Integrand size = 20, antiderivative size = 86 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx=-\frac {(B d-A e) (a+b x)^7}{8 e (b d-a e) (d+e x)^8}+\frac {(7 b B d+A b e-8 a B e) (a+b x)^7}{56 e (b d-a e)^2 (d+e x)^7} \]
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Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {79, 37} \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx=\frac {(a+b x)^7 (-8 a B e+A b e+7 b B d)}{56 e (d+e x)^7 (b d-a e)^2}-\frac {(a+b x)^7 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]
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Rule 37
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {(B d-A e) (a+b x)^7}{8 e (b d-a e) (d+e x)^8}+\frac {(7 b B d+A b e-8 a B e) \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{8 e (b d-a e)} \\ & = -\frac {(B d-A e) (a+b x)^7}{8 e (b d-a e) (d+e x)^8}+\frac {(7 b B d+A b e-8 a B e) (a+b x)^7}{56 e (b d-a e)^2 (d+e x)^7} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(597\) vs. \(2(86)=172\).
Time = 0.17 (sec) , antiderivative size = 597, normalized size of antiderivative = 6.94 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx=-\frac {a^6 e^6 (7 A e+B (d+8 e x))+2 a^5 b e^5 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+a^4 b^2 e^4 \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+4 a^3 b^3 e^3 \left (A e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+B \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+a^2 b^4 e^2 \left (3 A e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 B \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )+2 a b^5 e \left (A e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+3 B \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )+b^6 \left (A e \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )+7 B \left (d^7+8 d^6 e x+28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+28 d e^6 x^6+8 e^7 x^7\right )\right )}{56 e^8 (d+e x)^8} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(779\) vs. \(2(82)=164\).
Time = 0.74 (sec) , antiderivative size = 780, normalized size of antiderivative = 9.07
method | result | size |
risch | \(\frac {-\frac {b^{6} B \,x^{7}}{e}-\frac {b^{5} \left (A b e +6 B a e +7 B b d \right ) x^{6}}{2 e^{2}}-\frac {b^{4} \left (2 A a b \,e^{2}+A \,b^{2} d e +5 B \,a^{2} e^{2}+6 B a b d e +7 b^{2} B \,d^{2}\right ) x^{5}}{e^{3}}-\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}+2 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}+5 B \,a^{2} b d \,e^{2}+6 B a \,b^{2} d^{2} e +7 b^{3} B \,d^{3}\right ) x^{4}}{4 e^{4}}-\frac {b^{2} \left (4 A \,a^{3} b \,e^{4}+3 A \,a^{2} b^{2} d \,e^{3}+2 A a \,b^{3} d^{2} e^{2}+A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}+4 B \,a^{3} b d \,e^{3}+5 B \,a^{2} b^{2} d^{2} e^{2}+6 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) x^{3}}{e^{5}}-\frac {b \left (5 A \,a^{4} b \,e^{5}+4 A \,a^{3} b^{2} d \,e^{4}+3 A \,a^{2} b^{3} d^{2} e^{3}+2 A a \,b^{4} d^{3} e^{2}+A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}+3 B \,a^{4} b d \,e^{4}+4 B \,a^{3} b^{2} d^{2} e^{3}+5 B \,a^{2} b^{3} d^{3} e^{2}+6 B a \,b^{4} d^{4} e +7 B \,b^{5} d^{5}\right ) x^{2}}{2 e^{6}}-\frac {\left (6 A \,a^{5} b \,e^{6}+5 A \,a^{4} b^{2} d \,e^{5}+4 A \,a^{3} b^{3} d^{2} e^{4}+3 A \,a^{2} b^{4} d^{3} e^{3}+2 A a \,b^{5} d^{4} e^{2}+A \,b^{6} d^{5} e +B \,a^{6} e^{6}+2 B \,a^{5} b d \,e^{5}+3 B \,a^{4} b^{2} d^{2} e^{4}+4 B \,a^{3} b^{3} d^{3} e^{3}+5 B \,a^{2} b^{4} d^{4} e^{2}+6 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}\right ) x}{7 e^{7}}-\frac {7 A \,a^{6} e^{7}+6 A \,a^{5} b d \,e^{6}+5 A \,a^{4} b^{2} d^{2} e^{5}+4 A \,a^{3} b^{3} d^{3} e^{4}+3 A \,a^{2} b^{4} d^{4} e^{3}+2 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+2 B \,a^{5} b \,d^{2} e^{5}+3 B \,a^{4} b^{2} d^{3} e^{4}+4 B \,a^{3} b^{3} d^{4} e^{3}+5 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{56 e^{8}}}{\left (e x +d \right )^{8}}\) | \(780\) |
norman | \(\frac {-\frac {b^{6} B \,x^{7}}{e}-\frac {\left (A \,b^{6} e +6 B a \,b^{5} e +7 b^{6} B d \right ) x^{6}}{2 e^{2}}-\frac {\left (2 A a \,b^{5} e^{2}+A \,b^{6} d e +5 B \,a^{2} b^{4} e^{2}+6 B a \,b^{5} d e +7 b^{6} B \,d^{2}\right ) x^{5}}{e^{3}}-\frac {5 \left (3 A \,a^{2} b^{4} e^{3}+2 A a \,b^{5} d \,e^{2}+A \,b^{6} d^{2} e +4 B \,a^{3} b^{3} e^{3}+5 B \,a^{2} b^{4} d \,e^{2}+6 B a \,b^{5} d^{2} e +7 b^{6} B \,d^{3}\right ) x^{4}}{4 e^{4}}-\frac {\left (4 A \,a^{3} b^{3} e^{4}+3 A \,a^{2} b^{4} d \,e^{3}+2 A a \,b^{5} d^{2} e^{2}+A \,b^{6} d^{3} e +3 B \,a^{4} b^{2} e^{4}+4 B \,a^{3} b^{3} d \,e^{3}+5 B \,a^{2} b^{4} d^{2} e^{2}+6 B a \,b^{5} d^{3} e +7 b^{6} B \,d^{4}\right ) x^{3}}{e^{5}}-\frac {\left (5 A \,a^{4} b^{2} e^{5}+4 A \,a^{3} b^{3} d \,e^{4}+3 A \,a^{2} b^{4} d^{2} e^{3}+2 A a \,b^{5} d^{3} e^{2}+A \,b^{6} d^{4} e +2 B \,a^{5} b \,e^{5}+3 B \,a^{4} b^{2} d \,e^{4}+4 B \,a^{3} b^{3} d^{2} e^{3}+5 B \,a^{2} b^{4} d^{3} e^{2}+6 B a \,b^{5} d^{4} e +7 b^{6} B \,d^{5}\right ) x^{2}}{2 e^{6}}-\frac {\left (6 A \,a^{5} b \,e^{6}+5 A \,a^{4} b^{2} d \,e^{5}+4 A \,a^{3} b^{3} d^{2} e^{4}+3 A \,a^{2} b^{4} d^{3} e^{3}+2 A a \,b^{5} d^{4} e^{2}+A \,b^{6} d^{5} e +B \,a^{6} e^{6}+2 B \,a^{5} b d \,e^{5}+3 B \,a^{4} b^{2} d^{2} e^{4}+4 B \,a^{3} b^{3} d^{3} e^{3}+5 B \,a^{2} b^{4} d^{4} e^{2}+6 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}\right ) x}{7 e^{7}}-\frac {7 A \,a^{6} e^{7}+6 A \,a^{5} b d \,e^{6}+5 A \,a^{4} b^{2} d^{2} e^{5}+4 A \,a^{3} b^{3} d^{3} e^{4}+3 A \,a^{2} b^{4} d^{4} e^{3}+2 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+2 B \,a^{5} b \,d^{2} e^{5}+3 B \,a^{4} b^{2} d^{3} e^{4}+4 B \,a^{3} b^{3} d^{4} e^{3}+5 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{56 e^{8}}}{\left (e x +d \right )^{8}}\) | \(800\) |
default | \(-\frac {A \,a^{6} e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{8 e^{8} \left (e x +d \right )^{8}}-\frac {b^{6} B}{e^{8} \left (e x +d \right )}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{7 e^{8} \left (e x +d \right )^{7}}-\frac {b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{e^{8} \left (e x +d \right )^{5}}-\frac {b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right )}{e^{8} \left (e x +d \right )^{3}}-\frac {b^{5} \left (A b e +6 B a e -7 B b d \right )}{2 e^{8} \left (e x +d \right )^{2}}-\frac {b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{2 e^{8} \left (e x +d \right )^{6}}-\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right )}{4 e^{8} \left (e x +d \right )^{4}}\) | \(814\) |
gosper | \(-\frac {56 B \,x^{7} b^{6} e^{7}+28 A \,x^{6} b^{6} e^{7}+168 B \,x^{6} a \,b^{5} e^{7}+196 B \,x^{6} b^{6} d \,e^{6}+112 A \,x^{5} a \,b^{5} e^{7}+56 A \,x^{5} b^{6} d \,e^{6}+280 B \,x^{5} a^{2} b^{4} e^{7}+336 B \,x^{5} a \,b^{5} d \,e^{6}+392 B \,x^{5} b^{6} d^{2} e^{5}+210 A \,x^{4} a^{2} b^{4} e^{7}+140 A \,x^{4} a \,b^{5} d \,e^{6}+70 A \,x^{4} b^{6} d^{2} e^{5}+280 B \,x^{4} a^{3} b^{3} e^{7}+350 B \,x^{4} a^{2} b^{4} d \,e^{6}+420 B \,x^{4} a \,b^{5} d^{2} e^{5}+490 B \,x^{4} b^{6} d^{3} e^{4}+224 A \,x^{3} a^{3} b^{3} e^{7}+168 A \,x^{3} a^{2} b^{4} d \,e^{6}+112 A \,x^{3} a \,b^{5} d^{2} e^{5}+56 A \,x^{3} b^{6} d^{3} e^{4}+168 B \,x^{3} a^{4} b^{2} e^{7}+224 B \,x^{3} a^{3} b^{3} d \,e^{6}+280 B \,x^{3} a^{2} b^{4} d^{2} e^{5}+336 B \,x^{3} a \,b^{5} d^{3} e^{4}+392 B \,x^{3} b^{6} d^{4} e^{3}+140 A \,x^{2} a^{4} b^{2} e^{7}+112 A \,x^{2} a^{3} b^{3} d \,e^{6}+84 A \,x^{2} a^{2} b^{4} d^{2} e^{5}+56 A \,x^{2} a \,b^{5} d^{3} e^{4}+28 A \,x^{2} b^{6} d^{4} e^{3}+56 B \,x^{2} a^{5} b \,e^{7}+84 B \,x^{2} a^{4} b^{2} d \,e^{6}+112 B \,x^{2} a^{3} b^{3} d^{2} e^{5}+140 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+168 B \,x^{2} a \,b^{5} d^{4} e^{3}+196 B \,x^{2} b^{6} d^{5} e^{2}+48 A x \,a^{5} b \,e^{7}+40 A x \,a^{4} b^{2} d \,e^{6}+32 A x \,a^{3} b^{3} d^{2} e^{5}+24 A x \,a^{2} b^{4} d^{3} e^{4}+16 A x a \,b^{5} d^{4} e^{3}+8 A x \,b^{6} d^{5} e^{2}+8 B x \,a^{6} e^{7}+16 B x \,a^{5} b d \,e^{6}+24 B x \,a^{4} b^{2} d^{2} e^{5}+32 B x \,a^{3} b^{3} d^{3} e^{4}+40 B x \,a^{2} b^{4} d^{4} e^{3}+48 B x a \,b^{5} d^{5} e^{2}+56 B x \,b^{6} d^{6} e +7 A \,a^{6} e^{7}+6 A \,a^{5} b d \,e^{6}+5 A \,a^{4} b^{2} d^{2} e^{5}+4 A \,a^{3} b^{3} d^{3} e^{4}+3 A \,a^{2} b^{4} d^{4} e^{3}+2 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+2 B \,a^{5} b \,d^{2} e^{5}+3 B \,a^{4} b^{2} d^{3} e^{4}+4 B \,a^{3} b^{3} d^{4} e^{3}+5 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{56 \left (e x +d \right )^{8} e^{8}}\) | \(911\) |
parallelrisch | \(-\frac {56 B \,x^{7} b^{6} e^{7}+28 A \,x^{6} b^{6} e^{7}+168 B \,x^{6} a \,b^{5} e^{7}+196 B \,x^{6} b^{6} d \,e^{6}+112 A \,x^{5} a \,b^{5} e^{7}+56 A \,x^{5} b^{6} d \,e^{6}+280 B \,x^{5} a^{2} b^{4} e^{7}+336 B \,x^{5} a \,b^{5} d \,e^{6}+392 B \,x^{5} b^{6} d^{2} e^{5}+210 A \,x^{4} a^{2} b^{4} e^{7}+140 A \,x^{4} a \,b^{5} d \,e^{6}+70 A \,x^{4} b^{6} d^{2} e^{5}+280 B \,x^{4} a^{3} b^{3} e^{7}+350 B \,x^{4} a^{2} b^{4} d \,e^{6}+420 B \,x^{4} a \,b^{5} d^{2} e^{5}+490 B \,x^{4} b^{6} d^{3} e^{4}+224 A \,x^{3} a^{3} b^{3} e^{7}+168 A \,x^{3} a^{2} b^{4} d \,e^{6}+112 A \,x^{3} a \,b^{5} d^{2} e^{5}+56 A \,x^{3} b^{6} d^{3} e^{4}+168 B \,x^{3} a^{4} b^{2} e^{7}+224 B \,x^{3} a^{3} b^{3} d \,e^{6}+280 B \,x^{3} a^{2} b^{4} d^{2} e^{5}+336 B \,x^{3} a \,b^{5} d^{3} e^{4}+392 B \,x^{3} b^{6} d^{4} e^{3}+140 A \,x^{2} a^{4} b^{2} e^{7}+112 A \,x^{2} a^{3} b^{3} d \,e^{6}+84 A \,x^{2} a^{2} b^{4} d^{2} e^{5}+56 A \,x^{2} a \,b^{5} d^{3} e^{4}+28 A \,x^{2} b^{6} d^{4} e^{3}+56 B \,x^{2} a^{5} b \,e^{7}+84 B \,x^{2} a^{4} b^{2} d \,e^{6}+112 B \,x^{2} a^{3} b^{3} d^{2} e^{5}+140 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+168 B \,x^{2} a \,b^{5} d^{4} e^{3}+196 B \,x^{2} b^{6} d^{5} e^{2}+48 A x \,a^{5} b \,e^{7}+40 A x \,a^{4} b^{2} d \,e^{6}+32 A x \,a^{3} b^{3} d^{2} e^{5}+24 A x \,a^{2} b^{4} d^{3} e^{4}+16 A x a \,b^{5} d^{4} e^{3}+8 A x \,b^{6} d^{5} e^{2}+8 B x \,a^{6} e^{7}+16 B x \,a^{5} b d \,e^{6}+24 B x \,a^{4} b^{2} d^{2} e^{5}+32 B x \,a^{3} b^{3} d^{3} e^{4}+40 B x \,a^{2} b^{4} d^{4} e^{3}+48 B x a \,b^{5} d^{5} e^{2}+56 B x \,b^{6} d^{6} e +7 A \,a^{6} e^{7}+6 A \,a^{5} b d \,e^{6}+5 A \,a^{4} b^{2} d^{2} e^{5}+4 A \,a^{3} b^{3} d^{3} e^{4}+3 A \,a^{2} b^{4} d^{4} e^{3}+2 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+2 B \,a^{5} b \,d^{2} e^{5}+3 B \,a^{4} b^{2} d^{3} e^{4}+4 B \,a^{3} b^{3} d^{4} e^{3}+5 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{56 \left (e x +d \right )^{8} e^{8}}\) | \(911\) |
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Leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (82) = 164\).
Time = 0.23 (sec) , antiderivative size = 823, normalized size of antiderivative = 9.57 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx=-\frac {56 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 7 \, A a^{6} e^{7} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + {\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} + 28 \, {\left (7 \, B b^{6} d e^{6} + {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 56 \, {\left (7 \, B b^{6} d^{2} e^{5} + {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 70 \, {\left (7 \, B b^{6} d^{3} e^{4} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + {\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} + 56 \, {\left (7 \, B b^{6} d^{4} e^{3} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + {\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} + 28 \, {\left (7 \, B b^{6} d^{5} e^{2} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 8 \, {\left (7 \, B b^{6} d^{6} e + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + {\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}{56 \, {\left (e^{16} x^{8} + 8 \, d e^{15} x^{7} + 28 \, d^{2} e^{14} x^{6} + 56 \, d^{3} e^{13} x^{5} + 70 \, d^{4} e^{12} x^{4} + 56 \, d^{5} e^{11} x^{3} + 28 \, d^{6} e^{10} x^{2} + 8 \, d^{7} e^{9} x + d^{8} e^{8}\right )}} \]
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Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (82) = 164\).
Time = 0.24 (sec) , antiderivative size = 823, normalized size of antiderivative = 9.57 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx=-\frac {56 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 7 \, A a^{6} e^{7} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + {\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} + 28 \, {\left (7 \, B b^{6} d e^{6} + {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 56 \, {\left (7 \, B b^{6} d^{2} e^{5} + {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 70 \, {\left (7 \, B b^{6} d^{3} e^{4} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + {\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} + 56 \, {\left (7 \, B b^{6} d^{4} e^{3} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + {\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} + 28 \, {\left (7 \, B b^{6} d^{5} e^{2} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 8 \, {\left (7 \, B b^{6} d^{6} e + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + {\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}{56 \, {\left (e^{16} x^{8} + 8 \, d e^{15} x^{7} + 28 \, d^{2} e^{14} x^{6} + 56 \, d^{3} e^{13} x^{5} + 70 \, d^{4} e^{12} x^{4} + 56 \, d^{5} e^{11} x^{3} + 28 \, d^{6} e^{10} x^{2} + 8 \, d^{7} e^{9} x + d^{8} e^{8}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (82) = 164\).
Time = 0.28 (sec) , antiderivative size = 910, normalized size of antiderivative = 10.58 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx=-\frac {56 \, B b^{6} e^{7} x^{7} + 196 \, B b^{6} d e^{6} x^{6} + 168 \, B a b^{5} e^{7} x^{6} + 28 \, A b^{6} e^{7} x^{6} + 392 \, B b^{6} d^{2} e^{5} x^{5} + 336 \, B a b^{5} d e^{6} x^{5} + 56 \, A b^{6} d e^{6} x^{5} + 280 \, B a^{2} b^{4} e^{7} x^{5} + 112 \, A a b^{5} e^{7} x^{5} + 490 \, B b^{6} d^{3} e^{4} x^{4} + 420 \, B a b^{5} d^{2} e^{5} x^{4} + 70 \, A b^{6} d^{2} e^{5} x^{4} + 350 \, B a^{2} b^{4} d e^{6} x^{4} + 140 \, A a b^{5} d e^{6} x^{4} + 280 \, B a^{3} b^{3} e^{7} x^{4} + 210 \, A a^{2} b^{4} e^{7} x^{4} + 392 \, B b^{6} d^{4} e^{3} x^{3} + 336 \, B a b^{5} d^{3} e^{4} x^{3} + 56 \, A b^{6} d^{3} e^{4} x^{3} + 280 \, B a^{2} b^{4} d^{2} e^{5} x^{3} + 112 \, A a b^{5} d^{2} e^{5} x^{3} + 224 \, B a^{3} b^{3} d e^{6} x^{3} + 168 \, A a^{2} b^{4} d e^{6} x^{3} + 168 \, B a^{4} b^{2} e^{7} x^{3} + 224 \, A a^{3} b^{3} e^{7} x^{3} + 196 \, B b^{6} d^{5} e^{2} x^{2} + 168 \, B a b^{5} d^{4} e^{3} x^{2} + 28 \, A b^{6} d^{4} e^{3} x^{2} + 140 \, B a^{2} b^{4} d^{3} e^{4} x^{2} + 56 \, A a b^{5} d^{3} e^{4} x^{2} + 112 \, B a^{3} b^{3} d^{2} e^{5} x^{2} + 84 \, A a^{2} b^{4} d^{2} e^{5} x^{2} + 84 \, B a^{4} b^{2} d e^{6} x^{2} + 112 \, A a^{3} b^{3} d e^{6} x^{2} + 56 \, B a^{5} b e^{7} x^{2} + 140 \, A a^{4} b^{2} e^{7} x^{2} + 56 \, B b^{6} d^{6} e x + 48 \, B a b^{5} d^{5} e^{2} x + 8 \, A b^{6} d^{5} e^{2} x + 40 \, B a^{2} b^{4} d^{4} e^{3} x + 16 \, A a b^{5} d^{4} e^{3} x + 32 \, B a^{3} b^{3} d^{3} e^{4} x + 24 \, A a^{2} b^{4} d^{3} e^{4} x + 24 \, B a^{4} b^{2} d^{2} e^{5} x + 32 \, A a^{3} b^{3} d^{2} e^{5} x + 16 \, B a^{5} b d e^{6} x + 40 \, A a^{4} b^{2} d e^{6} x + 8 \, B a^{6} e^{7} x + 48 \, A a^{5} b e^{7} x + 7 \, B b^{6} d^{7} + 6 \, B a b^{5} d^{6} e + A b^{6} d^{6} e + 5 \, B a^{2} b^{4} d^{5} e^{2} + 2 \, A a b^{5} d^{5} e^{2} + 4 \, B a^{3} b^{3} d^{4} e^{3} + 3 \, A a^{2} b^{4} d^{4} e^{3} + 3 \, B a^{4} b^{2} d^{3} e^{4} + 4 \, A a^{3} b^{3} d^{3} e^{4} + 2 \, B a^{5} b d^{2} e^{5} + 5 \, A a^{4} b^{2} d^{2} e^{5} + B a^{6} d e^{6} + 6 \, A a^{5} b d e^{6} + 7 \, A a^{6} e^{7}}{56 \, {\left (e x + d\right )}^{8} e^{8}} \]
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Time = 0.28 (sec) , antiderivative size = 854, normalized size of antiderivative = 9.93 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx=-\frac {\frac {B\,a^6\,d\,e^6+7\,A\,a^6\,e^7+2\,B\,a^5\,b\,d^2\,e^5+6\,A\,a^5\,b\,d\,e^6+3\,B\,a^4\,b^2\,d^3\,e^4+5\,A\,a^4\,b^2\,d^2\,e^5+4\,B\,a^3\,b^3\,d^4\,e^3+4\,A\,a^3\,b^3\,d^3\,e^4+5\,B\,a^2\,b^4\,d^5\,e^2+3\,A\,a^2\,b^4\,d^4\,e^3+6\,B\,a\,b^5\,d^6\,e+2\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+A\,b^6\,d^6\,e}{56\,e^8}+\frac {x\,\left (B\,a^6\,e^6+2\,B\,a^5\,b\,d\,e^5+6\,A\,a^5\,b\,e^6+3\,B\,a^4\,b^2\,d^2\,e^4+5\,A\,a^4\,b^2\,d\,e^5+4\,B\,a^3\,b^3\,d^3\,e^3+4\,A\,a^3\,b^3\,d^2\,e^4+5\,B\,a^2\,b^4\,d^4\,e^2+3\,A\,a^2\,b^4\,d^3\,e^3+6\,B\,a\,b^5\,d^5\,e+2\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+A\,b^6\,d^5\,e\right )}{7\,e^7}+\frac {5\,b^3\,x^4\,\left (4\,B\,a^3\,e^3+5\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3+6\,B\,a\,b^2\,d^2\,e+2\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{4\,e^4}+\frac {b^5\,x^6\,\left (A\,b\,e+6\,B\,a\,e+7\,B\,b\,d\right )}{2\,e^2}+\frac {b\,x^2\,\left (2\,B\,a^5\,e^5+3\,B\,a^4\,b\,d\,e^4+5\,A\,a^4\,b\,e^5+4\,B\,a^3\,b^2\,d^2\,e^3+4\,A\,a^3\,b^2\,d\,e^4+5\,B\,a^2\,b^3\,d^3\,e^2+3\,A\,a^2\,b^3\,d^2\,e^3+6\,B\,a\,b^4\,d^4\,e+2\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+A\,b^5\,d^4\,e\right )}{2\,e^6}+\frac {b^2\,x^3\,\left (3\,B\,a^4\,e^4+4\,B\,a^3\,b\,d\,e^3+4\,A\,a^3\,b\,e^4+5\,B\,a^2\,b^2\,d^2\,e^2+3\,A\,a^2\,b^2\,d\,e^3+6\,B\,a\,b^3\,d^3\,e+2\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+A\,b^4\,d^3\,e\right )}{e^5}+\frac {b^4\,x^5\,\left (5\,B\,a^2\,e^2+6\,B\,a\,b\,d\,e+2\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+A\,b^2\,d\,e\right )}{e^3}+\frac {B\,b^6\,x^7}{e}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \]
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