Integrand size = 20, antiderivative size = 292 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{15}} \, dx=\frac {(b d-a e)^6 (B d-A e)}{14 e^8 (d+e x)^{14}}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{13 e^8 (d+e x)^{13}}+\frac {b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{4 e^8 (d+e x)^{12}}-\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{11 e^8 (d+e x)^{11}}+\frac {b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{2 e^8 (d+e x)^{10}}-\frac {b^4 (b d-a e) (7 b B d-2 A b e-5 a B e)}{3 e^8 (d+e x)^9}+\frac {b^5 (7 b B d-A b e-6 a B e)}{8 e^8 (d+e x)^8}-\frac {b^6 B}{7 e^8 (d+e x)^7} \]
[Out]
Time = 0.17 (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} \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{15}} \, dx=\frac {b^5 (-6 a B e-A b e+7 b B d)}{8 e^8 (d+e x)^8}-\frac {b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{3 e^8 (d+e x)^9}+\frac {b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{2 e^8 (d+e x)^{10}}-\frac {5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{11 e^8 (d+e x)^{11}}+\frac {b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{4 e^8 (d+e x)^{12}}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{13 e^8 (d+e x)^{13}}+\frac {(b d-a e)^6 (B d-A e)}{14 e^8 (d+e x)^{14}}-\frac {b^6 B}{7 e^8 (d+e x)^7} \]
[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)^{15}}+\frac {(-b d+a e)^5 (-7 b B d+6 A b e+a B e)}{e^7 (d+e x)^{14}}+\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)^{13}}-\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)^{12}}+\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)^{11}}-\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)^{10}}+\frac {b^5 (-7 b B d+A b e+6 a B e)}{e^7 (d+e x)^9}+\frac {b^6 B}{e^7 (d+e x)^8}\right ) \, dx \\ & = \frac {(b d-a e)^6 (B d-A e)}{14 e^8 (d+e x)^{14}}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{13 e^8 (d+e x)^{13}}+\frac {b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{4 e^8 (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)}{11 e^8 (d+e x)^{11}}+\frac {b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{2 e^8 (d+e x)^{10}}-\frac {b^4 (b d-a e) (7 b B d-2 A b e-5 a B e)}{3 e^8 (d+e x)^9}+\frac {b^5 (7 b B d-A b e-6 a B e)}{8 e^8 (d+e x)^8}-\frac {b^6 B}{7 e^8 (d+e x)^7} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(602\) vs. \(2(292)=584\).
Time = 0.20 (sec) , antiderivative size = 602, normalized size of antiderivative = 2.06 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{15}} \, dx=-\frac {132 a^6 e^6 (13 A e+B (d+14 e x))+132 a^5 b e^5 \left (6 A e (d+14 e x)+B \left (d^2+14 d e x+91 e^2 x^2\right )\right )+30 a^4 b^2 e^4 \left (11 A e \left (d^2+14 d e x+91 e^2 x^2\right )+3 B \left (d^3+14 d^2 e x+91 d e^2 x^2+364 e^3 x^3\right )\right )+24 a^3 b^3 e^3 \left (5 A e \left (d^3+14 d^2 e x+91 d e^2 x^2+364 e^3 x^3\right )+2 B \left (d^4+14 d^3 e x+91 d^2 e^2 x^2+364 d e^3 x^3+1001 e^4 x^4\right )\right )+4 a^2 b^4 e^2 \left (9 A e \left (d^4+14 d^3 e x+91 d^2 e^2 x^2+364 d e^3 x^3+1001 e^4 x^4\right )+5 B \left (d^5+14 d^4 e x+91 d^3 e^2 x^2+364 d^2 e^3 x^3+1001 d e^4 x^4+2002 e^5 x^5\right )\right )+2 a b^5 e \left (4 A e \left (d^5+14 d^4 e x+91 d^3 e^2 x^2+364 d^2 e^3 x^3+1001 d e^4 x^4+2002 e^5 x^5\right )+3 B \left (d^6+14 d^5 e x+91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+2002 d e^5 x^5+3003 e^6 x^6\right )\right )+b^6 \left (A e \left (d^6+14 d^5 e x+91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+2002 d e^5 x^5+3003 e^6 x^6\right )+B \left (d^7+14 d^6 e x+91 d^5 e^2 x^2+364 d^4 e^3 x^3+1001 d^3 e^4 x^4+2002 d^2 e^5 x^5+3003 d e^6 x^6+3432 e^7 x^7\right )\right )}{24024 e^8 (d+e x)^{14}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(774\) vs. \(2(276)=552\).
Time = 0.71 (sec) , antiderivative size = 775, normalized size of antiderivative = 2.65
method | result | size |
risch | \(\frac {-\frac {b^{6} B \,x^{7}}{7 e}-\frac {b^{5} \left (A b e +6 B a e +B b d \right ) x^{6}}{8 e^{2}}-\frac {b^{4} \left (8 A a b \,e^{2}+A \,b^{2} d e +20 B \,a^{2} e^{2}+6 B a b d e +b^{2} B \,d^{2}\right ) x^{5}}{12 e^{3}}-\frac {b^{3} \left (36 A \,a^{2} b \,e^{3}+8 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +48 B \,a^{3} e^{3}+20 B \,a^{2} b d \,e^{2}+6 B a \,b^{2} d^{2} e +b^{3} B \,d^{3}\right ) x^{4}}{24 e^{4}}-\frac {b^{2} \left (120 A \,a^{3} b \,e^{4}+36 A \,a^{2} b^{2} d \,e^{3}+8 A a \,b^{3} d^{2} e^{2}+A \,b^{4} d^{3} e +90 B \,a^{4} e^{4}+48 B \,a^{3} b d \,e^{3}+20 B \,a^{2} b^{2} d^{2} e^{2}+6 B a \,b^{3} d^{3} e +B \,b^{4} d^{4}\right ) x^{3}}{66 e^{5}}-\frac {b \left (330 A \,a^{4} b \,e^{5}+120 A \,a^{3} b^{2} d \,e^{4}+36 A \,a^{2} b^{3} d^{2} e^{3}+8 A a \,b^{4} d^{3} e^{2}+A \,b^{5} d^{4} e +132 B \,a^{5} e^{5}+90 B \,a^{4} b d \,e^{4}+48 B \,a^{3} b^{2} d^{2} e^{3}+20 B \,a^{2} b^{3} d^{3} e^{2}+6 B a \,b^{4} d^{4} e +B \,b^{5} d^{5}\right ) x^{2}}{264 e^{6}}-\frac {\left (792 A \,a^{5} b \,e^{6}+330 A \,a^{4} b^{2} d \,e^{5}+120 A \,a^{3} b^{3} d^{2} e^{4}+36 A \,a^{2} b^{4} d^{3} e^{3}+8 A a \,b^{5} d^{4} e^{2}+A \,b^{6} d^{5} e +132 B \,a^{6} e^{6}+132 B \,a^{5} b d \,e^{5}+90 B \,a^{4} b^{2} d^{2} e^{4}+48 B \,a^{3} b^{3} d^{3} e^{3}+20 B \,a^{2} b^{4} d^{4} e^{2}+6 B a \,b^{5} d^{5} e +b^{6} B \,d^{6}\right ) x}{1716 e^{7}}-\frac {1716 A \,a^{6} e^{7}+792 A \,a^{5} b d \,e^{6}+330 A \,a^{4} b^{2} d^{2} e^{5}+120 A \,a^{3} b^{3} d^{3} e^{4}+36 A \,a^{2} b^{4} d^{4} e^{3}+8 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e +132 B \,a^{6} d \,e^{6}+132 B \,a^{5} b \,d^{2} e^{5}+90 B \,a^{4} b^{2} d^{3} e^{4}+48 B \,a^{3} b^{3} d^{4} e^{3}+20 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e +b^{6} B \,d^{7}}{24024 e^{8}}}{\left (e x +d \right )^{14}}\) | \(775\) |
default | \(-\frac {b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right )}{2 e^{8} \left (e x +d \right )^{10}}-\frac {A \,a^{6} e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{14 e^{8} \left (e x +d \right )^{14}}-\frac {b^{5} \left (A b e +6 B a e -7 B b d \right )}{8 e^{8} \left (e x +d \right )^{8}}-\frac {5 b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{11 e^{8} \left (e x +d \right )^{11}}-\frac {b^{6} B}{7 e^{8} \left (e x +d \right )^{7}}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{13 e^{8} \left (e x +d \right )^{13}}-\frac {b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right )}{3 e^{8} \left (e x +d \right )^{9}}-\frac {b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{4 e^{8} \left (e x +d \right )^{12}}\) | \(814\) |
norman | \(\frac {-\frac {1716 A \,a^{6} e^{13}+792 A \,a^{5} b d \,e^{12}+330 A \,a^{4} b^{2} d^{2} e^{11}+120 A \,a^{3} b^{3} d^{3} e^{10}+36 A \,a^{2} b^{4} d^{4} e^{9}+8 A a \,b^{5} d^{5} e^{8}+A \,b^{6} d^{6} e^{7}+132 B \,a^{6} d \,e^{12}+132 B \,a^{5} b \,d^{2} e^{11}+90 B \,a^{4} b^{2} d^{3} e^{10}+48 B \,a^{3} b^{3} d^{4} e^{9}+20 B \,a^{2} b^{4} d^{5} e^{8}+6 B a \,b^{5} d^{6} e^{7}+B \,b^{6} d^{7} e^{6}}{24024 e^{14}}-\frac {\left (792 A \,a^{5} b \,e^{12}+330 A \,a^{4} b^{2} d \,e^{11}+120 A \,a^{3} b^{3} d^{2} e^{10}+36 A \,a^{2} b^{4} d^{3} e^{9}+8 A a \,b^{5} d^{4} e^{8}+A \,b^{6} d^{5} e^{7}+132 B \,a^{6} e^{12}+132 B \,a^{5} b d \,e^{11}+90 B \,a^{4} b^{2} d^{2} e^{10}+48 B \,a^{3} b^{3} d^{3} e^{9}+20 B \,a^{2} b^{4} d^{4} e^{8}+6 B a \,b^{5} d^{5} e^{7}+b^{6} B \,d^{6} e^{6}\right ) x}{1716 e^{13}}-\frac {\left (330 a^{4} b^{2} A \,e^{11}+120 A \,a^{3} b^{3} d \,e^{10}+36 A \,a^{2} b^{4} d^{2} e^{9}+8 A a \,b^{5} d^{3} e^{8}+A \,b^{6} d^{4} e^{7}+132 a^{5} b B \,e^{11}+90 B \,a^{4} b^{2} d \,e^{10}+48 B \,a^{3} b^{3} d^{2} e^{9}+20 B \,a^{2} b^{4} d^{3} e^{8}+6 B a \,b^{5} d^{4} e^{7}+B \,b^{6} d^{5} e^{6}\right ) x^{2}}{264 e^{12}}-\frac {\left (120 A \,a^{3} b^{3} e^{10}+36 A \,a^{2} b^{4} d \,e^{9}+8 A a \,b^{5} d^{2} e^{8}+A \,b^{6} d^{3} e^{7}+90 B \,a^{4} b^{2} e^{10}+48 B \,a^{3} b^{3} d \,e^{9}+20 B \,a^{2} b^{4} d^{2} e^{8}+6 B a \,b^{5} d^{3} e^{7}+B \,b^{6} d^{4} e^{6}\right ) x^{3}}{66 e^{11}}-\frac {\left (36 A \,a^{2} b^{4} e^{9}+8 A a \,b^{5} d \,e^{8}+A \,b^{6} d^{2} e^{7}+48 B \,a^{3} b^{3} e^{9}+20 B \,a^{2} b^{4} d \,e^{8}+6 B a \,b^{5} d^{2} e^{7}+b^{6} B \,d^{3} e^{6}\right ) x^{4}}{24 e^{10}}-\frac {\left (8 A a \,b^{5} e^{8}+A \,b^{6} d \,e^{7}+20 B \,a^{2} b^{4} e^{8}+6 B a \,b^{5} d \,e^{7}+b^{6} B \,d^{2} e^{6}\right ) x^{5}}{12 e^{9}}-\frac {\left (A \,b^{6} e^{7}+6 B a \,b^{5} e^{7}+b^{6} B d \,e^{6}\right ) x^{6}}{8 e^{8}}-\frac {b^{6} B \,x^{7}}{7 e}}{\left (e x +d \right )^{14}}\) | \(844\) |
gosper | \(-\frac {3432 B \,x^{7} b^{6} e^{7}+3003 A \,x^{6} b^{6} e^{7}+18018 B \,x^{6} a \,b^{5} e^{7}+3003 B \,x^{6} b^{6} d \,e^{6}+16016 A \,x^{5} a \,b^{5} e^{7}+2002 A \,x^{5} b^{6} d \,e^{6}+40040 B \,x^{5} a^{2} b^{4} e^{7}+12012 B \,x^{5} a \,b^{5} d \,e^{6}+2002 B \,x^{5} b^{6} d^{2} e^{5}+36036 A \,x^{4} a^{2} b^{4} e^{7}+8008 A \,x^{4} a \,b^{5} d \,e^{6}+1001 A \,x^{4} b^{6} d^{2} e^{5}+48048 B \,x^{4} a^{3} b^{3} e^{7}+20020 B \,x^{4} a^{2} b^{4} d \,e^{6}+6006 B \,x^{4} a \,b^{5} d^{2} e^{5}+1001 B \,x^{4} b^{6} d^{3} e^{4}+43680 A \,x^{3} a^{3} b^{3} e^{7}+13104 A \,x^{3} a^{2} b^{4} d \,e^{6}+2912 A \,x^{3} a \,b^{5} d^{2} e^{5}+364 A \,x^{3} b^{6} d^{3} e^{4}+32760 B \,x^{3} a^{4} b^{2} e^{7}+17472 B \,x^{3} a^{3} b^{3} d \,e^{6}+7280 B \,x^{3} a^{2} b^{4} d^{2} e^{5}+2184 B \,x^{3} a \,b^{5} d^{3} e^{4}+364 B \,x^{3} b^{6} d^{4} e^{3}+30030 A \,x^{2} a^{4} b^{2} e^{7}+10920 A \,x^{2} a^{3} b^{3} d \,e^{6}+3276 A \,x^{2} a^{2} b^{4} d^{2} e^{5}+728 A \,x^{2} a \,b^{5} d^{3} e^{4}+91 A \,x^{2} b^{6} d^{4} e^{3}+12012 B \,x^{2} a^{5} b \,e^{7}+8190 B \,x^{2} a^{4} b^{2} d \,e^{6}+4368 B \,x^{2} a^{3} b^{3} d^{2} e^{5}+1820 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+546 B \,x^{2} a \,b^{5} d^{4} e^{3}+91 B \,x^{2} b^{6} d^{5} e^{2}+11088 A x \,a^{5} b \,e^{7}+4620 A x \,a^{4} b^{2} d \,e^{6}+1680 A x \,a^{3} b^{3} d^{2} e^{5}+504 A x \,a^{2} b^{4} d^{3} e^{4}+112 A x a \,b^{5} d^{4} e^{3}+14 A x \,b^{6} d^{5} e^{2}+1848 B x \,a^{6} e^{7}+1848 B x \,a^{5} b d \,e^{6}+1260 B x \,a^{4} b^{2} d^{2} e^{5}+672 B x \,a^{3} b^{3} d^{3} e^{4}+280 B x \,a^{2} b^{4} d^{4} e^{3}+84 B x a \,b^{5} d^{5} e^{2}+14 B x \,b^{6} d^{6} e +1716 A \,a^{6} e^{7}+792 A \,a^{5} b d \,e^{6}+330 A \,a^{4} b^{2} d^{2} e^{5}+120 A \,a^{3} b^{3} d^{3} e^{4}+36 A \,a^{2} b^{4} d^{4} e^{3}+8 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e +132 B \,a^{6} d \,e^{6}+132 B \,a^{5} b \,d^{2} e^{5}+90 B \,a^{4} b^{2} d^{3} e^{4}+48 B \,a^{3} b^{3} d^{4} e^{3}+20 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e +b^{6} B \,d^{7}}{24024 e^{8} \left (e x +d \right )^{14}}\) | \(911\) |
parallelrisch | \(-\frac {3432 B \,b^{6} x^{7} e^{13}+3003 A \,b^{6} e^{13} x^{6}+18018 B a \,b^{5} e^{13} x^{6}+3003 B \,b^{6} d \,e^{12} x^{6}+16016 A a \,b^{5} e^{13} x^{5}+2002 A \,b^{6} d \,e^{12} x^{5}+40040 B \,a^{2} b^{4} e^{13} x^{5}+12012 B a \,b^{5} d \,e^{12} x^{5}+2002 B \,b^{6} d^{2} e^{11} x^{5}+36036 A \,a^{2} b^{4} e^{13} x^{4}+8008 A a \,b^{5} d \,e^{12} x^{4}+1001 A \,b^{6} d^{2} e^{11} x^{4}+48048 B \,a^{3} b^{3} e^{13} x^{4}+20020 B \,a^{2} b^{4} d \,e^{12} x^{4}+6006 B a \,b^{5} d^{2} e^{11} x^{4}+1001 B \,b^{6} d^{3} e^{10} x^{4}+43680 A \,a^{3} b^{3} e^{13} x^{3}+13104 A \,a^{2} b^{4} d \,e^{12} x^{3}+2912 A a \,b^{5} d^{2} e^{11} x^{3}+364 A \,b^{6} d^{3} e^{10} x^{3}+32760 B \,a^{4} b^{2} e^{13} x^{3}+17472 B \,a^{3} b^{3} d \,e^{12} x^{3}+7280 B \,a^{2} b^{4} d^{2} e^{11} x^{3}+2184 B a \,b^{5} d^{3} e^{10} x^{3}+364 B \,b^{6} d^{4} e^{9} x^{3}+30030 A \,a^{4} b^{2} e^{13} x^{2}+10920 A \,a^{3} b^{3} d \,e^{12} x^{2}+3276 A \,a^{2} b^{4} d^{2} e^{11} x^{2}+728 A a \,b^{5} d^{3} e^{10} x^{2}+91 A \,b^{6} d^{4} e^{9} x^{2}+12012 B \,a^{5} b \,e^{13} x^{2}+8190 B \,a^{4} b^{2} d \,e^{12} x^{2}+4368 B \,a^{3} b^{3} d^{2} e^{11} x^{2}+1820 B \,a^{2} b^{4} d^{3} e^{10} x^{2}+546 B a \,b^{5} d^{4} e^{9} x^{2}+91 B \,b^{6} d^{5} e^{8} x^{2}+11088 A \,a^{5} b \,e^{13} x +4620 A \,a^{4} b^{2} d \,e^{12} x +1680 A \,a^{3} b^{3} d^{2} e^{11} x +504 A \,a^{2} b^{4} d^{3} e^{10} x +112 A a \,b^{5} d^{4} e^{9} x +14 A \,b^{6} d^{5} e^{8} x +1848 B \,a^{6} e^{13} x +1848 B \,a^{5} b d \,e^{12} x +1260 B \,a^{4} b^{2} d^{2} e^{11} x +672 B \,a^{3} b^{3} d^{3} e^{10} x +280 B \,a^{2} b^{4} d^{4} e^{9} x +84 B a \,b^{5} d^{5} e^{8} x +14 B \,b^{6} d^{6} e^{7} x +1716 A \,a^{6} e^{13}+792 A \,a^{5} b d \,e^{12}+330 A \,a^{4} b^{2} d^{2} e^{11}+120 A \,a^{3} b^{3} d^{3} e^{10}+36 A \,a^{2} b^{4} d^{4} e^{9}+8 A a \,b^{5} d^{5} e^{8}+A \,b^{6} d^{6} e^{7}+132 B \,a^{6} d \,e^{12}+132 B \,a^{5} b \,d^{2} e^{11}+90 B \,a^{4} b^{2} d^{3} e^{10}+48 B \,a^{3} b^{3} d^{4} e^{9}+20 B \,a^{2} b^{4} d^{5} e^{8}+6 B a \,b^{5} d^{6} e^{7}+B \,b^{6} d^{7} e^{6}}{24024 e^{14} \left (e x +d \right )^{14}}\) | \(920\) |
[In]
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Leaf count of result is larger than twice the leaf count of optimal. 902 vs. \(2 (276) = 552\).
Time = 0.26 (sec) , antiderivative size = 902, normalized size of antiderivative = 3.09 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{15}} \, dx=-\frac {3432 \, B b^{6} e^{7} x^{7} + B b^{6} d^{7} + 1716 \, A a^{6} e^{7} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 4 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 12 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 30 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 66 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 132 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 3003 \, {\left (B b^{6} d e^{6} + {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 2002 \, {\left (B b^{6} d^{2} e^{5} + {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 4 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 1001 \, {\left (B b^{6} d^{3} e^{4} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 4 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 12 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 364 \, {\left (B b^{6} d^{4} e^{3} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 4 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 12 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 30 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 91 \, {\left (B b^{6} d^{5} e^{2} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 4 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 12 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 30 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 66 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 14 \, {\left (B b^{6} d^{6} e + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 4 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 12 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 30 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 66 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 132 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{24024 \, {\left (e^{22} x^{14} + 14 \, d e^{21} x^{13} + 91 \, d^{2} e^{20} x^{12} + 364 \, d^{3} e^{19} x^{11} + 1001 \, d^{4} e^{18} x^{10} + 2002 \, d^{5} e^{17} x^{9} + 3003 \, d^{6} e^{16} x^{8} + 3432 \, d^{7} e^{15} x^{7} + 3003 \, d^{8} e^{14} x^{6} + 2002 \, d^{9} e^{13} x^{5} + 1001 \, d^{10} e^{12} x^{4} + 364 \, d^{11} e^{11} x^{3} + 91 \, d^{12} e^{10} x^{2} + 14 \, d^{13} e^{9} x + d^{14} e^{8}\right )}} \]
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Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{15}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 902 vs. \(2 (276) = 552\).
Time = 0.27 (sec) , antiderivative size = 902, normalized size of antiderivative = 3.09 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{15}} \, dx=-\frac {3432 \, B b^{6} e^{7} x^{7} + B b^{6} d^{7} + 1716 \, A a^{6} e^{7} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 4 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 12 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 30 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 66 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 132 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 3003 \, {\left (B b^{6} d e^{6} + {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 2002 \, {\left (B b^{6} d^{2} e^{5} + {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 4 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 1001 \, {\left (B b^{6} d^{3} e^{4} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 4 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 12 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 364 \, {\left (B b^{6} d^{4} e^{3} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 4 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 12 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 30 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 91 \, {\left (B b^{6} d^{5} e^{2} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 4 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 12 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 30 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 66 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 14 \, {\left (B b^{6} d^{6} e + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 4 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 12 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 30 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 66 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 132 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{24024 \, {\left (e^{22} x^{14} + 14 \, d e^{21} x^{13} + 91 \, d^{2} e^{20} x^{12} + 364 \, d^{3} e^{19} x^{11} + 1001 \, d^{4} e^{18} x^{10} + 2002 \, d^{5} e^{17} x^{9} + 3003 \, d^{6} e^{16} x^{8} + 3432 \, d^{7} e^{15} x^{7} + 3003 \, d^{8} e^{14} x^{6} + 2002 \, d^{9} e^{13} x^{5} + 1001 \, d^{10} e^{12} x^{4} + 364 \, d^{11} e^{11} x^{3} + 91 \, d^{12} e^{10} x^{2} + 14 \, d^{13} e^{9} x + d^{14} e^{8}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (276) = 552\).
Time = 0.30 (sec) , antiderivative size = 910, normalized size of antiderivative = 3.12 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{15}} \, dx=-\frac {3432 \, B b^{6} e^{7} x^{7} + 3003 \, B b^{6} d e^{6} x^{6} + 18018 \, B a b^{5} e^{7} x^{6} + 3003 \, A b^{6} e^{7} x^{6} + 2002 \, B b^{6} d^{2} e^{5} x^{5} + 12012 \, B a b^{5} d e^{6} x^{5} + 2002 \, A b^{6} d e^{6} x^{5} + 40040 \, B a^{2} b^{4} e^{7} x^{5} + 16016 \, A a b^{5} e^{7} x^{5} + 1001 \, B b^{6} d^{3} e^{4} x^{4} + 6006 \, B a b^{5} d^{2} e^{5} x^{4} + 1001 \, A b^{6} d^{2} e^{5} x^{4} + 20020 \, B a^{2} b^{4} d e^{6} x^{4} + 8008 \, A a b^{5} d e^{6} x^{4} + 48048 \, B a^{3} b^{3} e^{7} x^{4} + 36036 \, A a^{2} b^{4} e^{7} x^{4} + 364 \, B b^{6} d^{4} e^{3} x^{3} + 2184 \, B a b^{5} d^{3} e^{4} x^{3} + 364 \, A b^{6} d^{3} e^{4} x^{3} + 7280 \, B a^{2} b^{4} d^{2} e^{5} x^{3} + 2912 \, A a b^{5} d^{2} e^{5} x^{3} + 17472 \, B a^{3} b^{3} d e^{6} x^{3} + 13104 \, A a^{2} b^{4} d e^{6} x^{3} + 32760 \, B a^{4} b^{2} e^{7} x^{3} + 43680 \, A a^{3} b^{3} e^{7} x^{3} + 91 \, B b^{6} d^{5} e^{2} x^{2} + 546 \, B a b^{5} d^{4} e^{3} x^{2} + 91 \, A b^{6} d^{4} e^{3} x^{2} + 1820 \, B a^{2} b^{4} d^{3} e^{4} x^{2} + 728 \, A a b^{5} d^{3} e^{4} x^{2} + 4368 \, B a^{3} b^{3} d^{2} e^{5} x^{2} + 3276 \, A a^{2} b^{4} d^{2} e^{5} x^{2} + 8190 \, B a^{4} b^{2} d e^{6} x^{2} + 10920 \, A a^{3} b^{3} d e^{6} x^{2} + 12012 \, B a^{5} b e^{7} x^{2} + 30030 \, A a^{4} b^{2} e^{7} x^{2} + 14 \, B b^{6} d^{6} e x + 84 \, B a b^{5} d^{5} e^{2} x + 14 \, A b^{6} d^{5} e^{2} x + 280 \, B a^{2} b^{4} d^{4} e^{3} x + 112 \, A a b^{5} d^{4} e^{3} x + 672 \, B a^{3} b^{3} d^{3} e^{4} x + 504 \, A a^{2} b^{4} d^{3} e^{4} x + 1260 \, B a^{4} b^{2} d^{2} e^{5} x + 1680 \, A a^{3} b^{3} d^{2} e^{5} x + 1848 \, B a^{5} b d e^{6} x + 4620 \, A a^{4} b^{2} d e^{6} x + 1848 \, B a^{6} e^{7} x + 11088 \, A a^{5} b e^{7} x + B b^{6} d^{7} + 6 \, B a b^{5} d^{6} e + A b^{6} d^{6} e + 20 \, B a^{2} b^{4} d^{5} e^{2} + 8 \, A a b^{5} d^{5} e^{2} + 48 \, B a^{3} b^{3} d^{4} e^{3} + 36 \, A a^{2} b^{4} d^{4} e^{3} + 90 \, B a^{4} b^{2} d^{3} e^{4} + 120 \, A a^{3} b^{3} d^{3} e^{4} + 132 \, B a^{5} b d^{2} e^{5} + 330 \, A a^{4} b^{2} d^{2} e^{5} + 132 \, B a^{6} d e^{6} + 792 \, A a^{5} b d e^{6} + 1716 \, A a^{6} e^{7}}{24024 \, {\left (e x + d\right )}^{14} e^{8}} \]
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Time = 2.60 (sec) , antiderivative size = 918, normalized size of antiderivative = 3.14 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{15}} \, dx=-\frac {\frac {132\,B\,a^6\,d\,e^6+1716\,A\,a^6\,e^7+132\,B\,a^5\,b\,d^2\,e^5+792\,A\,a^5\,b\,d\,e^6+90\,B\,a^4\,b^2\,d^3\,e^4+330\,A\,a^4\,b^2\,d^2\,e^5+48\,B\,a^3\,b^3\,d^4\,e^3+120\,A\,a^3\,b^3\,d^3\,e^4+20\,B\,a^2\,b^4\,d^5\,e^2+36\,A\,a^2\,b^4\,d^4\,e^3+6\,B\,a\,b^5\,d^6\,e+8\,A\,a\,b^5\,d^5\,e^2+B\,b^6\,d^7+A\,b^6\,d^6\,e}{24024\,e^8}+\frac {x\,\left (132\,B\,a^6\,e^6+132\,B\,a^5\,b\,d\,e^5+792\,A\,a^5\,b\,e^6+90\,B\,a^4\,b^2\,d^2\,e^4+330\,A\,a^4\,b^2\,d\,e^5+48\,B\,a^3\,b^3\,d^3\,e^3+120\,A\,a^3\,b^3\,d^2\,e^4+20\,B\,a^2\,b^4\,d^4\,e^2+36\,A\,a^2\,b^4\,d^3\,e^3+6\,B\,a\,b^5\,d^5\,e+8\,A\,a\,b^5\,d^4\,e^2+B\,b^6\,d^6+A\,b^6\,d^5\,e\right )}{1716\,e^7}+\frac {b^3\,x^4\,\left (48\,B\,a^3\,e^3+20\,B\,a^2\,b\,d\,e^2+36\,A\,a^2\,b\,e^3+6\,B\,a\,b^2\,d^2\,e+8\,A\,a\,b^2\,d\,e^2+B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{24\,e^4}+\frac {b^5\,x^6\,\left (A\,b\,e+6\,B\,a\,e+B\,b\,d\right )}{8\,e^2}+\frac {b\,x^2\,\left (132\,B\,a^5\,e^5+90\,B\,a^4\,b\,d\,e^4+330\,A\,a^4\,b\,e^5+48\,B\,a^3\,b^2\,d^2\,e^3+120\,A\,a^3\,b^2\,d\,e^4+20\,B\,a^2\,b^3\,d^3\,e^2+36\,A\,a^2\,b^3\,d^2\,e^3+6\,B\,a\,b^4\,d^4\,e+8\,A\,a\,b^4\,d^3\,e^2+B\,b^5\,d^5+A\,b^5\,d^4\,e\right )}{264\,e^6}+\frac {b^2\,x^3\,\left (90\,B\,a^4\,e^4+48\,B\,a^3\,b\,d\,e^3+120\,A\,a^3\,b\,e^4+20\,B\,a^2\,b^2\,d^2\,e^2+36\,A\,a^2\,b^2\,d\,e^3+6\,B\,a\,b^3\,d^3\,e+8\,A\,a\,b^3\,d^2\,e^2+B\,b^4\,d^4+A\,b^4\,d^3\,e\right )}{66\,e^5}+\frac {b^4\,x^5\,\left (20\,B\,a^2\,e^2+6\,B\,a\,b\,d\,e+8\,A\,a\,b\,e^2+B\,b^2\,d^2+A\,b^2\,d\,e\right )}{12\,e^3}+\frac {B\,b^6\,x^7}{7\,e}}{d^{14}+14\,d^{13}\,e\,x+91\,d^{12}\,e^2\,x^2+364\,d^{11}\,e^3\,x^3+1001\,d^{10}\,e^4\,x^4+2002\,d^9\,e^5\,x^5+3003\,d^8\,e^6\,x^6+3432\,d^7\,e^7\,x^7+3003\,d^6\,e^8\,x^8+2002\,d^5\,e^9\,x^9+1001\,d^4\,e^{10}\,x^{10}+364\,d^3\,e^{11}\,x^{11}+91\,d^2\,e^{12}\,x^{12}+14\,d\,e^{13}\,x^{13}+e^{14}\,x^{14}} \]
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