Integrand size = 20, antiderivative size = 235 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx=-\frac {(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac {(7 b B d+4 A b e-11 a B e) (a+b x)^7}{110 e (b d-a e)^2 (d+e x)^{10}}+\frac {b (7 b B d+4 A b e-11 a B e) (a+b x)^7}{330 e (b d-a e)^3 (d+e x)^9}+\frac {b^2 (7 b B d+4 A b e-11 a B e) (a+b x)^7}{1320 e (b d-a e)^4 (d+e x)^8}+\frac {b^3 (7 b B d+4 A b e-11 a B e) (a+b x)^7}{9240 e (b d-a e)^5 (d+e x)^7} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx=\frac {b^3 (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{9240 e (d+e x)^7 (b d-a e)^5}+\frac {b^2 (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{1320 e (d+e x)^8 (b d-a e)^4}+\frac {b (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{330 e (d+e x)^9 (b d-a e)^3}+\frac {(a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{110 e (d+e x)^{10} (b d-a e)^2}-\frac {(a+b x)^7 (B d-A e)}{11 e (d+e x)^{11} (b d-a e)} \]
[In]
[Out]
Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac {(7 b B d+4 A b e-11 a B e) \int \frac {(a+b x)^6}{(d+e x)^{11}} \, dx}{11 e (b d-a e)} \\ & = -\frac {(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac {(7 b B d+4 A b e-11 a B e) (a+b x)^7}{110 e (b d-a e)^2 (d+e x)^{10}}+\frac {(3 b (7 b B d+4 A b e-11 a B e)) \int \frac {(a+b x)^6}{(d+e x)^{10}} \, dx}{110 e (b d-a e)^2} \\ & = -\frac {(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac {(7 b B d+4 A b e-11 a B e) (a+b x)^7}{110 e (b d-a e)^2 (d+e x)^{10}}+\frac {b (7 b B d+4 A b e-11 a B e) (a+b x)^7}{330 e (b d-a e)^3 (d+e x)^9}+\frac {\left (b^2 (7 b B d+4 A b e-11 a B e)\right ) \int \frac {(a+b x)^6}{(d+e x)^9} \, dx}{165 e (b d-a e)^3} \\ & = -\frac {(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac {(7 b B d+4 A b e-11 a B e) (a+b x)^7}{110 e (b d-a e)^2 (d+e x)^{10}}+\frac {b (7 b B d+4 A b e-11 a B e) (a+b x)^7}{330 e (b d-a e)^3 (d+e x)^9}+\frac {b^2 (7 b B d+4 A b e-11 a B e) (a+b x)^7}{1320 e (b d-a e)^4 (d+e x)^8}+\frac {\left (b^3 (7 b B d+4 A b e-11 a B e)\right ) \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{1320 e (b d-a e)^4} \\ & = -\frac {(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac {(7 b B d+4 A b e-11 a B e) (a+b x)^7}{110 e (b d-a e)^2 (d+e x)^{10}}+\frac {b (7 b B d+4 A b e-11 a B e) (a+b x)^7}{330 e (b d-a e)^3 (d+e x)^9}+\frac {b^2 (7 b B d+4 A b e-11 a B e) (a+b x)^7}{1320 e (b d-a e)^4 (d+e x)^8}+\frac {b^3 (7 b B d+4 A b e-11 a B e) (a+b x)^7}{9240 e (b d-a e)^5 (d+e x)^7} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(605\) vs. \(2(235)=470\).
Time = 0.18 (sec) , antiderivative size = 605, normalized size of antiderivative = 2.57 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx=-\frac {84 a^6 e^6 (10 A e+B (d+11 e x))+56 a^5 b e^5 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+35 a^4 b^2 e^4 \left (8 A e \left (d^2+11 d e x+55 e^2 x^2\right )+3 B \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )\right )+20 a^3 b^3 e^3 \left (7 A e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+4 B \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )+10 a^2 b^4 e^2 \left (6 A e \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 B \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )\right )+4 a b^5 e \left (5 A e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+6 B \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )\right )+b^6 \left (4 A e \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )+7 B \left (d^7+11 d^6 e x+55 d^5 e^2 x^2+165 d^4 e^3 x^3+330 d^3 e^4 x^4+462 d^2 e^5 x^5+462 d e^6 x^6+330 e^7 x^7\right )\right )}{9240 e^8 (d+e x)^{11}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(788\) vs. \(2(225)=450\).
Time = 0.71 (sec) , antiderivative size = 789, normalized size of antiderivative = 3.36
method | result | size |
risch | \(\frac {-\frac {b^{6} B \,x^{7}}{4 e}-\frac {b^{5} \left (4 A b e +24 B a e +7 B b d \right ) x^{6}}{20 e^{2}}-\frac {b^{4} \left (20 A a b \,e^{2}+4 A \,b^{2} d e +50 B \,a^{2} e^{2}+24 B a b d e +7 b^{2} B \,d^{2}\right ) x^{5}}{20 e^{3}}-\frac {b^{3} \left (60 A \,a^{2} b \,e^{3}+20 A a \,b^{2} d \,e^{2}+4 A \,b^{3} d^{2} e +80 B \,a^{3} e^{3}+50 B \,a^{2} b d \,e^{2}+24 B a \,b^{2} d^{2} e +7 b^{3} B \,d^{3}\right ) x^{4}}{28 e^{4}}-\frac {b^{2} \left (140 A \,a^{3} b \,e^{4}+60 A \,a^{2} b^{2} d \,e^{3}+20 A a \,b^{3} d^{2} e^{2}+4 A \,b^{4} d^{3} e +105 B \,a^{4} e^{4}+80 B \,a^{3} b d \,e^{3}+50 B \,a^{2} b^{2} d^{2} e^{2}+24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) x^{3}}{56 e^{5}}-\frac {b \left (280 A \,a^{4} b \,e^{5}+140 A \,a^{3} b^{2} d \,e^{4}+60 A \,a^{2} b^{3} d^{2} e^{3}+20 A a \,b^{4} d^{3} e^{2}+4 A \,b^{5} d^{4} e +112 B \,a^{5} e^{5}+105 B \,a^{4} b d \,e^{4}+80 B \,a^{3} b^{2} d^{2} e^{3}+50 B \,a^{2} b^{3} d^{3} e^{2}+24 B a \,b^{4} d^{4} e +7 B \,b^{5} d^{5}\right ) x^{2}}{168 e^{6}}-\frac {\left (504 A \,a^{5} b \,e^{6}+280 A \,a^{4} b^{2} d \,e^{5}+140 A \,a^{3} b^{3} d^{2} e^{4}+60 A \,a^{2} b^{4} d^{3} e^{3}+20 A a \,b^{5} d^{4} e^{2}+4 A \,b^{6} d^{5} e +84 B \,a^{6} e^{6}+112 B \,a^{5} b d \,e^{5}+105 B \,a^{4} b^{2} d^{2} e^{4}+80 B \,a^{3} b^{3} d^{3} e^{3}+50 B \,a^{2} b^{4} d^{4} e^{2}+24 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}\right ) x}{840 e^{7}}-\frac {840 A \,a^{6} e^{7}+504 A \,a^{5} b d \,e^{6}+280 A \,a^{4} b^{2} d^{2} e^{5}+140 A \,a^{3} b^{3} d^{3} e^{4}+60 A \,a^{2} b^{4} d^{4} e^{3}+20 A a \,b^{5} d^{5} e^{2}+4 A \,b^{6} d^{6} e +84 B \,a^{6} d \,e^{6}+112 B \,a^{5} b \,d^{2} e^{5}+105 B \,a^{4} b^{2} d^{3} e^{4}+80 B \,a^{3} b^{3} d^{4} e^{3}+50 B \,a^{2} b^{4} d^{5} e^{2}+24 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{9240 e^{8}}}{\left (e x +d \right )^{11}}\) | \(789\) |
default | \(-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{10 e^{8} \left (e x +d \right )^{10}}-\frac {5 b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{8 e^{8} \left (e x +d \right )^{8}}-\frac {A \,a^{6} e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{11 e^{8} \left (e x +d \right )^{11}}-\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right )}{7 e^{8} \left (e x +d \right )^{7}}-\frac {b^{5} \left (A b e +6 B a e -7 B b d \right )}{5 e^{8} \left (e x +d \right )^{5}}-\frac {b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{3 e^{8} \left (e x +d \right )^{9}}-\frac {b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right )}{2 e^{8} \left (e x +d \right )^{6}}-\frac {b^{6} B}{4 e^{8} \left (e x +d \right )^{4}}\) | \(814\) |
norman | \(\frac {-\frac {b^{6} B \,x^{7}}{4 e}-\frac {\left (4 A \,b^{6} e^{4}+24 B a \,b^{5} e^{4}+7 b^{6} B d \,e^{3}\right ) x^{6}}{20 e^{5}}-\frac {\left (20 A a \,b^{5} e^{5}+4 A \,b^{6} d \,e^{4}+50 B \,a^{2} b^{4} e^{5}+24 B a \,b^{5} d \,e^{4}+7 b^{6} B \,d^{2} e^{3}\right ) x^{5}}{20 e^{6}}-\frac {\left (60 A \,a^{2} b^{4} e^{6}+20 A a \,b^{5} d \,e^{5}+4 A \,b^{6} d^{2} e^{4}+80 B \,a^{3} b^{3} e^{6}+50 B \,a^{2} b^{4} d \,e^{5}+24 B a \,b^{5} d^{2} e^{4}+7 b^{6} B \,d^{3} e^{3}\right ) x^{4}}{28 e^{7}}-\frac {\left (140 A \,a^{3} b^{3} e^{7}+60 A \,a^{2} b^{4} d \,e^{6}+20 A a \,b^{5} d^{2} e^{5}+4 A \,b^{6} d^{3} e^{4}+105 B \,a^{4} b^{2} e^{7}+80 B \,a^{3} b^{3} d \,e^{6}+50 B \,a^{2} b^{4} d^{2} e^{5}+24 B a \,b^{5} d^{3} e^{4}+7 b^{6} B \,d^{4} e^{3}\right ) x^{3}}{56 e^{8}}-\frac {\left (280 A \,a^{4} b^{2} e^{8}+140 A \,a^{3} b^{3} d \,e^{7}+60 A \,a^{2} b^{4} d^{2} e^{6}+20 A a \,b^{5} d^{3} e^{5}+4 A \,b^{6} d^{4} e^{4}+112 B \,a^{5} b \,e^{8}+105 B \,a^{4} b^{2} d \,e^{7}+80 B \,a^{3} b^{3} d^{2} e^{6}+50 B \,a^{2} b^{4} d^{3} e^{5}+24 B a \,b^{5} d^{4} e^{4}+7 b^{6} B \,d^{5} e^{3}\right ) x^{2}}{168 e^{9}}-\frac {\left (504 A \,a^{5} b \,e^{9}+280 A \,a^{4} b^{2} d \,e^{8}+140 A \,a^{3} b^{3} d^{2} e^{7}+60 A \,a^{2} b^{4} d^{3} e^{6}+20 A a \,b^{5} d^{4} e^{5}+4 A \,b^{6} d^{5} e^{4}+84 B \,a^{6} e^{9}+112 B \,a^{5} b d \,e^{8}+105 B \,a^{4} b^{2} d^{2} e^{7}+80 B \,a^{3} b^{3} d^{3} e^{6}+50 B \,a^{2} b^{4} d^{4} e^{5}+24 B a \,b^{5} d^{5} e^{4}+7 b^{6} B \,d^{6} e^{3}\right ) x}{840 e^{10}}-\frac {840 A \,a^{6} e^{10}+504 A \,a^{5} b d \,e^{9}+280 A \,a^{4} b^{2} d^{2} e^{8}+140 A \,a^{3} b^{3} d^{3} e^{7}+60 A \,a^{2} b^{4} d^{4} e^{6}+20 A a \,b^{5} d^{5} e^{5}+4 A \,b^{6} d^{6} e^{4}+84 B \,a^{6} d \,e^{9}+112 B \,a^{5} b \,d^{2} e^{8}+105 B \,a^{4} b^{2} d^{3} e^{7}+80 B \,a^{3} b^{3} d^{4} e^{6}+50 B \,a^{2} b^{4} d^{5} e^{5}+24 B a \,b^{5} d^{6} e^{4}+7 B \,b^{6} d^{7} e^{3}}{9240 e^{11}}}{\left (e x +d \right )^{11}}\) | \(858\) |
gosper | \(-\frac {2310 B \,x^{7} b^{6} e^{7}+1848 A \,x^{6} b^{6} e^{7}+11088 B \,x^{6} a \,b^{5} e^{7}+3234 B \,x^{6} b^{6} d \,e^{6}+9240 A \,x^{5} a \,b^{5} e^{7}+1848 A \,x^{5} b^{6} d \,e^{6}+23100 B \,x^{5} a^{2} b^{4} e^{7}+11088 B \,x^{5} a \,b^{5} d \,e^{6}+3234 B \,x^{5} b^{6} d^{2} e^{5}+19800 A \,x^{4} a^{2} b^{4} e^{7}+6600 A \,x^{4} a \,b^{5} d \,e^{6}+1320 A \,x^{4} b^{6} d^{2} e^{5}+26400 B \,x^{4} a^{3} b^{3} e^{7}+16500 B \,x^{4} a^{2} b^{4} d \,e^{6}+7920 B \,x^{4} a \,b^{5} d^{2} e^{5}+2310 B \,x^{4} b^{6} d^{3} e^{4}+23100 A \,x^{3} a^{3} b^{3} e^{7}+9900 A \,x^{3} a^{2} b^{4} d \,e^{6}+3300 A \,x^{3} a \,b^{5} d^{2} e^{5}+660 A \,x^{3} b^{6} d^{3} e^{4}+17325 B \,x^{3} a^{4} b^{2} e^{7}+13200 B \,x^{3} a^{3} b^{3} d \,e^{6}+8250 B \,x^{3} a^{2} b^{4} d^{2} e^{5}+3960 B \,x^{3} a \,b^{5} d^{3} e^{4}+1155 B \,x^{3} b^{6} d^{4} e^{3}+15400 A \,x^{2} a^{4} b^{2} e^{7}+7700 A \,x^{2} a^{3} b^{3} d \,e^{6}+3300 A \,x^{2} a^{2} b^{4} d^{2} e^{5}+1100 A \,x^{2} a \,b^{5} d^{3} e^{4}+220 A \,x^{2} b^{6} d^{4} e^{3}+6160 B \,x^{2} a^{5} b \,e^{7}+5775 B \,x^{2} a^{4} b^{2} d \,e^{6}+4400 B \,x^{2} a^{3} b^{3} d^{2} e^{5}+2750 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+1320 B \,x^{2} a \,b^{5} d^{4} e^{3}+385 B \,x^{2} b^{6} d^{5} e^{2}+5544 A x \,a^{5} b \,e^{7}+3080 A x \,a^{4} b^{2} d \,e^{6}+1540 A x \,a^{3} b^{3} d^{2} e^{5}+660 A x \,a^{2} b^{4} d^{3} e^{4}+220 A x a \,b^{5} d^{4} e^{3}+44 A x \,b^{6} d^{5} e^{2}+924 B x \,a^{6} e^{7}+1232 B x \,a^{5} b d \,e^{6}+1155 B x \,a^{4} b^{2} d^{2} e^{5}+880 B x \,a^{3} b^{3} d^{3} e^{4}+550 B x \,a^{2} b^{4} d^{4} e^{3}+264 B x a \,b^{5} d^{5} e^{2}+77 B x \,b^{6} d^{6} e +840 A \,a^{6} e^{7}+504 A \,a^{5} b d \,e^{6}+280 A \,a^{4} b^{2} d^{2} e^{5}+140 A \,a^{3} b^{3} d^{3} e^{4}+60 A \,a^{2} b^{4} d^{4} e^{3}+20 A a \,b^{5} d^{5} e^{2}+4 A \,b^{6} d^{6} e +84 B \,a^{6} d \,e^{6}+112 B \,a^{5} b \,d^{2} e^{5}+105 B \,a^{4} b^{2} d^{3} e^{4}+80 B \,a^{3} b^{3} d^{4} e^{3}+50 B \,a^{2} b^{4} d^{5} e^{2}+24 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{9240 e^{8} \left (e x +d \right )^{11}}\) | \(913\) |
parallelrisch | \(-\frac {2310 B \,b^{6} x^{7} e^{10}+1848 A \,b^{6} e^{10} x^{6}+11088 B a \,b^{5} e^{10} x^{6}+3234 B \,b^{6} d \,e^{9} x^{6}+9240 A a \,b^{5} e^{10} x^{5}+1848 A \,b^{6} d \,e^{9} x^{5}+23100 B \,a^{2} b^{4} e^{10} x^{5}+11088 B a \,b^{5} d \,e^{9} x^{5}+3234 B \,b^{6} d^{2} e^{8} x^{5}+19800 A \,a^{2} b^{4} e^{10} x^{4}+6600 A a \,b^{5} d \,e^{9} x^{4}+1320 A \,b^{6} d^{2} e^{8} x^{4}+26400 B \,a^{3} b^{3} e^{10} x^{4}+16500 B \,a^{2} b^{4} d \,e^{9} x^{4}+7920 B a \,b^{5} d^{2} e^{8} x^{4}+2310 B \,b^{6} d^{3} e^{7} x^{4}+23100 A \,a^{3} b^{3} e^{10} x^{3}+9900 A \,a^{2} b^{4} d \,e^{9} x^{3}+3300 A a \,b^{5} d^{2} e^{8} x^{3}+660 A \,b^{6} d^{3} e^{7} x^{3}+17325 B \,a^{4} b^{2} e^{10} x^{3}+13200 B \,a^{3} b^{3} d \,e^{9} x^{3}+8250 B \,a^{2} b^{4} d^{2} e^{8} x^{3}+3960 B a \,b^{5} d^{3} e^{7} x^{3}+1155 B \,b^{6} d^{4} e^{6} x^{3}+15400 A \,a^{4} b^{2} e^{10} x^{2}+7700 A \,a^{3} b^{3} d \,e^{9} x^{2}+3300 A \,a^{2} b^{4} d^{2} e^{8} x^{2}+1100 A a \,b^{5} d^{3} e^{7} x^{2}+220 A \,b^{6} d^{4} e^{6} x^{2}+6160 B \,a^{5} b \,e^{10} x^{2}+5775 B \,a^{4} b^{2} d \,e^{9} x^{2}+4400 B \,a^{3} b^{3} d^{2} e^{8} x^{2}+2750 B \,a^{2} b^{4} d^{3} e^{7} x^{2}+1320 B a \,b^{5} d^{4} e^{6} x^{2}+385 B \,b^{6} d^{5} e^{5} x^{2}+5544 A \,a^{5} b \,e^{10} x +3080 A \,a^{4} b^{2} d \,e^{9} x +1540 A \,a^{3} b^{3} d^{2} e^{8} x +660 A \,a^{2} b^{4} d^{3} e^{7} x +220 A a \,b^{5} d^{4} e^{6} x +44 A \,b^{6} d^{5} e^{5} x +924 B \,a^{6} e^{10} x +1232 B \,a^{5} b d \,e^{9} x +1155 B \,a^{4} b^{2} d^{2} e^{8} x +880 B \,a^{3} b^{3} d^{3} e^{7} x +550 B \,a^{2} b^{4} d^{4} e^{6} x +264 B a \,b^{5} d^{5} e^{5} x +77 B \,b^{6} d^{6} e^{4} x +840 A \,a^{6} e^{10}+504 A \,a^{5} b d \,e^{9}+280 A \,a^{4} b^{2} d^{2} e^{8}+140 A \,a^{3} b^{3} d^{3} e^{7}+60 A \,a^{2} b^{4} d^{4} e^{6}+20 A a \,b^{5} d^{5} e^{5}+4 A \,b^{6} d^{6} e^{4}+84 B \,a^{6} d \,e^{9}+112 B \,a^{5} b \,d^{2} e^{8}+105 B \,a^{4} b^{2} d^{3} e^{7}+80 B \,a^{3} b^{3} d^{4} e^{6}+50 B \,a^{2} b^{4} d^{5} e^{5}+24 B a \,b^{5} d^{6} e^{4}+7 B \,b^{6} d^{7} e^{3}}{9240 e^{11} \left (e x +d \right )^{11}}\) | \(922\) |
[In]
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Leaf count of result is larger than twice the leaf count of optimal. 883 vs. \(2 (225) = 450\).
Time = 0.24 (sec) , antiderivative size = 883, normalized size of antiderivative = 3.76 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx=-\frac {2310 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 840 \, A a^{6} e^{7} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 10 \, {\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} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 56 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 84 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 462 \, {\left (7 \, B b^{6} d e^{6} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 462 \, {\left (7 \, B b^{6} d^{2} e^{5} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 330 \, {\left (7 \, B b^{6} d^{3} e^{4} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 165 \, {\left (7 \, B b^{6} d^{4} e^{3} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 55 \, {\left (7 \, B b^{6} d^{5} e^{2} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 10 \, {\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} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 56 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 11 \, {\left (7 \, B b^{6} d^{6} e + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 10 \, {\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} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 56 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 84 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{9240 \, {\left (e^{19} x^{11} + 11 \, d e^{18} x^{10} + 55 \, d^{2} e^{17} x^{9} + 165 \, d^{3} e^{16} x^{8} + 330 \, d^{4} e^{15} x^{7} + 462 \, d^{5} e^{14} x^{6} + 462 \, d^{6} e^{13} x^{5} + 330 \, d^{7} e^{12} x^{4} + 165 \, d^{8} e^{11} x^{3} + 55 \, d^{9} e^{10} x^{2} + 11 \, d^{10} e^{9} x + d^{11} e^{8}\right )}} \]
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Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 883 vs. \(2 (225) = 450\).
Time = 0.23 (sec) , antiderivative size = 883, normalized size of antiderivative = 3.76 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx=-\frac {2310 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 840 \, A a^{6} e^{7} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 10 \, {\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} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 56 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 84 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 462 \, {\left (7 \, B b^{6} d e^{6} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 462 \, {\left (7 \, B b^{6} d^{2} e^{5} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 330 \, {\left (7 \, B b^{6} d^{3} e^{4} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 165 \, {\left (7 \, B b^{6} d^{4} e^{3} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 55 \, {\left (7 \, B b^{6} d^{5} e^{2} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 10 \, {\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} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 56 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 11 \, {\left (7 \, B b^{6} d^{6} e + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 10 \, {\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} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 56 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 84 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{9240 \, {\left (e^{19} x^{11} + 11 \, d e^{18} x^{10} + 55 \, d^{2} e^{17} x^{9} + 165 \, d^{3} e^{16} x^{8} + 330 \, d^{4} e^{15} x^{7} + 462 \, d^{5} e^{14} x^{6} + 462 \, d^{6} e^{13} x^{5} + 330 \, d^{7} e^{12} x^{4} + 165 \, d^{8} e^{11} x^{3} + 55 \, d^{9} e^{10} x^{2} + 11 \, d^{10} e^{9} x + d^{11} e^{8}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (225) = 450\).
Time = 0.29 (sec) , antiderivative size = 912, normalized size of antiderivative = 3.88 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx=-\frac {2310 \, B b^{6} e^{7} x^{7} + 3234 \, B b^{6} d e^{6} x^{6} + 11088 \, B a b^{5} e^{7} x^{6} + 1848 \, A b^{6} e^{7} x^{6} + 3234 \, B b^{6} d^{2} e^{5} x^{5} + 11088 \, B a b^{5} d e^{6} x^{5} + 1848 \, A b^{6} d e^{6} x^{5} + 23100 \, B a^{2} b^{4} e^{7} x^{5} + 9240 \, A a b^{5} e^{7} x^{5} + 2310 \, B b^{6} d^{3} e^{4} x^{4} + 7920 \, B a b^{5} d^{2} e^{5} x^{4} + 1320 \, A b^{6} d^{2} e^{5} x^{4} + 16500 \, B a^{2} b^{4} d e^{6} x^{4} + 6600 \, A a b^{5} d e^{6} x^{4} + 26400 \, B a^{3} b^{3} e^{7} x^{4} + 19800 \, A a^{2} b^{4} e^{7} x^{4} + 1155 \, B b^{6} d^{4} e^{3} x^{3} + 3960 \, B a b^{5} d^{3} e^{4} x^{3} + 660 \, A b^{6} d^{3} e^{4} x^{3} + 8250 \, B a^{2} b^{4} d^{2} e^{5} x^{3} + 3300 \, A a b^{5} d^{2} e^{5} x^{3} + 13200 \, B a^{3} b^{3} d e^{6} x^{3} + 9900 \, A a^{2} b^{4} d e^{6} x^{3} + 17325 \, B a^{4} b^{2} e^{7} x^{3} + 23100 \, A a^{3} b^{3} e^{7} x^{3} + 385 \, B b^{6} d^{5} e^{2} x^{2} + 1320 \, B a b^{5} d^{4} e^{3} x^{2} + 220 \, A b^{6} d^{4} e^{3} x^{2} + 2750 \, B a^{2} b^{4} d^{3} e^{4} x^{2} + 1100 \, A a b^{5} d^{3} e^{4} x^{2} + 4400 \, B a^{3} b^{3} d^{2} e^{5} x^{2} + 3300 \, A a^{2} b^{4} d^{2} e^{5} x^{2} + 5775 \, B a^{4} b^{2} d e^{6} x^{2} + 7700 \, A a^{3} b^{3} d e^{6} x^{2} + 6160 \, B a^{5} b e^{7} x^{2} + 15400 \, A a^{4} b^{2} e^{7} x^{2} + 77 \, B b^{6} d^{6} e x + 264 \, B a b^{5} d^{5} e^{2} x + 44 \, A b^{6} d^{5} e^{2} x + 550 \, B a^{2} b^{4} d^{4} e^{3} x + 220 \, A a b^{5} d^{4} e^{3} x + 880 \, B a^{3} b^{3} d^{3} e^{4} x + 660 \, A a^{2} b^{4} d^{3} e^{4} x + 1155 \, B a^{4} b^{2} d^{2} e^{5} x + 1540 \, A a^{3} b^{3} d^{2} e^{5} x + 1232 \, B a^{5} b d e^{6} x + 3080 \, A a^{4} b^{2} d e^{6} x + 924 \, B a^{6} e^{7} x + 5544 \, A a^{5} b e^{7} x + 7 \, B b^{6} d^{7} + 24 \, B a b^{5} d^{6} e + 4 \, A b^{6} d^{6} e + 50 \, B a^{2} b^{4} d^{5} e^{2} + 20 \, 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} + 105 \, B a^{4} b^{2} d^{3} e^{4} + 140 \, A a^{3} b^{3} d^{3} e^{4} + 112 \, B a^{5} b d^{2} e^{5} + 280 \, A a^{4} b^{2} d^{2} e^{5} + 84 \, B a^{6} d e^{6} + 504 \, A a^{5} b d e^{6} + 840 \, A a^{6} e^{7}}{9240 \, {\left (e x + d\right )}^{11} e^{8}} \]
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Time = 1.47 (sec) , antiderivative size = 899, normalized size of antiderivative = 3.83 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx=-\frac {\frac {84\,B\,a^6\,d\,e^6+840\,A\,a^6\,e^7+112\,B\,a^5\,b\,d^2\,e^5+504\,A\,a^5\,b\,d\,e^6+105\,B\,a^4\,b^2\,d^3\,e^4+280\,A\,a^4\,b^2\,d^2\,e^5+80\,B\,a^3\,b^3\,d^4\,e^3+140\,A\,a^3\,b^3\,d^3\,e^4+50\,B\,a^2\,b^4\,d^5\,e^2+60\,A\,a^2\,b^4\,d^4\,e^3+24\,B\,a\,b^5\,d^6\,e+20\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+4\,A\,b^6\,d^6\,e}{9240\,e^8}+\frac {x\,\left (84\,B\,a^6\,e^6+112\,B\,a^5\,b\,d\,e^5+504\,A\,a^5\,b\,e^6+105\,B\,a^4\,b^2\,d^2\,e^4+280\,A\,a^4\,b^2\,d\,e^5+80\,B\,a^3\,b^3\,d^3\,e^3+140\,A\,a^3\,b^3\,d^2\,e^4+50\,B\,a^2\,b^4\,d^4\,e^2+60\,A\,a^2\,b^4\,d^3\,e^3+24\,B\,a\,b^5\,d^5\,e+20\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+4\,A\,b^6\,d^5\,e\right )}{840\,e^7}+\frac {b^3\,x^4\,\left (80\,B\,a^3\,e^3+50\,B\,a^2\,b\,d\,e^2+60\,A\,a^2\,b\,e^3+24\,B\,a\,b^2\,d^2\,e+20\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+4\,A\,b^3\,d^2\,e\right )}{28\,e^4}+\frac {b^5\,x^6\,\left (4\,A\,b\,e+24\,B\,a\,e+7\,B\,b\,d\right )}{20\,e^2}+\frac {b\,x^2\,\left (112\,B\,a^5\,e^5+105\,B\,a^4\,b\,d\,e^4+280\,A\,a^4\,b\,e^5+80\,B\,a^3\,b^2\,d^2\,e^3+140\,A\,a^3\,b^2\,d\,e^4+50\,B\,a^2\,b^3\,d^3\,e^2+60\,A\,a^2\,b^3\,d^2\,e^3+24\,B\,a\,b^4\,d^4\,e+20\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+4\,A\,b^5\,d^4\,e\right )}{168\,e^6}+\frac {b^2\,x^3\,\left (105\,B\,a^4\,e^4+80\,B\,a^3\,b\,d\,e^3+140\,A\,a^3\,b\,e^4+50\,B\,a^2\,b^2\,d^2\,e^2+60\,A\,a^2\,b^2\,d\,e^3+24\,B\,a\,b^3\,d^3\,e+20\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+4\,A\,b^4\,d^3\,e\right )}{56\,e^5}+\frac {b^4\,x^5\,\left (50\,B\,a^2\,e^2+24\,B\,a\,b\,d\,e+20\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+4\,A\,b^2\,d\,e\right )}{20\,e^3}+\frac {B\,b^6\,x^7}{4\,e}}{d^{11}+11\,d^{10}\,e\,x+55\,d^9\,e^2\,x^2+165\,d^8\,e^3\,x^3+330\,d^7\,e^4\,x^4+462\,d^6\,e^5\,x^5+462\,d^5\,e^6\,x^6+330\,d^4\,e^7\,x^7+165\,d^3\,e^8\,x^8+55\,d^2\,e^9\,x^9+11\,d\,e^{10}\,x^{10}+e^{11}\,x^{11}} \]
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