\(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx\) [63]

Optimal result

Integrand size = 20, antiderivative size = 292 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx=\frac {(b d-a e)^6 (B d-A e)}{13 e^8 (d+e x)^{13}}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{12 e^8 (d+e x)^{12}}+\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{11 e^8 (d+e x)^{11}}-\frac {b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{2 e^8 (d+e x)^{10}}+\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{9 e^8 (d+e x)^9}-\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e)}{8 e^8 (d+e x)^8}+\frac {b^5 (7 b B d-A b e-6 a B e)}{7 e^8 (d+e x)^7}-\frac {b^6 B}{6 e^8 (d+e x)^6} \] Output:

1/13*(-a*e+b*d)^6*(-A*e+B*d)/e^8/(e*x+d)^13-1/12*(-a*e+b*d)^5*(-6*A*b*e-B* 
a*e+7*B*b*d)/e^8/(e*x+d)^12+3/11*b*(-a*e+b*d)^4*(-5*A*b*e-2*B*a*e+7*B*b*d) 
/e^8/(e*x+d)^11-1/2*b^2*(-a*e+b*d)^3*(-4*A*b*e-3*B*a*e+7*B*b*d)/e^8/(e*x+d 
)^10+5/9*b^3*(-a*e+b*d)^2*(-3*A*b*e-4*B*a*e+7*B*b*d)/e^8/(e*x+d)^9-3/8*b^4 
*(-a*e+b*d)*(-2*A*b*e-5*B*a*e+7*B*b*d)/e^8/(e*x+d)^8+1/7*b^5*(-A*b*e-6*B*a 
*e+7*B*b*d)/e^8/(e*x+d)^7-1/6*b^6*B/e^8/(e*x+d)^6
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(605\) vs. \(2(292)=584\).

Time = 0.17 (sec) , antiderivative size = 605, normalized size of antiderivative = 2.07 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx=-\frac {462 a^6 e^6 (12 A e+B (d+13 e x))+252 a^5 b e^5 \left (11 A e (d+13 e x)+2 B \left (d^2+13 d e x+78 e^2 x^2\right )\right )+126 a^4 b^2 e^4 \left (10 A e \left (d^2+13 d e x+78 e^2 x^2\right )+3 B \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )\right )+56 a^3 b^3 e^3 \left (9 A e \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )+4 B \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )\right )+21 a^2 b^4 e^2 \left (8 A e \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )+5 B \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )\right )+6 a b^5 e \left (7 A e \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )+6 B \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )\right )+b^6 \left (6 A e \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )+7 B \left (d^7+13 d^6 e x+78 d^5 e^2 x^2+286 d^4 e^3 x^3+715 d^3 e^4 x^4+1287 d^2 e^5 x^5+1716 d e^6 x^6+1716 e^7 x^7\right )\right )}{72072 e^8 (d+e x)^{13}} \] Input:

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^14,x]
 

Output:

-1/72072*(462*a^6*e^6*(12*A*e + B*(d + 13*e*x)) + 252*a^5*b*e^5*(11*A*e*(d 
 + 13*e*x) + 2*B*(d^2 + 13*d*e*x + 78*e^2*x^2)) + 126*a^4*b^2*e^4*(10*A*e* 
(d^2 + 13*d*e*x + 78*e^2*x^2) + 3*B*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286 
*e^3*x^3)) + 56*a^3*b^3*e^3*(9*A*e*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286* 
e^3*x^3) + 4*B*(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)) + 21*a^2*b^4*e^2*(8*A*e*(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) + 5*B*(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)) + 6*a*b^5*e*(7*A*e*(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) + 
6*B*(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)) + b^6*(6*A*e*(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) + 7*B*(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)))/(e^8*(d + e*x 
)^13)
 

Rubi [A] (verified)

Time = 0.66 (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.

\(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int \left (\frac {b^5 (6 a B e+A b e-7 b B d)}{e^7 (d+e x)^8}-\frac {3 b^4 (b d-a e) (5 a B e+2 A b e-7 b B d)}{e^7 (d+e x)^9}+\frac {5 b^3 (b d-a e)^2 (4 a B e+3 A b e-7 b B d)}{e^7 (d+e x)^{10}}-\frac {5 b^2 (b d-a e)^3 (3 a B e+4 A b e-7 b B d)}{e^7 (d+e x)^{11}}+\frac {3 b (b d-a e)^4 (2 a B e+5 A b e-7 b B d)}{e^7 (d+e x)^{12}}+\frac {(a e-b d)^5 (a B e+6 A b e-7 b B d)}{e^7 (d+e x)^{13}}+\frac {(a e-b d)^6 (A e-B d)}{e^7 (d+e x)^{14}}+\frac {b^6 B}{e^7 (d+e x)^7}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^14,x]
 

Output:

((b*d - a*e)^6*(B*d - A*e))/(13*e^8*(d + e*x)^13) - ((b*d - a*e)^5*(7*b*B* 
d - 6*A*b*e - a*B*e))/(12*e^8*(d + e*x)^12) + (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) - (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) + (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) - (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) + (b^5*(7*b*B*d - A*b*e - 6*a*B*e 
))/(7*e^8*(d + e*x)^7) - (b^6*B)/(6*e^8*(d + e*x)^6)
 

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	\(\frac {-\frac {b^{6} B \,x^{7}}{6 e}-\frac {b^{5} \left (6 A b e +36 B a e +7 B b d \right ) x^{6}}{42 e^{2}}-\frac {b^{4} \left (42 A a b \,e^{2}+6 A \,b^{2} d e +105 B \,a^{2} e^{2}+36 B a b d e +7 b^{2} B \,d^{2}\right ) x^{5}}{56 e^{3}}-\frac {5 b^{3} \left (168 A \,a^{2} b \,e^{3}+42 A a \,b^{2} d \,e^{2}+6 A \,b^{3} d^{2} e +224 B \,a^{3} e^{3}+105 B \,a^{2} b d \,e^{2}+36 B a \,b^{2} d^{2} e +7 b^{3} B \,d^{3}\right ) x^{4}}{504 e^{4}}-\frac {b^{2} \left (504 A \,a^{3} b \,e^{4}+168 A \,a^{2} b^{2} d \,e^{3}+42 A a \,b^{3} d^{2} e^{2}+6 A \,b^{4} d^{3} e +378 B \,a^{4} e^{4}+224 B \,a^{3} b d \,e^{3}+105 B \,a^{2} b^{2} d^{2} e^{2}+36 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) x^{3}}{252 e^{5}}-\frac {b \left (1260 A \,a^{4} b \,e^{5}+504 A \,a^{3} b^{2} d \,e^{4}+168 A \,a^{2} b^{3} d^{2} e^{3}+42 A a \,b^{4} d^{3} e^{2}+6 A \,b^{5} d^{4} e +504 B \,a^{5} e^{5}+378 B \,a^{4} b d \,e^{4}+224 B \,a^{3} b^{2} d^{2} e^{3}+105 B \,a^{2} b^{3} d^{3} e^{2}+36 B a \,b^{4} d^{4} e +7 B \,b^{5} d^{5}\right ) x^{2}}{924 e^{6}}-\frac {\left (2772 A \,a^{5} b \,e^{6}+1260 A \,a^{4} b^{2} d \,e^{5}+504 A \,a^{3} b^{3} d^{2} e^{4}+168 A \,a^{2} b^{4} d^{3} e^{3}+42 A a \,b^{5} d^{4} e^{2}+6 A \,b^{6} d^{5} e +462 B \,a^{6} e^{6}+504 B \,a^{5} b d \,e^{5}+378 B \,a^{4} b^{2} d^{2} e^{4}+224 B \,a^{3} b^{3} d^{3} e^{3}+105 B \,a^{2} b^{4} d^{4} e^{2}+36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}\right ) x}{5544 e^{7}}-\frac {5544 a^{6} A \,e^{7}+2772 A \,a^{5} b d \,e^{6}+1260 A \,a^{4} b^{2} d^{2} e^{5}+504 A \,a^{3} b^{3} d^{3} e^{4}+168 A \,a^{2} b^{4} d^{4} e^{3}+42 A a \,b^{5} d^{5} e^{2}+6 A \,b^{6} d^{6} e +462 B \,a^{6} d \,e^{6}+504 B \,a^{5} b \,d^{2} e^{5}+378 B \,a^{4} b^{2} d^{3} e^{4}+224 B \,a^{3} b^{3} d^{4} e^{3}+105 B \,a^{2} b^{4} d^{5} e^{2}+36 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{72072 e^{8}}}{\left (e x +d \right )^{13}}\)	\(789\)
default	\(-\frac {a^{6} A \,e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{13 e^{8} \left (e x +d \right )^{13}}-\frac {b^{6} B}{6 e^{8} \left (e x +d \right )^{6}}-\frac {b^{5} \left (A b e +6 B a e -7 B b d \right )}{7 e^{8} \left (e x +d \right )^{7}}-\frac {3 b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right )}{8 e^{8} \left (e x +d \right )^{8}}-\frac {3 b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{11 e^{8} \left (e x +d \right )^{11}}-\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right )}{9 e^{8} \left (e x +d \right )^{9}}-\frac {b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{2 e^{8} \left (e x +d \right )^{10}}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{12 e^{8} \left (e x +d \right )^{12}}\)	\(814\)
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 \,b^{6} d^{6} e^{5}\right ) x}{5544 e^{12}}-\frac {5544 a^{6} A \,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^{6} A \,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}}\)	\(913\)
orering	\(-\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^{6} A \,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^{6} A \,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}}\)	\(922\)

Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^14,x,method=_RETURNVERBOSE)
 

Output:

(-1/6*b^6*B/e*x^7-1/42*b^5/e^2*(6*A*b*e+36*B*a*e+7*B*b*d)*x^6-1/56*b^4/e^3 
*(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*b^2*d^2)*x^5-5/5 
04*b^3/e^4*(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+1 
05*B*a^2*b*d*e^2+36*B*a*b^2*d^2*e+7*B*b^3*d^3)*x^4-1/252*b^2/e^5*(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)*x^3 
-1/924*b/e^6*(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+4 
2*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)*x^2-1/5544 
/e^7*(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*b^6*d^6)*x-1/72072/e^8*(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*b^6*d^7))/(e*x+d)^13
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 905 vs. \(2 (276) = 552\).

Time = 0.13 (sec) , antiderivative size = 905, normalized size of antiderivative = 3.10 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^14,x, algorithm="fricas")
 

Output:

-1/72072*(12012*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 5544*A*a^6*e^7 + 6*(6*B*a*b^ 
5 + A*b^6)*d^6*e + 21*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 56*(4*B*a^3*b^3 
+ 3*A*a^2*b^4)*d^4*e^3 + 126*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 252*(2* 
B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + 462*(B*a^6 + 6*A*a^5*b)*d*e^6 + 1716*(7*B 
*b^6*d*e^6 + 6*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 1287*(7*B*b^6*d^2*e^5 + 6*(6 
*B*a*b^5 + A*b^6)*d*e^6 + 21*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 715*(7*B 
*b^6*d^3*e^4 + 6*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 21*(5*B*a^2*b^4 + 2*A*a*b^5 
)*d*e^6 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 286*(7*B*b^6*d^4*e^3 + 
 6*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 21*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 56 
*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 126*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)* 
x^3 + 78*(7*B*b^6*d^5*e^2 + 6*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 21*(5*B*a^2*b^ 
4 + 2*A*a*b^5)*d^3*e^4 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 126*(3*B 
*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 252*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 1 
3*(7*B*b^6*d^6*e + 6*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 21*(5*B*a^2*b^4 + 2*A*a 
*b^5)*d^4*e^3 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 126*(3*B*a^4*b^2 
+ 4*A*a^3*b^3)*d^2*e^5 + 252*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 462*(B*a^6 
+ 6*A*a^5*b)*e^7)*x)/(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)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**6*(B*x+A)/(e*x+d)**14,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 905 vs. \(2 (276) = 552\).

Time = 0.07 (sec) , antiderivative size = 905, normalized size of antiderivative = 3.10 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^14,x, algorithm="maxima")
 

Output:

-1/72072*(12012*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 5544*A*a^6*e^7 + 6*(6*B*a*b^ 
5 + A*b^6)*d^6*e + 21*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 56*(4*B*a^3*b^3 
+ 3*A*a^2*b^4)*d^4*e^3 + 126*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 252*(2* 
B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + 462*(B*a^6 + 6*A*a^5*b)*d*e^6 + 1716*(7*B 
*b^6*d*e^6 + 6*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 1287*(7*B*b^6*d^2*e^5 + 6*(6 
*B*a*b^5 + A*b^6)*d*e^6 + 21*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 715*(7*B 
*b^6*d^3*e^4 + 6*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 21*(5*B*a^2*b^4 + 2*A*a*b^5 
)*d*e^6 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 286*(7*B*b^6*d^4*e^3 + 
 6*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 21*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 56 
*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 126*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)* 
x^3 + 78*(7*B*b^6*d^5*e^2 + 6*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 21*(5*B*a^2*b^ 
4 + 2*A*a*b^5)*d^3*e^4 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 126*(3*B 
*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 252*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 1 
3*(7*B*b^6*d^6*e + 6*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 21*(5*B*a^2*b^4 + 2*A*a 
*b^5)*d^4*e^3 + 56*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 126*(3*B*a^4*b^2 
+ 4*A*a^3*b^3)*d^2*e^5 + 252*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 462*(B*a^6 
+ 6*A*a^5*b)*e^7)*x)/(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)
 

Giac [B] (verification not implemented)

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

Time = 0.13 (sec) , antiderivative size = 912, normalized size of antiderivative = 3.12 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^14,x, algorithm="giac")
 

Output:

-1/72072*(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^...
 

Mupad [B] (verification not implemented)

Time = 1.41 (sec) , antiderivative size = 921, normalized size of antiderivative = 3.15 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx=-\frac {\frac {462\,B\,a^6\,d\,e^6+5544\,A\,a^6\,e^7+504\,B\,a^5\,b\,d^2\,e^5+2772\,A\,a^5\,b\,d\,e^6+378\,B\,a^4\,b^2\,d^3\,e^4+1260\,A\,a^4\,b^2\,d^2\,e^5+224\,B\,a^3\,b^3\,d^4\,e^3+504\,A\,a^3\,b^3\,d^3\,e^4+105\,B\,a^2\,b^4\,d^5\,e^2+168\,A\,a^2\,b^4\,d^4\,e^3+36\,B\,a\,b^5\,d^6\,e+42\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+6\,A\,b^6\,d^6\,e}{72072\,e^8}+\frac {x\,\left (462\,B\,a^6\,e^6+504\,B\,a^5\,b\,d\,e^5+2772\,A\,a^5\,b\,e^6+378\,B\,a^4\,b^2\,d^2\,e^4+1260\,A\,a^4\,b^2\,d\,e^5+224\,B\,a^3\,b^3\,d^3\,e^3+504\,A\,a^3\,b^3\,d^2\,e^4+105\,B\,a^2\,b^4\,d^4\,e^2+168\,A\,a^2\,b^4\,d^3\,e^3+36\,B\,a\,b^5\,d^5\,e+42\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+6\,A\,b^6\,d^5\,e\right )}{5544\,e^7}+\frac {5\,b^3\,x^4\,\left (224\,B\,a^3\,e^3+105\,B\,a^2\,b\,d\,e^2+168\,A\,a^2\,b\,e^3+36\,B\,a\,b^2\,d^2\,e+42\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+6\,A\,b^3\,d^2\,e\right )}{504\,e^4}+\frac {b^5\,x^6\,\left (6\,A\,b\,e+36\,B\,a\,e+7\,B\,b\,d\right )}{42\,e^2}+\frac {b\,x^2\,\left (504\,B\,a^5\,e^5+378\,B\,a^4\,b\,d\,e^4+1260\,A\,a^4\,b\,e^5+224\,B\,a^3\,b^2\,d^2\,e^3+504\,A\,a^3\,b^2\,d\,e^4+105\,B\,a^2\,b^3\,d^3\,e^2+168\,A\,a^2\,b^3\,d^2\,e^3+36\,B\,a\,b^4\,d^4\,e+42\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+6\,A\,b^5\,d^4\,e\right )}{924\,e^6}+\frac {b^2\,x^3\,\left (378\,B\,a^4\,e^4+224\,B\,a^3\,b\,d\,e^3+504\,A\,a^3\,b\,e^4+105\,B\,a^2\,b^2\,d^2\,e^2+168\,A\,a^2\,b^2\,d\,e^3+36\,B\,a\,b^3\,d^3\,e+42\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+6\,A\,b^4\,d^3\,e\right )}{252\,e^5}+\frac {b^4\,x^5\,\left (105\,B\,a^2\,e^2+36\,B\,a\,b\,d\,e+42\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+6\,A\,b^2\,d\,e\right )}{56\,e^3}+\frac {B\,b^6\,x^7}{6\,e}}{d^{13}+13\,d^{12}\,e\,x+78\,d^{11}\,e^2\,x^2+286\,d^{10}\,e^3\,x^3+715\,d^9\,e^4\,x^4+1287\,d^8\,e^5\,x^5+1716\,d^7\,e^6\,x^6+1716\,d^6\,e^7\,x^7+1287\,d^5\,e^8\,x^8+715\,d^4\,e^9\,x^9+286\,d^3\,e^{10}\,x^{10}+78\,d^2\,e^{11}\,x^{11}+13\,d\,e^{12}\,x^{12}+e^{13}\,x^{13}} \] Input:

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^14,x)
 

Output:

-((5544*A*a^6*e^7 + 7*B*b^6*d^7 + 6*A*b^6*d^6*e + 462*B*a^6*d*e^6 + 42*A*a 
*b^5*d^5*e^2 + 504*B*a^5*b*d^2*e^5 + 168*A*a^2*b^4*d^4*e^3 + 504*A*a^3*b^3 
*d^3*e^4 + 1260*A*a^4*b^2*d^2*e^5 + 105*B*a^2*b^4*d^5*e^2 + 224*B*a^3*b^3* 
d^4*e^3 + 378*B*a^4*b^2*d^3*e^4 + 2772*A*a^5*b*d*e^6 + 36*B*a*b^5*d^6*e)/( 
72072*e^8) + (x*(462*B*a^6*e^6 + 7*B*b^6*d^6 + 2772*A*a^5*b*e^6 + 6*A*b^6* 
d^5*e + 42*A*a*b^5*d^4*e^2 + 1260*A*a^4*b^2*d*e^5 + 168*A*a^2*b^4*d^3*e^3 
+ 504*A*a^3*b^3*d^2*e^4 + 105*B*a^2*b^4*d^4*e^2 + 224*B*a^3*b^3*d^3*e^3 + 
378*B*a^4*b^2*d^2*e^4 + 36*B*a*b^5*d^5*e + 504*B*a^5*b*d*e^5))/(5544*e^7) 
+ (5*b^3*x^4*(224*B*a^3*e^3 + 7*B*b^3*d^3 + 168*A*a^2*b*e^3 + 6*A*b^3*d^2* 
e + 42*A*a*b^2*d*e^2 + 36*B*a*b^2*d^2*e + 105*B*a^2*b*d*e^2))/(504*e^4) + 
(b^5*x^6*(6*A*b*e + 36*B*a*e + 7*B*b*d))/(42*e^2) + (b*x^2*(504*B*a^5*e^5 
+ 7*B*b^5*d^5 + 1260*A*a^4*b*e^5 + 6*A*b^5*d^4*e + 42*A*a*b^4*d^3*e^2 + 50 
4*A*a^3*b^2*d*e^4 + 168*A*a^2*b^3*d^2*e^3 + 105*B*a^2*b^3*d^3*e^2 + 224*B* 
a^3*b^2*d^2*e^3 + 36*B*a*b^4*d^4*e + 378*B*a^4*b*d*e^4))/(924*e^6) + (b^2* 
x^3*(378*B*a^4*e^4 + 7*B*b^4*d^4 + 504*A*a^3*b*e^4 + 6*A*b^4*d^3*e + 42*A* 
a*b^3*d^2*e^2 + 168*A*a^2*b^2*d*e^3 + 105*B*a^2*b^2*d^2*e^2 + 36*B*a*b^3*d 
^3*e + 224*B*a^3*b*d*e^3))/(252*e^5) + (b^4*x^5*(105*B*a^2*e^2 + 7*B*b^2*d 
^2 + 42*A*a*b*e^2 + 6*A*b^2*d*e + 36*B*a*b*d*e))/(56*e^3) + (B*b^6*x^7)/(6 
*e))/(d^13 + e^13*x^13 + 13*d*e^12*x^12 + 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...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 629, normalized size of antiderivative = 2.15 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx=\frac {-1716 b^{7} e^{7} x^{7}-10296 a \,b^{6} e^{7} x^{6}-1716 b^{7} d \,e^{6} x^{6}-27027 a^{2} b^{5} e^{7} x^{5}-7722 a \,b^{6} d \,e^{6} x^{5}-1287 b^{7} d^{2} e^{5} x^{5}-40040 a^{3} b^{4} e^{7} x^{4}-15015 a^{2} b^{5} d \,e^{6} x^{4}-4290 a \,b^{6} d^{2} e^{5} x^{4}-715 b^{7} d^{3} e^{4} x^{4}-36036 a^{4} b^{3} e^{7} x^{3}-16016 a^{3} b^{4} d \,e^{6} x^{3}-6006 a^{2} b^{5} d^{2} e^{5} x^{3}-1716 a \,b^{6} d^{3} e^{4} x^{3}-286 b^{7} d^{4} e^{3} x^{3}-19656 a^{5} b^{2} e^{7} x^{2}-9828 a^{4} b^{3} d \,e^{6} x^{2}-4368 a^{3} b^{4} d^{2} e^{5} x^{2}-1638 a^{2} b^{5} d^{3} e^{4} x^{2}-468 a \,b^{6} d^{4} e^{3} x^{2}-78 b^{7} d^{5} e^{2} x^{2}-6006 a^{6} b \,e^{7} x -3276 a^{5} b^{2} d \,e^{6} x -1638 a^{4} b^{3} d^{2} e^{5} x -728 a^{3} b^{4} d^{3} e^{4} x -273 a^{2} b^{5} d^{4} e^{3} x -78 a \,b^{6} d^{5} e^{2} x -13 b^{7} d^{6} e x -792 a^{7} e^{7}-462 a^{6} b d \,e^{6}-252 a^{5} b^{2} d^{2} e^{5}-126 a^{4} b^{3} d^{3} e^{4}-56 a^{3} b^{4} d^{4} e^{3}-21 a^{2} b^{5} d^{5} e^{2}-6 a \,b^{6} d^{6} e -b^{7} d^{7}}{10296 e^{8} \left (e^{13} x^{13}+13 d \,e^{12} x^{12}+78 d^{2} e^{11} x^{11}+286 d^{3} e^{10} x^{10}+715 d^{4} e^{9} x^{9}+1287 d^{5} e^{8} x^{8}+1716 d^{6} e^{7} x^{7}+1716 d^{7} e^{6} x^{6}+1287 d^{8} e^{5} x^{5}+715 d^{9} e^{4} x^{4}+286 d^{10} e^{3} x^{3}+78 d^{11} e^{2} x^{2}+13 d^{12} e x +d^{13}\right )} \] Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^14,x)
 

Output:

( - 792*a**7*e**7 - 462*a**6*b*d*e**6 - 6006*a**6*b*e**7*x - 252*a**5*b**2 
*d**2*e**5 - 3276*a**5*b**2*d*e**6*x - 19656*a**5*b**2*e**7*x**2 - 126*a** 
4*b**3*d**3*e**4 - 1638*a**4*b**3*d**2*e**5*x - 9828*a**4*b**3*d*e**6*x**2 
 - 36036*a**4*b**3*e**7*x**3 - 56*a**3*b**4*d**4*e**3 - 728*a**3*b**4*d**3 
*e**4*x - 4368*a**3*b**4*d**2*e**5*x**2 - 16016*a**3*b**4*d*e**6*x**3 - 40 
040*a**3*b**4*e**7*x**4 - 21*a**2*b**5*d**5*e**2 - 273*a**2*b**5*d**4*e**3 
*x - 1638*a**2*b**5*d**3*e**4*x**2 - 6006*a**2*b**5*d**2*e**5*x**3 - 15015 
*a**2*b**5*d*e**6*x**4 - 27027*a**2*b**5*e**7*x**5 - 6*a*b**6*d**6*e - 78* 
a*b**6*d**5*e**2*x - 468*a*b**6*d**4*e**3*x**2 - 1716*a*b**6*d**3*e**4*x** 
3 - 4290*a*b**6*d**2*e**5*x**4 - 7722*a*b**6*d*e**6*x**5 - 10296*a*b**6*e* 
*7*x**6 - b**7*d**7 - 13*b**7*d**6*e*x - 78*b**7*d**5*e**2*x**2 - 286*b**7 
*d**4*e**3*x**3 - 715*b**7*d**3*e**4*x**4 - 1287*b**7*d**2*e**5*x**5 - 171 
6*b**7*d*e**6*x**6 - 1716*b**7*e**7*x**7)/(10296*e**8*(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))