{2}}{e^{5}}-\frac {150 b^{4} \ln \left (e x +d \right ) B \,a^{2} d^{3}}{e^{6}}+\frac {90 b^{5} \ln \left (e x +d \right ) B a \,d^{4}}{e^{7}}+\frac {5 b^{4} B \,a^{2} x^{3}}{e^{3}}+\frac {2 b^{6} B \,d^{2} x^{3}}{e^{5}}+\frac {3 b^{5} B a \,x^{4}}{2 e^{3}}-\frac {3 b^{6} B d \,x^{4}}{4 e^{4}}+\frac {15 b^{2} B \,a^{4} x}{e^{3}}+\frac {15 b^{6} B \,d^{4} x}{e^{7}}+\frac {15 b^{2} \ln \left (e x +d \right ) A \,a^{4}}{e^{3}}+\frac {15 b^{6} \ln \left (e x +d \right ) A \,d^{4}}{e^{7}}+\frac {6 b \ln \left (e x +d \right ) B \,a^{5}}{e^{3}}-\frac {21 b^{6} \ln \left (e x +d \right ) B \,d^{5}}{e^{8}}+\frac {\left (-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}\right ) x -\frac {A \,a^{6} e^{7}+6 A \,a^{5} b d \,e^{6}-45 A \,a^{4} b^{2} d^{2} e^{5}+100 A \,a^{3} b^{3} d^{3} e^{4}-105 A \,a^{2} b^{4} d^{4} e^{3}+54 A a \,b^{5} d^{5} e^{2}-11 A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}-18 B \,a^{5} b \,d^{2} e^{5}+75 B \,a^{4} b^{2} d^{3} e^{4}-140 B \,a^{3} b^{3} d^{4} e^{3}+135 B \,a^{2} b^{4} d^{5} e^{2}-66 B a \,b^{5} d^{6} e +13 b^{6} B \,d^{7}}{2 e}}{e^{7} \left (e x +d \right )^{2}}-\frac {60 b^{3} \ln \left (e x +d \right ) A \,a^{3} d}{e^{4}}+\frac {90 b^{4} \ln \left (e x +d \right ) A \,a^{2} d^{2}}{e^{5}}-\frac {60 b^{5} \ln \left (e x +d \right ) A a \,d^{3}}{e^{6}}+\frac {10 b^{3} B \,a^{3} x^{2}}{e^{3}}-\frac {5 b^{6} B \,d^{3} x^{2}}{e^{6}}+\frac {20 b^{3} A \,a^{3} x}{e^{3}}-\frac {10 b^{6} A \,d^{3} x}{e^{6}}+\frac {2 b^{5} A a \,x^{3}}{e^{3}}-\frac {b^{6} A d \,x^{3}}{e^{4}}+\frac {15 b^{4} A \,a^{2} x^{2}}{2 e^{3}}+\frac {3 b^{6} A \,d^{2} x^{2}}{e^{5}}+\frac {18 b^{5} B a \,d^{2} x^{2}}{e^{5}}-\frac {45 b^{4} A \,a^{2} d x}{e^{4}}+\frac {36 b^{5} A a \,d^{2} x}{e^{5}}-\frac {9 b^{5} A a d \,x^{2}}{e^{4}}-\frac {45 b^{4} B \,a^{2} d \,x^{2}}{2 e^{4}}-\frac {6 b^{5} B a d \,x^{3}}{e^{4}}-\frac {60 b^{3} B \,a^{3} d x}{e^{4}}\) | \(917\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1415\) |
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[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1177 vs. \(2 (268) = 536\).
Time = 0.24 (sec) , antiderivative size = 1177, normalized size of antiderivative = 4.26 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx=\text {Too large to display} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 821 vs. \(2 (280) = 560\).
Time = 7.20 (sec) , antiderivative size = 821, normalized size of antiderivative = 2.97 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx=\frac {B b^{6} x^{5}}{5 e^{3}} + \frac {3 b \left (a e - b d\right )^{4} \cdot \left (5 A b e + 2 B a e - 7 B b d\right ) \log {\left (d + e x \right )}}{e^{8}} + x^{4} \left (\frac {A b^{6}}{4 e^{3}} + \frac {3 B a b^{5}}{2 e^{3}} - \frac {3 B b^{6} d}{4 e^{4}}\right ) + x^{3} \cdot \left (\frac {2 A a b^{5}}{e^{3}} - \frac {A b^{6} d}{e^{4}} + \frac {5 B a^{2} b^{4}}{e^{3}} - \frac {6 B a b^{5} d}{e^{4}} + \frac {2 B b^{6} d^{2}}{e^{5}}\right ) + x^{2} \cdot \left (\frac {15 A a^{2} b^{4}}{2 e^{3}} - \frac {9 A a b^{5} d}{e^{4}} + \frac {3 A b^{6} d^{2}}{e^{5}} + \frac {10 B a^{3} b^{3}}{e^{3}} - \frac {45 B a^{2} b^{4} d}{2 e^{4}} + \frac {18 B a b^{5} d^{2}}{e^{5}} - \frac {5 B b^{6} d^{3}}{e^{6}}\right ) + x \left (\frac {20 A a^{3} b^{3}}{e^{3}} - \frac {45 A a^{2} b^{4} d}{e^{4}} + \frac {36 A a b^{5} d^{2}}{e^{5}} - \frac {10 A b^{6} d^{3}}{e^{6}} + \frac {15 B a^{4} b^{2}}{e^{3}} - \frac {60 B a^{3} b^{3} d}{e^{4}} + \frac {90 B a^{2} b^{4} d^{2}}{e^{5}} - \frac {60 B a b^{5} d^{3}}{e^{6}} + \frac {15 B b^{6} d^{4}}{e^{7}}\right ) + \frac {- A a^{6} e^{7} - 6 A a^{5} b d e^{6} + 45 A a^{4} b^{2} d^{2} e^{5} - 100 A a^{3} b^{3} d^{3} e^{4} + 105 A a^{2} b^{4} d^{4} e^{3} - 54 A a b^{5} d^{5} e^{2} + 11 A b^{6} d^{6} e - B a^{6} d e^{6} + 18 B a^{5} b d^{2} e^{5} - 75 B a^{4} b^{2} d^{3} e^{4} + 140 B a^{3} b^{3} d^{4} e^{3} - 135 B a^{2} b^{4} d^{5} e^{2} + 66 B a b^{5} d^{6} e - 13 B b^{6} d^{7} + x \left (- 12 A a^{5} b e^{7} + 60 A a^{4} b^{2} d e^{6} - 120 A a^{3} b^{3} d^{2} e^{5} + 120 A a^{2} b^{4} d^{3} e^{4} - 60 A a b^{5} d^{4} e^{3} + 12 A b^{6} d^{5} e^{2} - 2 B a^{6} e^{7} + 24 B a^{5} b d e^{6} - 90 B a^{4} b^{2} d^{2} e^{5} + 160 B a^{3} b^{3} d^{3} e^{4} - 150 B a^{2} b^{4} d^{4} e^{3} + 72 B a b^{5} d^{5} e^{2} - 14 B b^{6} d^{6} e\right )}{2 d^{2} e^{8} + 4 d e^{9} x + 2 e^{10} x^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 779 vs. \(2 (268) = 536\).
Time = 0.22 (sec) , antiderivative size = 779, normalized size of antiderivative = 2.82 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx=-\frac {13 \, B b^{6} d^{7} + A a^{6} e^{7} - 11 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 27 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 35 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 25 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 9 \, {\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} + 2 \, {\left (7 \, B b^{6} d^{6} e - 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 15 \, {\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} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} - 6 \, {\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}{2 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} + \frac {4 \, B b^{6} e^{4} x^{5} - 5 \, {\left (3 \, B b^{6} d e^{3} - {\left (6 \, B a b^{5} + A b^{6}\right )} e^{4}\right )} x^{4} + 20 \, {\left (2 \, B b^{6} d^{2} e^{2} - {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{3} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{4}\right )} x^{3} - 10 \, {\left (10 \, B b^{6} d^{3} e - 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{2} + 9 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{3} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{4}\right )} x^{2} + 20 \, {\left (15 \, B b^{6} d^{4} - 10 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e + 18 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{2} - 15 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{4}\right )} x}{20 \, e^{7}} - \frac {3 \, {\left (7 \, B b^{6} d^{5} - 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{2} - 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{4} - {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{5}\right )} \log \left (e x + d\right )}{e^{8}} \]
[In]
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Leaf count of result is larger than twice the leaf count of optimal. 862 vs. \(2 (268) = 536\).
Time = 0.28 (sec) , antiderivative size = 862, normalized size of antiderivative = 3.12 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx=-\frac {3 \, {\left (7 \, B b^{6} d^{5} - 30 \, B a b^{5} d^{4} e - 5 \, A b^{6} d^{4} e + 50 \, B a^{2} b^{4} d^{3} e^{2} + 20 \, A a b^{5} d^{3} e^{2} - 40 \, B a^{3} b^{3} d^{2} e^{3} - 30 \, A a^{2} b^{4} d^{2} e^{3} + 15 \, B a^{4} b^{2} d e^{4} + 20 \, A a^{3} b^{3} d e^{4} - 2 \, B a^{5} b e^{5} - 5 \, A a^{4} b^{2} e^{5}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} - \frac {13 \, B b^{6} d^{7} - 66 \, B a b^{5} d^{6} e - 11 \, A b^{6} d^{6} e + 135 \, B a^{2} b^{4} d^{5} e^{2} + 54 \, A a b^{5} d^{5} e^{2} - 140 \, B a^{3} b^{3} d^{4} e^{3} - 105 \, A a^{2} b^{4} d^{4} e^{3} + 75 \, B a^{4} b^{2} d^{3} e^{4} + 100 \, A a^{3} b^{3} d^{3} e^{4} - 18 \, B a^{5} b d^{2} e^{5} - 45 \, A a^{4} b^{2} d^{2} e^{5} + B a^{6} d e^{6} + 6 \, A a^{5} b d e^{6} + A a^{6} e^{7} + 2 \, {\left (7 \, B b^{6} d^{6} e - 36 \, B a b^{5} d^{5} e^{2} - 6 \, A b^{6} d^{5} e^{2} + 75 \, B a^{2} b^{4} d^{4} e^{3} + 30 \, 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} + 45 \, B a^{4} b^{2} d^{2} e^{5} + 60 \, A a^{3} b^{3} d^{2} e^{5} - 12 \, B a^{5} b d e^{6} - 30 \, A a^{4} b^{2} d e^{6} + B a^{6} e^{7} + 6 \, A a^{5} b e^{7}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{8}} + \frac {4 \, B b^{6} e^{12} x^{5} - 15 \, B b^{6} d e^{11} x^{4} + 30 \, B a b^{5} e^{12} x^{4} + 5 \, A b^{6} e^{12} x^{4} + 40 \, B b^{6} d^{2} e^{10} x^{3} - 120 \, B a b^{5} d e^{11} x^{3} - 20 \, A b^{6} d e^{11} x^{3} + 100 \, B a^{2} b^{4} e^{12} x^{3} + 40 \, A a b^{5} e^{12} x^{3} - 100 \, B b^{6} d^{3} e^{9} x^{2} + 360 \, B a b^{5} d^{2} e^{10} x^{2} + 60 \, A b^{6} d^{2} e^{10} x^{2} - 450 \, B a^{2} b^{4} d e^{11} x^{2} - 180 \, A a b^{5} d e^{11} x^{2} + 200 \, B a^{3} b^{3} e^{12} x^{2} + 150 \, A a^{2} b^{4} e^{12} x^{2} + 300 \, B b^{6} d^{4} e^{8} x - 1200 \, B a b^{5} d^{3} e^{9} x - 200 \, A b^{6} d^{3} e^{9} x + 1800 \, B a^{2} b^{4} d^{2} e^{10} x + 720 \, A a b^{5} d^{2} e^{10} x - 1200 \, B a^{3} b^{3} d e^{11} x - 900 \, A a^{2} b^{4} d e^{11} x + 300 \, B a^{4} b^{2} e^{12} x + 400 \, A a^{3} b^{3} e^{12} x}{20 \, e^{15}} \]
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Time = 0.19 (sec) , antiderivative size = 1053, normalized size of antiderivative = 3.82 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx=x\,\left (\frac {3\,d\,\left (\frac {3\,d^2\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^3}-\frac {3\,B\,b^6\,d}{e^4}\right )}{e^2}-\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^3}-\frac {3\,B\,b^6\,d}{e^4}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e^3}+\frac {3\,B\,b^6\,d^2}{e^5}\right )}{e}-\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{e^3}+\frac {B\,b^6\,d^3}{e^6}\right )}{e}-\frac {d^3\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^3}-\frac {3\,B\,b^6\,d}{e^4}\right )}{e^3}+\frac {3\,d^2\,\left (\frac {3\,d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^3}-\frac {3\,B\,b^6\,d}{e^4}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e^3}+\frac {3\,B\,b^6\,d^2}{e^5}\right )}{e^2}+\frac {5\,a^3\,b^2\,\left (4\,A\,b+3\,B\,a\right )}{e^3}\right )-x^3\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^3}-\frac {3\,B\,b^6\,d}{e^4}\right )}{e}-\frac {a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e^3}+\frac {B\,b^6\,d^2}{e^5}\right )-x^2\,\left (\frac {3\,d^2\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^3}-\frac {3\,B\,b^6\,d}{e^4}\right )}{2\,e^2}-\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^3}-\frac {3\,B\,b^6\,d}{e^4}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e^3}+\frac {3\,B\,b^6\,d^2}{e^5}\right )}{2\,e}-\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{2\,e^3}+\frac {B\,b^6\,d^3}{2\,e^6}\right )-\frac {\frac {B\,a^6\,d\,e^6+A\,a^6\,e^7-18\,B\,a^5\,b\,d^2\,e^5+6\,A\,a^5\,b\,d\,e^6+75\,B\,a^4\,b^2\,d^3\,e^4-45\,A\,a^4\,b^2\,d^2\,e^5-140\,B\,a^3\,b^3\,d^4\,e^3+100\,A\,a^3\,b^3\,d^3\,e^4+135\,B\,a^2\,b^4\,d^5\,e^2-105\,A\,a^2\,b^4\,d^4\,e^3-66\,B\,a\,b^5\,d^6\,e+54\,A\,a\,b^5\,d^5\,e^2+13\,B\,b^6\,d^7-11\,A\,b^6\,d^6\,e}{2\,e}+x\,\left (B\,a^6\,e^6-12\,B\,a^5\,b\,d\,e^5+6\,A\,a^5\,b\,e^6+45\,B\,a^4\,b^2\,d^2\,e^4-30\,A\,a^4\,b^2\,d\,e^5-80\,B\,a^3\,b^3\,d^3\,e^3+60\,A\,a^3\,b^3\,d^2\,e^4+75\,B\,a^2\,b^4\,d^4\,e^2-60\,A\,a^2\,b^4\,d^3\,e^3-36\,B\,a\,b^5\,d^5\,e+30\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6-6\,A\,b^6\,d^5\,e\right )}{d^2\,e^7+2\,d\,e^8\,x+e^9\,x^2}+x^4\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{4\,e^3}-\frac {3\,B\,b^6\,d}{4\,e^4}\right )+\frac {\ln \left (d+e\,x\right )\,\left (6\,B\,a^5\,b\,e^5-45\,B\,a^4\,b^2\,d\,e^4+15\,A\,a^4\,b^2\,e^5+120\,B\,a^3\,b^3\,d^2\,e^3-60\,A\,a^3\,b^3\,d\,e^4-150\,B\,a^2\,b^4\,d^3\,e^2+90\,A\,a^2\,b^4\,d^2\,e^3+90\,B\,a\,b^5\,d^4\,e-60\,A\,a\,b^5\,d^3\,e^2-21\,B\,b^6\,d^5+15\,A\,b^6\,d^4\,e\right )}{e^8}+\frac {B\,b^6\,x^5}{5\,e^3} \]
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