\(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx\) [62]

Optimal result

Integrand size = 20, antiderivative size = 292 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx=\frac {(b d-a e)^6 (B d-A e)}{12 e^8 (d+e x)^{12}}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{11 e^8 (d+e x)^{11}}+\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{10 e^8 (d+e x)^{10}}-\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{9 e^8 (d+e x)^9}+\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{8 e^8 (d+e x)^8}-\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e)}{7 e^8 (d+e x)^7}+\frac {b^5 (7 b B d-A b e-6 a B e)}{6 e^8 (d+e x)^6}-\frac {b^6 B}{5 e^8 (d+e x)^5} \] Output:

1/12*(-a*e+b*d)^6*(-A*e+B*d)/e^8/(e*x+d)^12-1/11*(-a*e+b*d)^5*(-6*A*b*e-B* 
a*e+7*B*b*d)/e^8/(e*x+d)^11+3/10*b*(-a*e+b*d)^4*(-5*A*b*e-2*B*a*e+7*B*b*d) 
/e^8/(e*x+d)^10-5/9*b^2*(-a*e+b*d)^3*(-4*A*b*e-3*B*a*e+7*B*b*d)/e^8/(e*x+d 
)^9+5/8*b^3*(-a*e+b*d)^2*(-3*A*b*e-4*B*a*e+7*B*b*d)/e^8/(e*x+d)^8-3/7*b^4* 
(-a*e+b*d)*(-2*A*b*e-5*B*a*e+7*B*b*d)/e^8/(e*x+d)^7+1/6*b^5*(-A*b*e-6*B*a* 
e+7*B*b*d)/e^8/(e*x+d)^6-1/5*b^6*B/e^8/(e*x+d)^5
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(600\) vs. \(2(292)=584\).

Time = 0.16 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.05 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx=-\frac {210 a^6 e^6 (11 A e+B (d+12 e x))+252 a^5 b e^5 \left (5 A e (d+12 e x)+B \left (d^2+12 d e x+66 e^2 x^2\right )\right )+210 a^4 b^2 e^4 \left (3 A e \left (d^2+12 d e x+66 e^2 x^2\right )+B \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )\right )+140 a^3 b^3 e^3 \left (2 A e \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )+B \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )\right )+15 a^2 b^4 e^2 \left (7 A e \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )+5 B \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )\right )+30 a b^5 e \left (A e \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )+B \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )\right )+b^6 \left (5 A e \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )+7 B \left (d^7+12 d^6 e x+66 d^5 e^2 x^2+220 d^4 e^3 x^3+495 d^3 e^4 x^4+792 d^2 e^5 x^5+924 d e^6 x^6+792 e^7 x^7\right )\right )}{27720 e^8 (d+e x)^{12}} \] Input:

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^13,x]
 

Output:

-1/27720*(210*a^6*e^6*(11*A*e + B*(d + 12*e*x)) + 252*a^5*b*e^5*(5*A*e*(d 
+ 12*e*x) + B*(d^2 + 12*d*e*x + 66*e^2*x^2)) + 210*a^4*b^2*e^4*(3*A*e*(d^2 
 + 12*d*e*x + 66*e^2*x^2) + B*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x 
^3)) + 140*a^3*b^3*e^3*(2*A*e*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x 
^3) + B*(d^4 + 12*d^3*e*x + 66*d^2*e^2*x^2 + 220*d*e^3*x^3 + 495*e^4*x^4)) 
 + 15*a^2*b^4*e^2*(7*A*e*(d^4 + 12*d^3*e*x + 66*d^2*e^2*x^2 + 220*d*e^3*x^ 
3 + 495*e^4*x^4) + 5*B*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^ 
3 + 495*d*e^4*x^4 + 792*e^5*x^5)) + 30*a*b^5*e*(A*e*(d^5 + 12*d^4*e*x + 66 
*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5) + B*(d^6 + 1 
2*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5 
*x^5 + 924*e^6*x^6)) + b^6*(5*A*e*(d^6 + 12*d^5*e*x + 66*d^4*e^2*x^2 + 220 
*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924*e^6*x^6) + 7*B*(d^7 + 
 12*d^6*e*x + 66*d^5*e^2*x^2 + 220*d^4*e^3*x^3 + 495*d^3*e^4*x^4 + 792*d^2 
*e^5*x^5 + 924*d*e^6*x^6 + 792*e^7*x^7)))/(e^8*(d + e*x)^12)
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 292, 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 = {86, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int \left (\frac {b^5 (6 a B e+A b e-7 b B d)}{e^7 (d+e x)^7}-\frac {3 b^4 (b d-a e) (5 a B e+2 A b e-7 b B d)}{e^7 (d+e x)^8}+\frac {5 b^3 (b d-a e)^2 (4 a B e+3 A b e-7 b B d)}{e^7 (d+e x)^9}-\frac {5 b^2 (b d-a e)^3 (3 a B e+4 A b e-7 b B d)}{e^7 (d+e x)^{10}}+\frac {3 b (b d-a e)^4 (2 a B e+5 A b e-7 b B d)}{e^7 (d+e x)^{11}}+\frac {(a e-b d)^5 (a B e+6 A b e-7 b B d)}{e^7 (d+e x)^{12}}+\frac {(a e-b d)^6 (A e-B d)}{e^7 (d+e x)^{13}}+\frac {b^6 B}{e^7 (d+e x)^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b^5 (-6 a B e-A b e+7 b B d)}{6 e^8 (d+e x)^6}-\frac {3 b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{7 e^8 (d+e x)^7}+\frac {5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{8 e^8 (d+e x)^8}-\frac {5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{9 e^8 (d+e x)^9}+\frac {3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{10 e^8 (d+e x)^{10}}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{11 e^8 (d+e x)^{11}}+\frac {(b d-a e)^6 (B d-A e)}{12 e^8 (d+e x)^{12}}-\frac {b^6 B}{5 e^8 (d+e x)^5}\)

Input:

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^13,x]
 

Output:

((b*d - a*e)^6*(B*d - A*e))/(12*e^8*(d + e*x)^12) - ((b*d - a*e)^5*(7*b*B* 
d - 6*A*b*e - a*B*e))/(11*e^8*(d + e*x)^11) + (3*b*(b*d - a*e)^4*(7*b*B*d 
- 5*A*b*e - 2*a*B*e))/(10*e^8*(d + e*x)^10) - (5*b^2*(b*d - a*e)^3*(7*b*B* 
d - 4*A*b*e - 3*a*B*e))/(9*e^8*(d + e*x)^9) + (5*b^3*(b*d - a*e)^2*(7*b*B* 
d - 3*A*b*e - 4*a*B*e))/(8*e^8*(d + e*x)^8) - (3*b^4*(b*d - a*e)*(7*b*B*d 
- 2*A*b*e - 5*a*B*e))/(7*e^8*(d + e*x)^7) + (b^5*(7*b*B*d - A*b*e - 6*a*B* 
e))/(6*e^8*(d + e*x)^6) - (b^6*B)/(5*e^8*(d + e*x)^5)
 

Defintions of rubi rules used

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(788\) vs. \(2(276)=552\).

Time = 0.25 (sec) , antiderivative size = 789, normalized size of antiderivative = 2.70

method	result	size
risch	\(\frac {-\frac {b^{6} B \,x^{7}}{5 e}-\frac {b^{5} \left (5 A b e +30 B a e +7 B b d \right ) x^{6}}{30 e^{2}}-\frac {b^{4} \left (30 A a b \,e^{2}+5 A \,b^{2} d e +75 B \,a^{2} e^{2}+30 B a b d e +7 b^{2} B \,d^{2}\right ) x^{5}}{35 e^{3}}-\frac {b^{3} \left (105 A \,a^{2} b \,e^{3}+30 A a \,b^{2} d \,e^{2}+5 A \,b^{3} d^{2} e +140 B \,a^{3} e^{3}+75 B \,a^{2} b d \,e^{2}+30 B a \,b^{2} d^{2} e +7 b^{3} B \,d^{3}\right ) x^{4}}{56 e^{4}}-\frac {b^{2} \left (280 A \,a^{3} b \,e^{4}+105 A \,a^{2} b^{2} d \,e^{3}+30 A a \,b^{3} d^{2} e^{2}+5 A \,b^{4} d^{3} e +210 B \,a^{4} e^{4}+140 B \,a^{3} b d \,e^{3}+75 B \,a^{2} b^{2} d^{2} e^{2}+30 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) x^{3}}{126 e^{5}}-\frac {b \left (630 A \,a^{4} b \,e^{5}+280 A \,a^{3} b^{2} d \,e^{4}+105 A \,a^{2} b^{3} d^{2} e^{3}+30 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +252 B \,a^{5} e^{5}+210 B \,a^{4} b d \,e^{4}+140 B \,a^{3} b^{2} d^{2} e^{3}+75 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e +7 B \,b^{5} d^{5}\right ) x^{2}}{420 e^{6}}-\frac {\left (1260 A \,a^{5} b \,e^{6}+630 A \,a^{4} b^{2} d \,e^{5}+280 A \,a^{3} b^{3} d^{2} e^{4}+105 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}+5 A \,b^{6} d^{5} e +210 B \,a^{6} e^{6}+252 B \,a^{5} b d \,e^{5}+210 B \,a^{4} b^{2} d^{2} e^{4}+140 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}+30 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}\right ) x}{2310 e^{7}}-\frac {2310 a^{6} A \,e^{7}+1260 A \,a^{5} b d \,e^{6}+630 A \,a^{4} b^{2} d^{2} e^{5}+280 A \,a^{3} b^{3} d^{3} e^{4}+105 A \,a^{2} b^{4} d^{4} e^{3}+30 A a \,b^{5} d^{5} e^{2}+5 A \,b^{6} d^{6} e +210 B \,a^{6} d \,e^{6}+252 B \,a^{5} b \,d^{2} e^{5}+210 B \,a^{4} b^{2} d^{3} e^{4}+140 B \,a^{3} b^{3} d^{4} e^{3}+75 B \,a^{2} b^{4} d^{5} e^{2}+30 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{27720 e^{8}}}{\left (e x +d \right )^{12}}\)	\(789\)
default	\(-\frac {b^{5} \left (A b e +6 B a e -7 B b d \right )}{6 e^{8} \left (e x +d \right )^{6}}-\frac {3 b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right )}{7 e^{8} \left (e x +d \right )^{7}}-\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right )}{8 e^{8} \left (e x +d \right )^{8}}-\frac {b^{6} B}{5 e^{8} \left (e x +d \right )^{5}}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{11 e^{8} \left (e x +d \right )^{11}}-\frac {5 b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{9 e^{8} \left (e x +d \right )^{9}}-\frac {3 b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{10 e^{8} \left (e x +d \right )^{10}}-\frac {a^{6} A \,e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{12 e^{8} \left (e x +d \right )^{12}}\)	\(814\)
norman	\(\frac {-\frac {b^{6} B \,x^{7}}{5 e}-\frac {\left (5 A \,b^{6} e^{5}+30 B a \,b^{5} e^{5}+7 b^{6} B d \,e^{4}\right ) x^{6}}{30 e^{6}}-\frac {\left (30 A a \,b^{5} e^{6}+5 A \,b^{6} d \,e^{5}+75 B \,a^{2} b^{4} e^{6}+30 B a \,b^{5} d \,e^{5}+7 b^{6} B \,d^{2} e^{4}\right ) x^{5}}{35 e^{7}}-\frac {\left (105 A \,a^{2} b^{4} e^{7}+30 A a \,b^{5} d \,e^{6}+5 A \,b^{6} d^{2} e^{5}+140 B \,a^{3} b^{3} e^{7}+75 B \,a^{2} b^{4} d \,e^{6}+30 B a \,b^{5} d^{2} e^{5}+7 b^{6} B \,d^{3} e^{4}\right ) x^{4}}{56 e^{8}}-\frac {\left (280 A \,a^{3} b^{3} e^{8}+105 A \,a^{2} b^{4} d \,e^{7}+30 A a \,b^{5} d^{2} e^{6}+5 A \,b^{6} d^{3} e^{5}+210 B \,a^{4} b^{2} e^{8}+140 B \,a^{3} b^{3} d \,e^{7}+75 B \,a^{2} b^{4} d^{2} e^{6}+30 B a \,b^{5} d^{3} e^{5}+7 b^{6} B \,d^{4} e^{4}\right ) x^{3}}{126 e^{9}}-\frac {\left (630 A \,a^{4} b^{2} e^{9}+280 A \,a^{3} b^{3} d \,e^{8}+105 A \,a^{2} b^{4} d^{2} e^{7}+30 A a \,b^{5} d^{3} e^{6}+5 A \,b^{6} d^{4} e^{5}+252 B \,a^{5} b \,e^{9}+210 B \,a^{4} b^{2} d \,e^{8}+140 B \,a^{3} b^{3} d^{2} e^{7}+75 B \,a^{2} b^{4} d^{3} e^{6}+30 B a \,b^{5} d^{4} e^{5}+7 B \,b^{6} d^{5} e^{4}\right ) x^{2}}{420 e^{10}}-\frac {\left (1260 A \,a^{5} b \,e^{10}+630 A \,a^{4} b^{2} d \,e^{9}+280 A \,a^{3} b^{3} d^{2} e^{8}+105 A \,a^{2} b^{4} d^{3} e^{7}+30 A a \,b^{5} d^{4} e^{6}+5 A \,b^{6} d^{5} e^{5}+210 B \,a^{6} e^{10}+252 B \,a^{5} b d \,e^{9}+210 B \,a^{4} b^{2} d^{2} e^{8}+140 B \,a^{3} b^{3} d^{3} e^{7}+75 B \,a^{2} b^{4} d^{4} e^{6}+30 B a \,b^{5} d^{5} e^{5}+7 b^{6} B \,d^{6} e^{4}\right ) x}{2310 e^{11}}-\frac {2310 a^{6} A \,e^{11}+1260 A \,a^{5} b d \,e^{10}+630 A \,a^{4} b^{2} d^{2} e^{9}+280 A \,a^{3} b^{3} d^{3} e^{8}+105 A \,a^{2} b^{4} d^{4} e^{7}+30 A a \,b^{5} d^{5} e^{6}+5 A \,b^{6} d^{6} e^{5}+210 B \,a^{6} d \,e^{10}+252 B \,a^{5} b \,d^{2} e^{9}+210 B \,a^{4} b^{2} d^{3} e^{8}+140 B \,a^{3} b^{3} d^{4} e^{7}+75 B \,a^{2} b^{4} d^{5} e^{6}+30 B a \,b^{5} d^{6} e^{5}+7 B \,b^{6} d^{7} e^{4}}{27720 e^{12}}}{\left (e x +d \right )^{12}}\)	\(858\)
gosper	\(-\frac {5544 B \,x^{7} b^{6} e^{7}+4620 A \,x^{6} b^{6} e^{7}+27720 B \,x^{6} a \,b^{5} e^{7}+6468 B \,x^{6} b^{6} d \,e^{6}+23760 A \,x^{5} a \,b^{5} e^{7}+3960 A \,x^{5} b^{6} d \,e^{6}+59400 B \,x^{5} a^{2} b^{4} e^{7}+23760 B \,x^{5} a \,b^{5} d \,e^{6}+5544 B \,x^{5} b^{6} d^{2} e^{5}+51975 A \,x^{4} a^{2} b^{4} e^{7}+14850 A \,x^{4} a \,b^{5} d \,e^{6}+2475 A \,x^{4} b^{6} d^{2} e^{5}+69300 B \,x^{4} a^{3} b^{3} e^{7}+37125 B \,x^{4} a^{2} b^{4} d \,e^{6}+14850 B \,x^{4} a \,b^{5} d^{2} e^{5}+3465 B \,x^{4} b^{6} d^{3} e^{4}+61600 A \,x^{3} a^{3} b^{3} e^{7}+23100 A \,x^{3} a^{2} b^{4} d \,e^{6}+6600 A \,x^{3} a \,b^{5} d^{2} e^{5}+1100 A \,x^{3} b^{6} d^{3} e^{4}+46200 B \,x^{3} a^{4} b^{2} e^{7}+30800 B \,x^{3} a^{3} b^{3} d \,e^{6}+16500 B \,x^{3} a^{2} b^{4} d^{2} e^{5}+6600 B \,x^{3} a \,b^{5} d^{3} e^{4}+1540 B \,x^{3} b^{6} d^{4} e^{3}+41580 A \,x^{2} a^{4} b^{2} e^{7}+18480 A \,x^{2} a^{3} b^{3} d \,e^{6}+6930 A \,x^{2} a^{2} b^{4} d^{2} e^{5}+1980 A \,x^{2} a \,b^{5} d^{3} e^{4}+330 A \,x^{2} b^{6} d^{4} e^{3}+16632 B \,x^{2} a^{5} b \,e^{7}+13860 B \,x^{2} a^{4} b^{2} d \,e^{6}+9240 B \,x^{2} a^{3} b^{3} d^{2} e^{5}+4950 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+1980 B \,x^{2} a \,b^{5} d^{4} e^{3}+462 B \,x^{2} b^{6} d^{5} e^{2}+15120 A x \,a^{5} b \,e^{7}+7560 A x \,a^{4} b^{2} d \,e^{6}+3360 A x \,a^{3} b^{3} d^{2} e^{5}+1260 A x \,a^{2} b^{4} d^{3} e^{4}+360 A x a \,b^{5} d^{4} e^{3}+60 A x \,b^{6} d^{5} e^{2}+2520 B x \,a^{6} e^{7}+3024 B x \,a^{5} b d \,e^{6}+2520 B x \,a^{4} b^{2} d^{2} e^{5}+1680 B x \,a^{3} b^{3} d^{3} e^{4}+900 B x \,a^{2} b^{4} d^{4} e^{3}+360 B x a \,b^{5} d^{5} e^{2}+84 B x \,b^{6} d^{6} e +2310 a^{6} A \,e^{7}+1260 A \,a^{5} b d \,e^{6}+630 A \,a^{4} b^{2} d^{2} e^{5}+280 A \,a^{3} b^{3} d^{3} e^{4}+105 A \,a^{2} b^{4} d^{4} e^{3}+30 A a \,b^{5} d^{5} e^{2}+5 A \,b^{6} d^{6} e +210 B \,a^{6} d \,e^{6}+252 B \,a^{5} b \,d^{2} e^{5}+210 B \,a^{4} b^{2} d^{3} e^{4}+140 B \,a^{3} b^{3} d^{4} e^{3}+75 B \,a^{2} b^{4} d^{5} e^{2}+30 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{27720 e^{8} \left (e x +d \right )^{12}}\)	\(913\)
orering	\(-\frac {5544 B \,x^{7} b^{6} e^{7}+4620 A \,x^{6} b^{6} e^{7}+27720 B \,x^{6} a \,b^{5} e^{7}+6468 B \,x^{6} b^{6} d \,e^{6}+23760 A \,x^{5} a \,b^{5} e^{7}+3960 A \,x^{5} b^{6} d \,e^{6}+59400 B \,x^{5} a^{2} b^{4} e^{7}+23760 B \,x^{5} a \,b^{5} d \,e^{6}+5544 B \,x^{5} b^{6} d^{2} e^{5}+51975 A \,x^{4} a^{2} b^{4} e^{7}+14850 A \,x^{4} a \,b^{5} d \,e^{6}+2475 A \,x^{4} b^{6} d^{2} e^{5}+69300 B \,x^{4} a^{3} b^{3} e^{7}+37125 B \,x^{4} a^{2} b^{4} d \,e^{6}+14850 B \,x^{4} a \,b^{5} d^{2} e^{5}+3465 B \,x^{4} b^{6} d^{3} e^{4}+61600 A \,x^{3} a^{3} b^{3} e^{7}+23100 A \,x^{3} a^{2} b^{4} d \,e^{6}+6600 A \,x^{3} a \,b^{5} d^{2} e^{5}+1100 A \,x^{3} b^{6} d^{3} e^{4}+46200 B \,x^{3} a^{4} b^{2} e^{7}+30800 B \,x^{3} a^{3} b^{3} d \,e^{6}+16500 B \,x^{3} a^{2} b^{4} d^{2} e^{5}+6600 B \,x^{3} a \,b^{5} d^{3} e^{4}+1540 B \,x^{3} b^{6} d^{4} e^{3}+41580 A \,x^{2} a^{4} b^{2} e^{7}+18480 A \,x^{2} a^{3} b^{3} d \,e^{6}+6930 A \,x^{2} a^{2} b^{4} d^{2} e^{5}+1980 A \,x^{2} a \,b^{5} d^{3} e^{4}+330 A \,x^{2} b^{6} d^{4} e^{3}+16632 B \,x^{2} a^{5} b \,e^{7}+13860 B \,x^{2} a^{4} b^{2} d \,e^{6}+9240 B \,x^{2} a^{3} b^{3} d^{2} e^{5}+4950 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+1980 B \,x^{2} a \,b^{5} d^{4} e^{3}+462 B \,x^{2} b^{6} d^{5} e^{2}+15120 A x \,a^{5} b \,e^{7}+7560 A x \,a^{4} b^{2} d \,e^{6}+3360 A x \,a^{3} b^{3} d^{2} e^{5}+1260 A x \,a^{2} b^{4} d^{3} e^{4}+360 A x a \,b^{5} d^{4} e^{3}+60 A x \,b^{6} d^{5} e^{2}+2520 B x \,a^{6} e^{7}+3024 B x \,a^{5} b d \,e^{6}+2520 B x \,a^{4} b^{2} d^{2} e^{5}+1680 B x \,a^{3} b^{3} d^{3} e^{4}+900 B x \,a^{2} b^{4} d^{4} e^{3}+360 B x a \,b^{5} d^{5} e^{2}+84 B x \,b^{6} d^{6} e +2310 a^{6} A \,e^{7}+1260 A \,a^{5} b d \,e^{6}+630 A \,a^{4} b^{2} d^{2} e^{5}+280 A \,a^{3} b^{3} d^{3} e^{4}+105 A \,a^{2} b^{4} d^{4} e^{3}+30 A a \,b^{5} d^{5} e^{2}+5 A \,b^{6} d^{6} e +210 B \,a^{6} d \,e^{6}+252 B \,a^{5} b \,d^{2} e^{5}+210 B \,a^{4} b^{2} d^{3} e^{4}+140 B \,a^{3} b^{3} d^{4} e^{3}+75 B \,a^{2} b^{4} d^{5} e^{2}+30 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{27720 e^{8} \left (e x +d \right )^{12}}\)	\(913\)
parallelrisch	\(-\frac {5544 B \,b^{6} x^{7} e^{11}+4620 A \,b^{6} e^{11} x^{6}+27720 B a \,b^{5} e^{11} x^{6}+6468 B \,b^{6} d \,e^{10} x^{6}+23760 A a \,b^{5} e^{11} x^{5}+3960 A \,b^{6} d \,e^{10} x^{5}+59400 B \,a^{2} b^{4} e^{11} x^{5}+23760 B a \,b^{5} d \,e^{10} x^{5}+5544 B \,b^{6} d^{2} e^{9} x^{5}+51975 A \,a^{2} b^{4} e^{11} x^{4}+14850 A a \,b^{5} d \,e^{10} x^{4}+2475 A \,b^{6} d^{2} e^{9} x^{4}+69300 B \,a^{3} b^{3} e^{11} x^{4}+37125 B \,a^{2} b^{4} d \,e^{10} x^{4}+14850 B a \,b^{5} d^{2} e^{9} x^{4}+3465 B \,b^{6} d^{3} e^{8} x^{4}+61600 A \,a^{3} b^{3} e^{11} x^{3}+23100 A \,a^{2} b^{4} d \,e^{10} x^{3}+6600 A a \,b^{5} d^{2} e^{9} x^{3}+1100 A \,b^{6} d^{3} e^{8} x^{3}+46200 B \,a^{4} b^{2} e^{11} x^{3}+30800 B \,a^{3} b^{3} d \,e^{10} x^{3}+16500 B \,a^{2} b^{4} d^{2} e^{9} x^{3}+6600 B a \,b^{5} d^{3} e^{8} x^{3}+1540 B \,b^{6} d^{4} e^{7} x^{3}+41580 A \,a^{4} b^{2} e^{11} x^{2}+18480 A \,a^{3} b^{3} d \,e^{10} x^{2}+6930 A \,a^{2} b^{4} d^{2} e^{9} x^{2}+1980 A a \,b^{5} d^{3} e^{8} x^{2}+330 A \,b^{6} d^{4} e^{7} x^{2}+16632 B \,a^{5} b \,e^{11} x^{2}+13860 B \,a^{4} b^{2} d \,e^{10} x^{2}+9240 B \,a^{3} b^{3} d^{2} e^{9} x^{2}+4950 B \,a^{2} b^{4} d^{3} e^{8} x^{2}+1980 B a \,b^{5} d^{4} e^{7} x^{2}+462 B \,b^{6} d^{5} e^{6} x^{2}+15120 A \,a^{5} b \,e^{11} x +7560 A \,a^{4} b^{2} d \,e^{10} x +3360 A \,a^{3} b^{3} d^{2} e^{9} x +1260 A \,a^{2} b^{4} d^{3} e^{8} x +360 A a \,b^{5} d^{4} e^{7} x +60 A \,b^{6} d^{5} e^{6} x +2520 B \,a^{6} e^{11} x +3024 B \,a^{5} b d \,e^{10} x +2520 B \,a^{4} b^{2} d^{2} e^{9} x +1680 B \,a^{3} b^{3} d^{3} e^{8} x +900 B \,a^{2} b^{4} d^{4} e^{7} x +360 B a \,b^{5} d^{5} e^{6} x +84 B \,b^{6} d^{6} e^{5} x +2310 a^{6} A \,e^{11}+1260 A \,a^{5} b d \,e^{10}+630 A \,a^{4} b^{2} d^{2} e^{9}+280 A \,a^{3} b^{3} d^{3} e^{8}+105 A \,a^{2} b^{4} d^{4} e^{7}+30 A a \,b^{5} d^{5} e^{6}+5 A \,b^{6} d^{6} e^{5}+210 B \,a^{6} d \,e^{10}+252 B \,a^{5} b \,d^{2} e^{9}+210 B \,a^{4} b^{2} d^{3} e^{8}+140 B \,a^{3} b^{3} d^{4} e^{7}+75 B \,a^{2} b^{4} d^{5} e^{6}+30 B a \,b^{5} d^{6} e^{5}+7 B \,b^{6} d^{7} e^{4}}{27720 e^{12} \left (e x +d \right )^{12}}\)	\(922\)

Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^13,x,method=_RETURNVERBOSE)
 

Output:

(-1/5*b^6*B/e*x^7-1/30*b^5/e^2*(5*A*b*e+30*B*a*e+7*B*b*d)*x^6-1/35*b^4/e^3 
*(30*A*a*b*e^2+5*A*b^2*d*e+75*B*a^2*e^2+30*B*a*b*d*e+7*B*b^2*d^2)*x^5-1/56 
/e^4*b^3*(105*A*a^2*b*e^3+30*A*a*b^2*d*e^2+5*A*b^3*d^2*e+140*B*a^3*e^3+75* 
B*a^2*b*d*e^2+30*B*a*b^2*d^2*e+7*B*b^3*d^3)*x^4-1/126*b^2/e^5*(280*A*a^3*b 
*e^4+105*A*a^2*b^2*d*e^3+30*A*a*b^3*d^2*e^2+5*A*b^4*d^3*e+210*B*a^4*e^4+14 
0*B*a^3*b*d*e^3+75*B*a^2*b^2*d^2*e^2+30*B*a*b^3*d^3*e+7*B*b^4*d^4)*x^3-1/4 
20*b/e^6*(630*A*a^4*b*e^5+280*A*a^3*b^2*d*e^4+105*A*a^2*b^3*d^2*e^3+30*A*a 
*b^4*d^3*e^2+5*A*b^5*d^4*e+252*B*a^5*e^5+210*B*a^4*b*d*e^4+140*B*a^3*b^2*d 
^2*e^3+75*B*a^2*b^3*d^3*e^2+30*B*a*b^4*d^4*e+7*B*b^5*d^5)*x^2-1/2310/e^7*( 
1260*A*a^5*b*e^6+630*A*a^4*b^2*d*e^5+280*A*a^3*b^3*d^2*e^4+105*A*a^2*b^4*d 
^3*e^3+30*A*a*b^5*d^4*e^2+5*A*b^6*d^5*e+210*B*a^6*e^6+252*B*a^5*b*d*e^5+21 
0*B*a^4*b^2*d^2*e^4+140*B*a^3*b^3*d^3*e^3+75*B*a^2*b^4*d^4*e^2+30*B*a*b^5* 
d^5*e+7*B*b^6*d^6)*x-1/27720/e^8*(2310*A*a^6*e^7+1260*A*a^5*b*d*e^6+630*A* 
a^4*b^2*d^2*e^5+280*A*a^3*b^3*d^3*e^4+105*A*a^2*b^4*d^4*e^3+30*A*a*b^5*d^5 
*e^2+5*A*b^6*d^6*e+210*B*a^6*d*e^6+252*B*a^5*b*d^2*e^5+210*B*a^4*b^2*d^3*e 
^4+140*B*a^3*b^3*d^4*e^3+75*B*a^2*b^4*d^5*e^2+30*B*a*b^5*d^6*e+7*B*b^6*d^7 
))/(e*x+d)^12
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 894 vs. \(2 (276) = 552\).

Time = 0.10 (sec) , antiderivative size = 894, normalized size of antiderivative = 3.06 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^13,x, algorithm="fricas")
 

Output:

-1/27720*(5544*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 2310*A*a^6*e^7 + 5*(6*B*a*b^5 
 + A*b^6)*d^6*e + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 35*(4*B*a^3*b^3 + 
 3*A*a^2*b^4)*d^4*e^3 + 70*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 126*(2*B* 
a^5*b + 5*A*a^4*b^2)*d^2*e^5 + 210*(B*a^6 + 6*A*a^5*b)*d*e^6 + 924*(7*B*b^ 
6*d*e^6 + 5*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 792*(7*B*b^6*d^2*e^5 + 5*(6*B*a 
*b^5 + A*b^6)*d*e^6 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 495*(7*B*b^6 
*d^3*e^4 + 5*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d* 
e^6 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 220*(7*B*b^6*d^4*e^3 + 5*( 
6*B*a*b^5 + A*b^6)*d^3*e^4 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 35*(4* 
B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 70*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 
 66*(7*B*b^6*d^5*e^2 + 5*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 15*(5*B*a^2*b^4 + 2 
*A*a*b^5)*d^3*e^4 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 70*(3*B*a^4*b 
^2 + 4*A*a^3*b^3)*d*e^6 + 126*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 12*(7*B 
*b^6*d^6*e + 5*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)* 
d^4*e^3 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 70*(3*B*a^4*b^2 + 4*A*a 
^3*b^3)*d^2*e^5 + 126*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 210*(B*a^6 + 6*A*a 
^5*b)*e^7)*x)/(e^20*x^12 + 12*d*e^19*x^11 + 66*d^2*e^18*x^10 + 220*d^3*e^1 
7*x^9 + 495*d^4*e^16*x^8 + 792*d^5*e^15*x^7 + 924*d^6*e^14*x^6 + 792*d^7*e 
^13*x^5 + 495*d^8*e^12*x^4 + 220*d^9*e^11*x^3 + 66*d^10*e^10*x^2 + 12*d^11 
*e^9*x + d^12*e^8)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**6*(B*x+A)/(e*x+d)**13,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 894 vs. \(2 (276) = 552\).

Time = 0.08 (sec) , antiderivative size = 894, normalized size of antiderivative = 3.06 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^13,x, algorithm="maxima")
 

Output:

-1/27720*(5544*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 2310*A*a^6*e^7 + 5*(6*B*a*b^5 
 + A*b^6)*d^6*e + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 35*(4*B*a^3*b^3 + 
 3*A*a^2*b^4)*d^4*e^3 + 70*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 126*(2*B* 
a^5*b + 5*A*a^4*b^2)*d^2*e^5 + 210*(B*a^6 + 6*A*a^5*b)*d*e^6 + 924*(7*B*b^ 
6*d*e^6 + 5*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 792*(7*B*b^6*d^2*e^5 + 5*(6*B*a 
*b^5 + A*b^6)*d*e^6 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 495*(7*B*b^6 
*d^3*e^4 + 5*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d* 
e^6 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 220*(7*B*b^6*d^4*e^3 + 5*( 
6*B*a*b^5 + A*b^6)*d^3*e^4 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 35*(4* 
B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 70*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 
 66*(7*B*b^6*d^5*e^2 + 5*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 15*(5*B*a^2*b^4 + 2 
*A*a*b^5)*d^3*e^4 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 70*(3*B*a^4*b 
^2 + 4*A*a^3*b^3)*d*e^6 + 126*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 12*(7*B 
*b^6*d^6*e + 5*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)* 
d^4*e^3 + 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 70*(3*B*a^4*b^2 + 4*A*a 
^3*b^3)*d^2*e^5 + 126*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 210*(B*a^6 + 6*A*a 
^5*b)*e^7)*x)/(e^20*x^12 + 12*d*e^19*x^11 + 66*d^2*e^18*x^10 + 220*d^3*e^1 
7*x^9 + 495*d^4*e^16*x^8 + 792*d^5*e^15*x^7 + 924*d^6*e^14*x^6 + 792*d^7*e 
^13*x^5 + 495*d^8*e^12*x^4 + 220*d^9*e^11*x^3 + 66*d^10*e^10*x^2 + 12*d^11 
*e^9*x + d^12*e^8)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (276) = 552\).

Time = 0.12 (sec) , antiderivative size = 912, normalized size of antiderivative = 3.12 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^13,x, algorithm="giac")
 

Output:

-1/27720*(5544*B*b^6*e^7*x^7 + 6468*B*b^6*d*e^6*x^6 + 27720*B*a*b^5*e^7*x^ 
6 + 4620*A*b^6*e^7*x^6 + 5544*B*b^6*d^2*e^5*x^5 + 23760*B*a*b^5*d*e^6*x^5 
+ 3960*A*b^6*d*e^6*x^5 + 59400*B*a^2*b^4*e^7*x^5 + 23760*A*a*b^5*e^7*x^5 + 
 3465*B*b^6*d^3*e^4*x^4 + 14850*B*a*b^5*d^2*e^5*x^4 + 2475*A*b^6*d^2*e^5*x 
^4 + 37125*B*a^2*b^4*d*e^6*x^4 + 14850*A*a*b^5*d*e^6*x^4 + 69300*B*a^3*b^3 
*e^7*x^4 + 51975*A*a^2*b^4*e^7*x^4 + 1540*B*b^6*d^4*e^3*x^3 + 6600*B*a*b^5 
*d^3*e^4*x^3 + 1100*A*b^6*d^3*e^4*x^3 + 16500*B*a^2*b^4*d^2*e^5*x^3 + 6600 
*A*a*b^5*d^2*e^5*x^3 + 30800*B*a^3*b^3*d*e^6*x^3 + 23100*A*a^2*b^4*d*e^6*x 
^3 + 46200*B*a^4*b^2*e^7*x^3 + 61600*A*a^3*b^3*e^7*x^3 + 462*B*b^6*d^5*e^2 
*x^2 + 1980*B*a*b^5*d^4*e^3*x^2 + 330*A*b^6*d^4*e^3*x^2 + 4950*B*a^2*b^4*d 
^3*e^4*x^2 + 1980*A*a*b^5*d^3*e^4*x^2 + 9240*B*a^3*b^3*d^2*e^5*x^2 + 6930* 
A*a^2*b^4*d^2*e^5*x^2 + 13860*B*a^4*b^2*d*e^6*x^2 + 18480*A*a^3*b^3*d*e^6* 
x^2 + 16632*B*a^5*b*e^7*x^2 + 41580*A*a^4*b^2*e^7*x^2 + 84*B*b^6*d^6*e*x + 
 360*B*a*b^5*d^5*e^2*x + 60*A*b^6*d^5*e^2*x + 900*B*a^2*b^4*d^4*e^3*x + 36 
0*A*a*b^5*d^4*e^3*x + 1680*B*a^3*b^3*d^3*e^4*x + 1260*A*a^2*b^4*d^3*e^4*x 
+ 2520*B*a^4*b^2*d^2*e^5*x + 3360*A*a^3*b^3*d^2*e^5*x + 3024*B*a^5*b*d*e^6 
*x + 7560*A*a^4*b^2*d*e^6*x + 2520*B*a^6*e^7*x + 15120*A*a^5*b*e^7*x + 7*B 
*b^6*d^7 + 30*B*a*b^5*d^6*e + 5*A*b^6*d^6*e + 75*B*a^2*b^4*d^5*e^2 + 30*A* 
a*b^5*d^5*e^2 + 140*B*a^3*b^3*d^4*e^3 + 105*A*a^2*b^4*d^4*e^3 + 210*B*a^4* 
b^2*d^3*e^4 + 280*A*a^3*b^3*d^3*e^4 + 252*B*a^5*b*d^2*e^5 + 630*A*a^4*b...
 

Mupad [B] (verification not implemented)

Time = 1.21 (sec) , antiderivative size = 910, normalized size of antiderivative = 3.12 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx=-\frac {\frac {210\,B\,a^6\,d\,e^6+2310\,A\,a^6\,e^7+252\,B\,a^5\,b\,d^2\,e^5+1260\,A\,a^5\,b\,d\,e^6+210\,B\,a^4\,b^2\,d^3\,e^4+630\,A\,a^4\,b^2\,d^2\,e^5+140\,B\,a^3\,b^3\,d^4\,e^3+280\,A\,a^3\,b^3\,d^3\,e^4+75\,B\,a^2\,b^4\,d^5\,e^2+105\,A\,a^2\,b^4\,d^4\,e^3+30\,B\,a\,b^5\,d^6\,e+30\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+5\,A\,b^6\,d^6\,e}{27720\,e^8}+\frac {x\,\left (210\,B\,a^6\,e^6+252\,B\,a^5\,b\,d\,e^5+1260\,A\,a^5\,b\,e^6+210\,B\,a^4\,b^2\,d^2\,e^4+630\,A\,a^4\,b^2\,d\,e^5+140\,B\,a^3\,b^3\,d^3\,e^3+280\,A\,a^3\,b^3\,d^2\,e^4+75\,B\,a^2\,b^4\,d^4\,e^2+105\,A\,a^2\,b^4\,d^3\,e^3+30\,B\,a\,b^5\,d^5\,e+30\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+5\,A\,b^6\,d^5\,e\right )}{2310\,e^7}+\frac {b^3\,x^4\,\left (140\,B\,a^3\,e^3+75\,B\,a^2\,b\,d\,e^2+105\,A\,a^2\,b\,e^3+30\,B\,a\,b^2\,d^2\,e+30\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+5\,A\,b^3\,d^2\,e\right )}{56\,e^4}+\frac {b^5\,x^6\,\left (5\,A\,b\,e+30\,B\,a\,e+7\,B\,b\,d\right )}{30\,e^2}+\frac {b\,x^2\,\left (252\,B\,a^5\,e^5+210\,B\,a^4\,b\,d\,e^4+630\,A\,a^4\,b\,e^5+140\,B\,a^3\,b^2\,d^2\,e^3+280\,A\,a^3\,b^2\,d\,e^4+75\,B\,a^2\,b^3\,d^3\,e^2+105\,A\,a^2\,b^3\,d^2\,e^3+30\,B\,a\,b^4\,d^4\,e+30\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+5\,A\,b^5\,d^4\,e\right )}{420\,e^6}+\frac {b^2\,x^3\,\left (210\,B\,a^4\,e^4+140\,B\,a^3\,b\,d\,e^3+280\,A\,a^3\,b\,e^4+75\,B\,a^2\,b^2\,d^2\,e^2+105\,A\,a^2\,b^2\,d\,e^3+30\,B\,a\,b^3\,d^3\,e+30\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+5\,A\,b^4\,d^3\,e\right )}{126\,e^5}+\frac {b^4\,x^5\,\left (75\,B\,a^2\,e^2+30\,B\,a\,b\,d\,e+30\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+5\,A\,b^2\,d\,e\right )}{35\,e^3}+\frac {B\,b^6\,x^7}{5\,e}}{d^{12}+12\,d^{11}\,e\,x+66\,d^{10}\,e^2\,x^2+220\,d^9\,e^3\,x^3+495\,d^8\,e^4\,x^4+792\,d^7\,e^5\,x^5+924\,d^6\,e^6\,x^6+792\,d^5\,e^7\,x^7+495\,d^4\,e^8\,x^8+220\,d^3\,e^9\,x^9+66\,d^2\,e^{10}\,x^{10}+12\,d\,e^{11}\,x^{11}+e^{12}\,x^{12}} \] Input:

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^13,x)
 

Output:

-((2310*A*a^6*e^7 + 7*B*b^6*d^7 + 5*A*b^6*d^6*e + 210*B*a^6*d*e^6 + 30*A*a 
*b^5*d^5*e^2 + 252*B*a^5*b*d^2*e^5 + 105*A*a^2*b^4*d^4*e^3 + 280*A*a^3*b^3 
*d^3*e^4 + 630*A*a^4*b^2*d^2*e^5 + 75*B*a^2*b^4*d^5*e^2 + 140*B*a^3*b^3*d^ 
4*e^3 + 210*B*a^4*b^2*d^3*e^4 + 1260*A*a^5*b*d*e^6 + 30*B*a*b^5*d^6*e)/(27 
720*e^8) + (x*(210*B*a^6*e^6 + 7*B*b^6*d^6 + 1260*A*a^5*b*e^6 + 5*A*b^6*d^ 
5*e + 30*A*a*b^5*d^4*e^2 + 630*A*a^4*b^2*d*e^5 + 105*A*a^2*b^4*d^3*e^3 + 2 
80*A*a^3*b^3*d^2*e^4 + 75*B*a^2*b^4*d^4*e^2 + 140*B*a^3*b^3*d^3*e^3 + 210* 
B*a^4*b^2*d^2*e^4 + 30*B*a*b^5*d^5*e + 252*B*a^5*b*d*e^5))/(2310*e^7) + (b 
^3*x^4*(140*B*a^3*e^3 + 7*B*b^3*d^3 + 105*A*a^2*b*e^3 + 5*A*b^3*d^2*e + 30 
*A*a*b^2*d*e^2 + 30*B*a*b^2*d^2*e + 75*B*a^2*b*d*e^2))/(56*e^4) + (b^5*x^6 
*(5*A*b*e + 30*B*a*e + 7*B*b*d))/(30*e^2) + (b*x^2*(252*B*a^5*e^5 + 7*B*b^ 
5*d^5 + 630*A*a^4*b*e^5 + 5*A*b^5*d^4*e + 30*A*a*b^4*d^3*e^2 + 280*A*a^3*b 
^2*d*e^4 + 105*A*a^2*b^3*d^2*e^3 + 75*B*a^2*b^3*d^3*e^2 + 140*B*a^3*b^2*d^ 
2*e^3 + 30*B*a*b^4*d^4*e + 210*B*a^4*b*d*e^4))/(420*e^6) + (b^2*x^3*(210*B 
*a^4*e^4 + 7*B*b^4*d^4 + 280*A*a^3*b*e^4 + 5*A*b^4*d^3*e + 30*A*a*b^3*d^2* 
e^2 + 105*A*a^2*b^2*d*e^3 + 75*B*a^2*b^2*d^2*e^2 + 30*B*a*b^3*d^3*e + 140* 
B*a^3*b*d*e^3))/(126*e^5) + (b^4*x^5*(75*B*a^2*e^2 + 7*B*b^2*d^2 + 30*A*a* 
b*e^2 + 5*A*b^2*d*e + 30*B*a*b*d*e))/(35*e^3) + (B*b^6*x^7)/(5*e))/(d^12 + 
 e^12*x^12 + 12*d*e^11*x^11 + 66*d^10*e^2*x^2 + 220*d^9*e^3*x^3 + 495*d^8* 
e^4*x^4 + 792*d^7*e^5*x^5 + 924*d^6*e^6*x^6 + 792*d^5*e^7*x^7 + 495*d^4...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.12 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx=\frac {-792 b^{7} e^{7} x^{7}-4620 a \,b^{6} e^{7} x^{6}-924 b^{7} d \,e^{6} x^{6}-11880 a^{2} b^{5} e^{7} x^{5}-3960 a \,b^{6} d \,e^{6} x^{5}-792 b^{7} d^{2} e^{5} x^{5}-17325 a^{3} b^{4} e^{7} x^{4}-7425 a^{2} b^{5} d \,e^{6} x^{4}-2475 a \,b^{6} d^{2} e^{5} x^{4}-495 b^{7} d^{3} e^{4} x^{4}-15400 a^{4} b^{3} e^{7} x^{3}-7700 a^{3} b^{4} d \,e^{6} x^{3}-3300 a^{2} b^{5} d^{2} e^{5} x^{3}-1100 a \,b^{6} d^{3} e^{4} x^{3}-220 b^{7} d^{4} e^{3} x^{3}-8316 a^{5} b^{2} e^{7} x^{2}-4620 a^{4} b^{3} d \,e^{6} x^{2}-2310 a^{3} b^{4} d^{2} e^{5} x^{2}-990 a^{2} b^{5} d^{3} e^{4} x^{2}-330 a \,b^{6} d^{4} e^{3} x^{2}-66 b^{7} d^{5} e^{2} x^{2}-2520 a^{6} b \,e^{7} x -1512 a^{5} b^{2} d \,e^{6} x -840 a^{4} b^{3} d^{2} e^{5} x -420 a^{3} b^{4} d^{3} e^{4} x -180 a^{2} b^{5} d^{4} e^{3} x -60 a \,b^{6} d^{5} e^{2} x -12 b^{7} d^{6} e x -330 a^{7} e^{7}-210 a^{6} b d \,e^{6}-126 a^{5} b^{2} d^{2} e^{5}-70 a^{4} b^{3} d^{3} e^{4}-35 a^{3} b^{4} d^{4} e^{3}-15 a^{2} b^{5} d^{5} e^{2}-5 a \,b^{6} d^{6} e -b^{7} d^{7}}{3960 e^{8} \left (e^{12} x^{12}+12 d \,e^{11} x^{11}+66 d^{2} e^{10} x^{10}+220 d^{3} e^{9} x^{9}+495 d^{4} e^{8} x^{8}+792 d^{5} e^{7} x^{7}+924 d^{6} e^{6} x^{6}+792 d^{7} e^{5} x^{5}+495 d^{8} e^{4} x^{4}+220 d^{9} e^{3} x^{3}+66 d^{10} e^{2} x^{2}+12 d^{11} e x +d^{12}\right )} \] Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^13,x)
 

Output:

( - 330*a**7*e**7 - 210*a**6*b*d*e**6 - 2520*a**6*b*e**7*x - 126*a**5*b**2 
*d**2*e**5 - 1512*a**5*b**2*d*e**6*x - 8316*a**5*b**2*e**7*x**2 - 70*a**4* 
b**3*d**3*e**4 - 840*a**4*b**3*d**2*e**5*x - 4620*a**4*b**3*d*e**6*x**2 - 
15400*a**4*b**3*e**7*x**3 - 35*a**3*b**4*d**4*e**3 - 420*a**3*b**4*d**3*e* 
*4*x - 2310*a**3*b**4*d**2*e**5*x**2 - 7700*a**3*b**4*d*e**6*x**3 - 17325* 
a**3*b**4*e**7*x**4 - 15*a**2*b**5*d**5*e**2 - 180*a**2*b**5*d**4*e**3*x - 
 990*a**2*b**5*d**3*e**4*x**2 - 3300*a**2*b**5*d**2*e**5*x**3 - 7425*a**2* 
b**5*d*e**6*x**4 - 11880*a**2*b**5*e**7*x**5 - 5*a*b**6*d**6*e - 60*a*b**6 
*d**5*e**2*x - 330*a*b**6*d**4*e**3*x**2 - 1100*a*b**6*d**3*e**4*x**3 - 24 
75*a*b**6*d**2*e**5*x**4 - 3960*a*b**6*d*e**6*x**5 - 4620*a*b**6*e**7*x**6 
 - b**7*d**7 - 12*b**7*d**6*e*x - 66*b**7*d**5*e**2*x**2 - 220*b**7*d**4*e 
**3*x**3 - 495*b**7*d**3*e**4*x**4 - 792*b**7*d**2*e**5*x**5 - 924*b**7*d* 
e**6*x**6 - 792*b**7*e**7*x**7)/(3960*e**8*(d**12 + 12*d**11*e*x + 66*d**1 
0*e**2*x**2 + 220*d**9*e**3*x**3 + 495*d**8*e**4*x**4 + 792*d**7*e**5*x**5 
 + 924*d**6*e**6*x**6 + 792*d**5*e**7*x**7 + 495*d**4*e**8*x**8 + 220*d**3 
*e**9*x**9 + 66*d**2*e**10*x**10 + 12*d*e**11*x**11 + e**12*x**12))