Integrand size = 20, antiderivative size = 292 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx=\frac {(b d-a e)^6 (B d-A e)}{13 e^8 (d+e x)^{13}}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{12 e^8 (d+e x)^{12}}+\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{11 e^8 (d+e x)^{11}}-\frac {b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{2 e^8 (d+e x)^{10}}+\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{9 e^8 (d+e x)^9}-\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e)}{8 e^8 (d+e x)^8}+\frac {b^5 (7 b B d-A b e-6 a B e)}{7 e^8 (d+e x)^7}-\frac {b^6 B}{6 e^8 (d+e x)^6} \]
[Out]
Time = 0.18 (sec) , antiderivative size = 292, 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} \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx=\frac {b^5 (-6 a B e-A b e+7 b B d)}{7 e^8 (d+e x)^7}-\frac {3 b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{8 e^8 (d+e x)^8}+\frac {5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{9 e^8 (d+e x)^9}-\frac {b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{2 e^8 (d+e x)^{10}}+\frac {3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{11 e^8 (d+e x)^{11}}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{12 e^8 (d+e x)^{12}}+\frac {(b d-a e)^6 (B d-A e)}{13 e^8 (d+e x)^{13}}-\frac {b^6 B}{6 e^8 (d+e x)^6} \]
[In]
[Out]
Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e)^6 (-B d+A e)}{e^7 (d+e x)^{14}}+\frac {(-b d+a e)^5 (-7 b B d+6 A b e+a B e)}{e^7 (d+e x)^{13}}+\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)^{12}}-\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)^{11}}+\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)^{10}}-\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)^9}+\frac {b^5 (-7 b B d+A b e+6 a B e)}{e^7 (d+e x)^8}+\frac {b^6 B}{e^7 (d+e x)^7}\right ) \, dx \\ & = \frac {(b d-a e)^6 (B d-A e)}{13 e^8 (d+e x)^{13}}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{12 e^8 (d+e x)^{12}}+\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{11 e^8 (d+e x)^{11}}-\frac {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)^{10}}+\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{9 e^8 (d+e x)^9}-\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e)}{8 e^8 (d+e x)^8}+\frac {b^5 (7 b B d-A b e-6 a B e)}{7 e^8 (d+e x)^7}-\frac {b^6 B}{6 e^8 (d+e x)^6} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(605\) vs. \(2(292)=584\).
Time = 0.20 (sec) , antiderivative size = 605, normalized size of antiderivative = 2.07 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx=-\frac {462 a^6 e^6 (12 A e+B (d+13 e x))+252 a^5 b e^5 \left (11 A e (d+13 e x)+2 B \left (d^2+13 d e x+78 e^2 x^2\right )\right )+126 a^4 b^2 e^4 \left (10 A e \left (d^2+13 d e x+78 e^2 x^2\right )+3 B \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )\right )+56 a^3 b^3 e^3 \left (9 A e \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )+4 B \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )\right )+21 a^2 b^4 e^2 \left (8 A e \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )+5 B \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )\right )+6 a b^5 e \left (7 A e \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )+6 B \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )\right )+b^6 \left (6 A e \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )+7 B \left (d^7+13 d^6 e x+78 d^5 e^2 x^2+286 d^4 e^3 x^3+715 d^3 e^4 x^4+1287 d^2 e^5 x^5+1716 d e^6 x^6+1716 e^7 x^7\right )\right )}{72072 e^8 (d+e x)^{13}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(788\) vs. \(2(276)=552\).
Time = 0.71 (sec) , antiderivative size = 789, normalized size of antiderivative = 2.70
method | result | size |
risch | \(\frac {-\frac {b^{6} B \,x^{7}}{6 e}-\frac {b^{5} \left (6 A b e +36 B a e +7 B b d \right ) x^{6}}{42 e^{2}}-\frac {b^{4} \left (42 A a b \,e^{2}+6 A \,b^{2} d e +105 B \,a^{2} e^{2}+36 B a b d e +7 b^{2} B \,d^{2}\right ) x^{5}}{56 e^{3}}-\frac {5 b^{3} \left (168 A \,a^{2} b \,e^{3}+42 A a \,b^{2} d \,e^{2}+6 A \,b^{3} d^{2} e +224 B \,a^{3} e^{3}+105 B \,a^{2} b d \,e^{2}+36 B a \,b^{2} d^{2} e +7 b^{3} B \,d^{3}\right ) x^{4}}{504 e^{4}}-\frac {b^{2} \left (504 A \,a^{3} b \,e^{4}+168 A \,a^{2} b^{2} d \,e^{3}+42 A a \,b^{3} d^{2} e^{2}+6 A \,b^{4} d^{3} e +378 B \,a^{4} e^{4}+224 B \,a^{3} b d \,e^{3}+105 B \,a^{2} b^{2} d^{2} e^{2}+36 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) x^{3}}{252 e^{5}}-\frac {b \left (1260 A \,a^{4} b \,e^{5}+504 A \,a^{3} b^{2} d \,e^{4}+168 A \,a^{2} b^{3} d^{2} e^{3}+42 A a \,b^{4} d^{3} e^{2}+6 A \,b^{5} d^{4} e +504 B \,a^{5} e^{5}+378 B \,a^{4} b d \,e^{4}+224 B \,a^{3} b^{2} d^{2} e^{3}+105 B \,a^{2} b^{3} d^{3} e^{2}+36 B a \,b^{4} d^{4} e +7 B \,b^{5} d^{5}\right ) x^{2}}{924 e^{6}}-\frac {\left (2772 A \,a^{5} b \,e^{6}+1260 A \,a^{4} b^{2} d \,e^{5}+504 A \,a^{3} b^{3} d^{2} e^{4}+168 A \,a^{2} b^{4} d^{3} e^{3}+42 A a \,b^{5} d^{4} e^{2}+6 A \,b^{6} d^{5} e +462 B \,a^{6} e^{6}+504 B \,a^{5} b d \,e^{5}+378 B \,a^{4} b^{2} d^{2} e^{4}+224 B \,a^{3} b^{3} d^{3} e^{3}+105 B \,a^{2} b^{4} d^{4} e^{2}+36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}\right ) x}{5544 e^{7}}-\frac {5544 A \,a^{6} e^{7}+2772 A \,a^{5} b d \,e^{6}+1260 A \,a^{4} b^{2} d^{2} e^{5}+504 A \,a^{3} b^{3} d^{3} e^{4}+168 A \,a^{2} b^{4} d^{4} e^{3}+42 A a \,b^{5} d^{5} e^{2}+6 A \,b^{6} d^{6} e +462 B \,a^{6} d \,e^{6}+504 B \,a^{5} b \,d^{2} e^{5}+378 B \,a^{4} b^{2} d^{3} e^{4}+224 B \,a^{3} b^{3} d^{4} e^{3}+105 B \,a^{2} b^{4} d^{5} e^{2}+36 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{72072 e^{8}}}{\left (e x +d \right )^{13}}\) | \(789\) |
default | \(-\frac {b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{2 e^{8} \left (e x +d \right )^{10}}-\frac {3 b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right )}{8 e^{8} \left (e x +d \right )^{8}}-\frac {3 b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{11 e^{8} \left (e x +d \right )^{11}}-\frac {b^{5} \left (A b e +6 B a e -7 B b d \right )}{7 e^{8} \left (e x +d \right )^{7}}-\frac {A \,a^{6} e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{13 e^{8} \left (e x +d \right )^{13}}-\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right )}{9 e^{8} \left (e x +d \right )^{9}}-\frac {b^{6} B}{6 e^{8} \left (e x +d \right )^{6}}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{12 e^{8} \left (e x +d \right )^{12}}\) | \(814\) |
norman | \(\frac {-\frac {b^{6} B \,x^{7}}{6 e}-\frac {\left (6 A \,b^{6} e^{6}+36 B a \,b^{5} e^{6}+7 b^{6} B d \,e^{5}\right ) x^{6}}{42 e^{7}}-\frac {\left (42 A a \,b^{5} e^{7}+6 A \,b^{6} d \,e^{6}+105 B \,a^{2} b^{4} e^{7}+36 B a \,b^{5} d \,e^{6}+7 b^{6} B \,d^{2} e^{5}\right ) x^{5}}{56 e^{8}}-\frac {5 \left (168 A \,a^{2} b^{4} e^{8}+42 A a \,b^{5} d \,e^{7}+6 A \,b^{6} d^{2} e^{6}+224 B \,a^{3} b^{3} e^{8}+105 B \,a^{2} b^{4} d \,e^{7}+36 B a \,b^{5} d^{2} e^{6}+7 b^{6} B \,d^{3} e^{5}\right ) x^{4}}{504 e^{9}}-\frac {\left (504 A \,a^{3} b^{3} e^{9}+168 A \,a^{2} b^{4} d \,e^{8}+42 A a \,b^{5} d^{2} e^{7}+6 A \,b^{6} d^{3} e^{6}+378 B \,a^{4} b^{2} e^{9}+224 B \,a^{3} b^{3} d \,e^{8}+105 B \,a^{2} b^{4} d^{2} e^{7}+36 B a \,b^{5} d^{3} e^{6}+7 B \,b^{6} d^{4} e^{5}\right ) x^{3}}{252 e^{10}}-\frac {\left (1260 A \,a^{4} b^{2} e^{10}+504 A \,a^{3} b^{3} d \,e^{9}+168 A \,a^{2} b^{4} d^{2} e^{8}+42 A a \,b^{5} d^{3} e^{7}+6 A \,b^{6} d^{4} e^{6}+504 B \,a^{5} b \,e^{10}+378 B \,a^{4} b^{2} d \,e^{9}+224 B \,a^{3} b^{3} d^{2} e^{8}+105 B \,a^{2} b^{4} d^{3} e^{7}+36 B a \,b^{5} d^{4} e^{6}+7 B \,b^{6} d^{5} e^{5}\right ) x^{2}}{924 e^{11}}-\frac {\left (2772 A \,a^{5} b \,e^{11}+1260 A \,a^{4} b^{2} d \,e^{10}+504 A \,a^{3} b^{3} d^{2} e^{9}+168 A \,a^{2} b^{4} d^{3} e^{8}+42 A a \,b^{5} d^{4} e^{7}+6 A \,b^{6} d^{5} e^{6}+462 B \,a^{6} e^{11}+504 B \,a^{5} b d \,e^{10}+378 B \,a^{4} b^{2} d^{2} e^{9}+224 B \,a^{3} b^{3} d^{3} e^{8}+105 B \,a^{2} b^{4} d^{4} e^{7}+36 B a \,b^{5} d^{5} e^{6}+7 b^{6} B \,d^{6} e^{5}\right ) x}{5544 e^{12}}-\frac {5544 A \,a^{6} e^{12}+2772 A \,a^{5} b d \,e^{11}+1260 A \,a^{4} b^{2} d^{2} e^{10}+504 A \,a^{3} b^{3} d^{3} e^{9}+168 A \,a^{2} b^{4} d^{4} e^{8}+42 A a \,b^{5} d^{5} e^{7}+6 A \,b^{6} d^{6} e^{6}+462 B \,a^{6} d \,e^{11}+504 B \,a^{5} b \,d^{2} e^{10}+378 B \,a^{4} b^{2} d^{3} e^{9}+224 B \,a^{3} b^{3} d^{4} e^{8}+105 B \,a^{2} b^{4} d^{5} e^{7}+36 B a \,b^{5} d^{6} e^{6}+7 B \,b^{6} d^{7} e^{5}}{72072 e^{13}}}{\left (e x +d \right )^{13}}\) | \(858\) |
gosper | \(-\frac {12012 B \,x^{7} b^{6} e^{7}+10296 A \,x^{6} b^{6} e^{7}+61776 B \,x^{6} a \,b^{5} e^{7}+12012 B \,x^{6} b^{6} d \,e^{6}+54054 A \,x^{5} a \,b^{5} e^{7}+7722 A \,x^{5} b^{6} d \,e^{6}+135135 B \,x^{5} a^{2} b^{4} e^{7}+46332 B \,x^{5} a \,b^{5} d \,e^{6}+9009 B \,x^{5} b^{6} d^{2} e^{5}+120120 A \,x^{4} a^{2} b^{4} e^{7}+30030 A \,x^{4} a \,b^{5} d \,e^{6}+4290 A \,x^{4} b^{6} d^{2} e^{5}+160160 B \,x^{4} a^{3} b^{3} e^{7}+75075 B \,x^{4} a^{2} b^{4} d \,e^{6}+25740 B \,x^{4} a \,b^{5} d^{2} e^{5}+5005 B \,x^{4} b^{6} d^{3} e^{4}+144144 A \,x^{3} a^{3} b^{3} e^{7}+48048 A \,x^{3} a^{2} b^{4} d \,e^{6}+12012 A \,x^{3} a \,b^{5} d^{2} e^{5}+1716 A \,x^{3} b^{6} d^{3} e^{4}+108108 B \,x^{3} a^{4} b^{2} e^{7}+64064 B \,x^{3} a^{3} b^{3} d \,e^{6}+30030 B \,x^{3} a^{2} b^{4} d^{2} e^{5}+10296 B \,x^{3} a \,b^{5} d^{3} e^{4}+2002 B \,x^{3} b^{6} d^{4} e^{3}+98280 A \,x^{2} a^{4} b^{2} e^{7}+39312 A \,x^{2} a^{3} b^{3} d \,e^{6}+13104 A \,x^{2} a^{2} b^{4} d^{2} e^{5}+3276 A \,x^{2} a \,b^{5} d^{3} e^{4}+468 A \,x^{2} b^{6} d^{4} e^{3}+39312 B \,x^{2} a^{5} b \,e^{7}+29484 B \,x^{2} a^{4} b^{2} d \,e^{6}+17472 B \,x^{2} a^{3} b^{3} d^{2} e^{5}+8190 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+2808 B \,x^{2} a \,b^{5} d^{4} e^{3}+546 B \,x^{2} b^{6} d^{5} e^{2}+36036 A x \,a^{5} b \,e^{7}+16380 A x \,a^{4} b^{2} d \,e^{6}+6552 A x \,a^{3} b^{3} d^{2} e^{5}+2184 A x \,a^{2} b^{4} d^{3} e^{4}+546 A x a \,b^{5} d^{4} e^{3}+78 A x \,b^{6} d^{5} e^{2}+6006 B x \,a^{6} e^{7}+6552 B x \,a^{5} b d \,e^{6}+4914 B x \,a^{4} b^{2} d^{2} e^{5}+2912 B x \,a^{3} b^{3} d^{3} e^{4}+1365 B x \,a^{2} b^{4} d^{4} e^{3}+468 B x a \,b^{5} d^{5} e^{2}+91 B x \,b^{6} d^{6} e +5544 A \,a^{6} e^{7}+2772 A \,a^{5} b d \,e^{6}+1260 A \,a^{4} b^{2} d^{2} e^{5}+504 A \,a^{3} b^{3} d^{3} e^{4}+168 A \,a^{2} b^{4} d^{4} e^{3}+42 A a \,b^{5} d^{5} e^{2}+6 A \,b^{6} d^{6} e +462 B \,a^{6} d \,e^{6}+504 B \,a^{5} b \,d^{2} e^{5}+378 B \,a^{4} b^{2} d^{3} e^{4}+224 B \,a^{3} b^{3} d^{4} e^{3}+105 B \,a^{2} b^{4} d^{5} e^{2}+36 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{72072 e^{8} \left (e x +d \right )^{13}}\) | \(913\) |
parallelrisch | \(-\frac {12012 B \,b^{6} x^{7} e^{12}+10296 A \,b^{6} e^{12} x^{6}+61776 B a \,b^{5} e^{12} x^{6}+12012 B \,b^{6} d \,e^{11} x^{6}+54054 A a \,b^{5} e^{12} x^{5}+7722 A \,b^{6} d \,e^{11} x^{5}+135135 B \,a^{2} b^{4} e^{12} x^{5}+46332 B a \,b^{5} d \,e^{11} x^{5}+9009 B \,b^{6} d^{2} e^{10} x^{5}+120120 A \,a^{2} b^{4} e^{12} x^{4}+30030 A a \,b^{5} d \,e^{11} x^{4}+4290 A \,b^{6} d^{2} e^{10} x^{4}+160160 B \,a^{3} b^{3} e^{12} x^{4}+75075 B \,a^{2} b^{4} d \,e^{11} x^{4}+25740 B a \,b^{5} d^{2} e^{10} x^{4}+5005 B \,b^{6} d^{3} e^{9} x^{4}+144144 A \,a^{3} b^{3} e^{12} x^{3}+48048 A \,a^{2} b^{4} d \,e^{11} x^{3}+12012 A a \,b^{5} d^{2} e^{10} x^{3}+1716 A \,b^{6} d^{3} e^{9} x^{3}+108108 B \,a^{4} b^{2} e^{12} x^{3}+64064 B \,a^{3} b^{3} d \,e^{11} x^{3}+30030 B \,a^{2} b^{4} d^{2} e^{10} x^{3}+10296 B a \,b^{5} d^{3} e^{9} x^{3}+2002 B \,b^{6} d^{4} e^{8} x^{3}+98280 A \,a^{4} b^{2} e^{12} x^{2}+39312 A \,a^{3} b^{3} d \,e^{11} x^{2}+13104 A \,a^{2} b^{4} d^{2} e^{10} x^{2}+3276 A a \,b^{5} d^{3} e^{9} x^{2}+468 A \,b^{6} d^{4} e^{8} x^{2}+39312 B \,a^{5} b \,e^{12} x^{2}+29484 B \,a^{4} b^{2} d \,e^{11} x^{2}+17472 B \,a^{3} b^{3} d^{2} e^{10} x^{2}+8190 B \,a^{2} b^{4} d^{3} e^{9} x^{2}+2808 B a \,b^{5} d^{4} e^{8} x^{2}+546 B \,b^{6} d^{5} e^{7} x^{2}+36036 A \,a^{5} b \,e^{12} x +16380 A \,a^{4} b^{2} d \,e^{11} x +6552 A \,a^{3} b^{3} d^{2} e^{10} x +2184 A \,a^{2} b^{4} d^{3} e^{9} x +546 A a \,b^{5} d^{4} e^{8} x +78 A \,b^{6} d^{5} e^{7} x +6006 B \,a^{6} e^{12} x +6552 B \,a^{5} b d \,e^{11} x +4914 B \,a^{4} b^{2} d^{2} e^{10} x +2912 B \,a^{3} b^{3} d^{3} e^{9} x +1365 B \,a^{2} b^{4} d^{4} e^{8} x +468 B a \,b^{5} d^{5} e^{7} x +91 B \,b^{6} d^{6} e^{6} x +5544 A \,a^{6} e^{12}+2772 A \,a^{5} b d \,e^{11}+1260 A \,a^{4} b^{2} d^{2} e^{10}+504 A \,a^{3} b^{3} d^{3} e^{9}+168 A \,a^{2} b^{4} d^{4} e^{8}+42 A a \,b^{5} d^{5} e^{7}+6 A \,b^{6} d^{6} e^{6}+462 B \,a^{6} d \,e^{11}+504 B \,a^{5} b \,d^{2} e^{10}+378 B \,a^{4} b^{2} d^{3} e^{9}+224 B \,a^{3} b^{3} d^{4} e^{8}+105 B \,a^{2} b^{4} d^{5} e^{7}+36 B a \,b^{5} d^{6} e^{6}+7 B \,b^{6} d^{7} e^{5}}{72072 e^{13} \left (e x +d \right )^{13}}\) | \(922\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 905 vs. \(2 (276) = 552\).
Time = 0.26 (sec) , antiderivative size = 905, normalized size of antiderivative = 3.10 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx=-\frac {12012 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 5544 \, A a^{6} e^{7} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 21 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 56 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 126 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 252 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 462 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 1716 \, {\left (7 \, B b^{6} d e^{6} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 1287 \, {\left (7 \, B b^{6} d^{2} e^{5} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 21 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 715 \, {\left (7 \, B b^{6} d^{3} e^{4} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 21 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 56 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 286 \, {\left (7 \, B b^{6} d^{4} e^{3} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 21 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 56 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 126 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 78 \, {\left (7 \, B b^{6} d^{5} e^{2} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 21 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 56 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 126 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 252 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 13 \, {\left (7 \, B b^{6} d^{6} e + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 21 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 56 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 126 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 252 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 462 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{72072 \, {\left (e^{21} x^{13} + 13 \, d e^{20} x^{12} + 78 \, d^{2} e^{19} x^{11} + 286 \, d^{3} e^{18} x^{10} + 715 \, d^{4} e^{17} x^{9} + 1287 \, d^{5} e^{16} x^{8} + 1716 \, d^{6} e^{15} x^{7} + 1716 \, d^{7} e^{14} x^{6} + 1287 \, d^{8} e^{13} x^{5} + 715 \, d^{9} e^{12} x^{4} + 286 \, d^{10} e^{11} x^{3} + 78 \, d^{11} e^{10} x^{2} + 13 \, d^{12} e^{9} x + d^{13} e^{8}\right )}} \]
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Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 905 vs. \(2 (276) = 552\).
Time = 0.25 (sec) , antiderivative size = 905, normalized size of antiderivative = 3.10 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx=-\frac {12012 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 5544 \, A a^{6} e^{7} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 21 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 56 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 126 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 252 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 462 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 1716 \, {\left (7 \, B b^{6} d e^{6} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 1287 \, {\left (7 \, B b^{6} d^{2} e^{5} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 21 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 715 \, {\left (7 \, B b^{6} d^{3} e^{4} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 21 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 56 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 286 \, {\left (7 \, B b^{6} d^{4} e^{3} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 21 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 56 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 126 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 78 \, {\left (7 \, B b^{6} d^{5} e^{2} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 21 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 56 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 126 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 252 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 13 \, {\left (7 \, B b^{6} d^{6} e + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 21 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 56 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 126 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 252 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 462 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{72072 \, {\left (e^{21} x^{13} + 13 \, d e^{20} x^{12} + 78 \, d^{2} e^{19} x^{11} + 286 \, d^{3} e^{18} x^{10} + 715 \, d^{4} e^{17} x^{9} + 1287 \, d^{5} e^{16} x^{8} + 1716 \, d^{6} e^{15} x^{7} + 1716 \, d^{7} e^{14} x^{6} + 1287 \, d^{8} e^{13} x^{5} + 715 \, d^{9} e^{12} x^{4} + 286 \, d^{10} e^{11} x^{3} + 78 \, d^{11} e^{10} x^{2} + 13 \, d^{12} e^{9} x + d^{13} e^{8}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (276) = 552\).
Time = 0.30 (sec) , antiderivative size = 912, normalized size of antiderivative = 3.12 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx=-\frac {12012 \, B b^{6} e^{7} x^{7} + 12012 \, B b^{6} d e^{6} x^{6} + 61776 \, B a b^{5} e^{7} x^{6} + 10296 \, A b^{6} e^{7} x^{6} + 9009 \, B b^{6} d^{2} e^{5} x^{5} + 46332 \, B a b^{5} d e^{6} x^{5} + 7722 \, A b^{6} d e^{6} x^{5} + 135135 \, B a^{2} b^{4} e^{7} x^{5} + 54054 \, A a b^{5} e^{7} x^{5} + 5005 \, B b^{6} d^{3} e^{4} x^{4} + 25740 \, B a b^{5} d^{2} e^{5} x^{4} + 4290 \, A b^{6} d^{2} e^{5} x^{4} + 75075 \, B a^{2} b^{4} d e^{6} x^{4} + 30030 \, A a b^{5} d e^{6} x^{4} + 160160 \, B a^{3} b^{3} e^{7} x^{4} + 120120 \, A a^{2} b^{4} e^{7} x^{4} + 2002 \, B b^{6} d^{4} e^{3} x^{3} + 10296 \, B a b^{5} d^{3} e^{4} x^{3} + 1716 \, A b^{6} d^{3} e^{4} x^{3} + 30030 \, B a^{2} b^{4} d^{2} e^{5} x^{3} + 12012 \, A a b^{5} d^{2} e^{5} x^{3} + 64064 \, B a^{3} b^{3} d e^{6} x^{3} + 48048 \, A a^{2} b^{4} d e^{6} x^{3} + 108108 \, B a^{4} b^{2} e^{7} x^{3} + 144144 \, A a^{3} b^{3} e^{7} x^{3} + 546 \, B b^{6} d^{5} e^{2} x^{2} + 2808 \, B a b^{5} d^{4} e^{3} x^{2} + 468 \, A b^{6} d^{4} e^{3} x^{2} + 8190 \, B a^{2} b^{4} d^{3} e^{4} x^{2} + 3276 \, A a b^{5} d^{3} e^{4} x^{2} + 17472 \, B a^{3} b^{3} d^{2} e^{5} x^{2} + 13104 \, A a^{2} b^{4} d^{2} e^{5} x^{2} + 29484 \, B a^{4} b^{2} d e^{6} x^{2} + 39312 \, A a^{3} b^{3} d e^{6} x^{2} + 39312 \, B a^{5} b e^{7} x^{2} + 98280 \, A a^{4} b^{2} e^{7} x^{2} + 91 \, B b^{6} d^{6} e x + 468 \, B a b^{5} d^{5} e^{2} x + 78 \, A b^{6} d^{5} e^{2} x + 1365 \, B a^{2} b^{4} d^{4} e^{3} x + 546 \, A a b^{5} d^{4} e^{3} x + 2912 \, B a^{3} b^{3} d^{3} e^{4} x + 2184 \, A a^{2} b^{4} d^{3} e^{4} x + 4914 \, B a^{4} b^{2} d^{2} e^{5} x + 6552 \, A a^{3} b^{3} d^{2} e^{5} x + 6552 \, B a^{5} b d e^{6} x + 16380 \, A a^{4} b^{2} d e^{6} x + 6006 \, B a^{6} e^{7} x + 36036 \, A a^{5} b e^{7} x + 7 \, B b^{6} d^{7} + 36 \, B a b^{5} d^{6} e + 6 \, A b^{6} d^{6} e + 105 \, B a^{2} b^{4} d^{5} e^{2} + 42 \, A a b^{5} d^{5} e^{2} + 224 \, B a^{3} b^{3} d^{4} e^{3} + 168 \, A a^{2} b^{4} d^{4} e^{3} + 378 \, B a^{4} b^{2} d^{3} e^{4} + 504 \, A a^{3} b^{3} d^{3} e^{4} + 504 \, B a^{5} b d^{2} e^{5} + 1260 \, A a^{4} b^{2} d^{2} e^{5} + 462 \, B a^{6} d e^{6} + 2772 \, A a^{5} b d e^{6} + 5544 \, A a^{6} e^{7}}{72072 \, {\left (e x + d\right )}^{13} e^{8}} \]
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Time = 0.85 (sec) , antiderivative size = 921, normalized size of antiderivative = 3.15 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx=-\frac {\frac {462\,B\,a^6\,d\,e^6+5544\,A\,a^6\,e^7+504\,B\,a^5\,b\,d^2\,e^5+2772\,A\,a^5\,b\,d\,e^6+378\,B\,a^4\,b^2\,d^3\,e^4+1260\,A\,a^4\,b^2\,d^2\,e^5+224\,B\,a^3\,b^3\,d^4\,e^3+504\,A\,a^3\,b^3\,d^3\,e^4+105\,B\,a^2\,b^4\,d^5\,e^2+168\,A\,a^2\,b^4\,d^4\,e^3+36\,B\,a\,b^5\,d^6\,e+42\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+6\,A\,b^6\,d^6\,e}{72072\,e^8}+\frac {x\,\left (462\,B\,a^6\,e^6+504\,B\,a^5\,b\,d\,e^5+2772\,A\,a^5\,b\,e^6+378\,B\,a^4\,b^2\,d^2\,e^4+1260\,A\,a^4\,b^2\,d\,e^5+224\,B\,a^3\,b^3\,d^3\,e^3+504\,A\,a^3\,b^3\,d^2\,e^4+105\,B\,a^2\,b^4\,d^4\,e^2+168\,A\,a^2\,b^4\,d^3\,e^3+36\,B\,a\,b^5\,d^5\,e+42\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+6\,A\,b^6\,d^5\,e\right )}{5544\,e^7}+\frac {5\,b^3\,x^4\,\left (224\,B\,a^3\,e^3+105\,B\,a^2\,b\,d\,e^2+168\,A\,a^2\,b\,e^3+36\,B\,a\,b^2\,d^2\,e+42\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+6\,A\,b^3\,d^2\,e\right )}{504\,e^4}+\frac {b^5\,x^6\,\left (6\,A\,b\,e+36\,B\,a\,e+7\,B\,b\,d\right )}{42\,e^2}+\frac {b\,x^2\,\left (504\,B\,a^5\,e^5+378\,B\,a^4\,b\,d\,e^4+1260\,A\,a^4\,b\,e^5+224\,B\,a^3\,b^2\,d^2\,e^3+504\,A\,a^3\,b^2\,d\,e^4+105\,B\,a^2\,b^3\,d^3\,e^2+168\,A\,a^2\,b^3\,d^2\,e^3+36\,B\,a\,b^4\,d^4\,e+42\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+6\,A\,b^5\,d^4\,e\right )}{924\,e^6}+\frac {b^2\,x^3\,\left (378\,B\,a^4\,e^4+224\,B\,a^3\,b\,d\,e^3+504\,A\,a^3\,b\,e^4+105\,B\,a^2\,b^2\,d^2\,e^2+168\,A\,a^2\,b^2\,d\,e^3+36\,B\,a\,b^3\,d^3\,e+42\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+6\,A\,b^4\,d^3\,e\right )}{252\,e^5}+\frac {b^4\,x^5\,\left (105\,B\,a^2\,e^2+36\,B\,a\,b\,d\,e+42\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+6\,A\,b^2\,d\,e\right )}{56\,e^3}+\frac {B\,b^6\,x^7}{6\,e}}{d^{13}+13\,d^{12}\,e\,x+78\,d^{11}\,e^2\,x^2+286\,d^{10}\,e^3\,x^3+715\,d^9\,e^4\,x^4+1287\,d^8\,e^5\,x^5+1716\,d^7\,e^6\,x^6+1716\,d^6\,e^7\,x^7+1287\,d^5\,e^8\,x^8+715\,d^4\,e^9\,x^9+286\,d^3\,e^{10}\,x^{10}+78\,d^2\,e^{11}\,x^{11}+13\,d\,e^{12}\,x^{12}+e^{13}\,x^{13}} \]
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