\(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx\) [58]

Optimal result

Integrand size = 20, antiderivative size = 86 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx=-\frac {(B d-A e) (a+b x)^7}{8 e (b d-a e) (d+e x)^8}+\frac {(7 b B d+A b e-8 a B e) (a+b x)^7}{56 e (b d-a e)^2 (d+e x)^7} \] Output:

-1/8*(-A*e+B*d)*(b*x+a)^7/e/(-a*e+b*d)/(e*x+d)^8+1/56*(A*b*e-8*B*a*e+7*B*b 
*d)*(b*x+a)^7/e/(-a*e+b*d)^2/(e*x+d)^7
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(597\) vs. \(2(86)=172\).

Time = 0.16 (sec) , antiderivative size = 597, normalized size of antiderivative = 6.94 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx=-\frac {a^6 e^6 (7 A e+B (d+8 e x))+2 a^5 b e^5 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+a^4 b^2 e^4 \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+4 a^3 b^3 e^3 \left (A e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+B \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+a^2 b^4 e^2 \left (3 A e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 B \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )+2 a b^5 e \left (A e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+3 B \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )+b^6 \left (A e \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )+7 B \left (d^7+8 d^6 e x+28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+28 d e^6 x^6+8 e^7 x^7\right )\right )}{56 e^8 (d+e x)^8} \] Input:

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^9,x]
 

Output:

-1/56*(a^6*e^6*(7*A*e + B*(d + 8*e*x)) + 2*a^5*b*e^5*(3*A*e*(d + 8*e*x) + 
B*(d^2 + 8*d*e*x + 28*e^2*x^2)) + a^4*b^2*e^4*(5*A*e*(d^2 + 8*d*e*x + 28*e 
^2*x^2) + 3*B*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3)) + 4*a^3*b^3*e 
^3*(A*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + B*(d^4 + 8*d^3*e*x 
 + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4)) + a^2*b^4*e^2*(3*A*e*(d^4 
+ 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*B*(d^5 + 8*d 
^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)) + 2 
*a*b^5*e*(A*e*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^ 
4*x^4 + 56*e^5*x^5) + 3*B*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x 
^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6)) + b^6*(A*e*(d^6 + 8*d^5* 
e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28 
*e^6*x^6) + 7*B*(d^7 + 8*d^6*e*x + 28*d^5*e^2*x^2 + 56*d^4*e^3*x^3 + 70*d^ 
3*e^4*x^4 + 56*d^2*e^5*x^5 + 28*d*e^6*x^6 + 8*e^7*x^7)))/(e^8*(d + e*x)^8)
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 86, 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 = {87, 48}

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.

\(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-8 a B e+A b e+7 b B d) \int \frac {(a+b x)^6}{(d+e x)^8}dx}{8 e (b d-a e)}-\frac {(a+b x)^7 (B d-A e)}{8 e (d+e x)^8 (b d-a e)}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {(a+b x)^7 (-8 a B e+A b e+7 b B d)}{56 e (d+e x)^7 (b d-a e)^2}-\frac {(a+b x)^7 (B d-A e)}{8 e (d+e x)^8 (b d-a e)}\)

Input:

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^9,x]
 

Output:

-1/8*((B*d - A*e)*(a + b*x)^7)/(e*(b*d - a*e)*(d + e*x)^8) + ((7*b*B*d + A 
*b*e - 8*a*B*e)*(a + b*x)^7)/(56*e*(b*d - a*e)^2*(d + e*x)^7)
 

Defintions of rubi rules used

rule 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(779\) vs. \(2(82)=164\).

Time = 0.24 (sec) , antiderivative size = 780, normalized size of antiderivative = 9.07

method	result	size
risch	\(\frac {-\frac {b^{6} B \,x^{7}}{e}-\frac {b^{5} \left (A b e +6 B a e +7 B b d \right ) x^{6}}{2 e^{2}}-\frac {b^{4} \left (2 A a b \,e^{2}+A \,b^{2} d e +5 B \,a^{2} e^{2}+6 B a b d e +7 b^{2} B \,d^{2}\right ) x^{5}}{e^{3}}-\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}+2 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}+5 B \,a^{2} b d \,e^{2}+6 B a \,b^{2} d^{2} e +7 b^{3} B \,d^{3}\right ) x^{4}}{4 e^{4}}-\frac {b^{2} \left (4 A \,a^{3} b \,e^{4}+3 A \,a^{2} b^{2} d \,e^{3}+2 A a \,b^{3} d^{2} e^{2}+A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}+4 B \,a^{3} b d \,e^{3}+5 B \,a^{2} b^{2} d^{2} e^{2}+6 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) x^{3}}{e^{5}}-\frac {b \left (5 A \,a^{4} b \,e^{5}+4 A \,a^{3} b^{2} d \,e^{4}+3 A \,a^{2} b^{3} d^{2} e^{3}+2 A a \,b^{4} d^{3} e^{2}+A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}+3 B \,a^{4} b d \,e^{4}+4 B \,a^{3} b^{2} d^{2} e^{3}+5 B \,a^{2} b^{3} d^{3} e^{2}+6 B a \,b^{4} d^{4} e +7 B \,b^{5} d^{5}\right ) x^{2}}{2 e^{6}}-\frac {\left (6 A \,a^{5} b \,e^{6}+5 A \,a^{4} b^{2} d \,e^{5}+4 A \,a^{3} b^{3} d^{2} e^{4}+3 A \,a^{2} b^{4} d^{3} e^{3}+2 A a \,b^{5} d^{4} e^{2}+A \,b^{6} d^{5} e +B \,a^{6} e^{6}+2 B \,a^{5} b d \,e^{5}+3 B \,a^{4} b^{2} d^{2} e^{4}+4 B \,a^{3} b^{3} d^{3} e^{3}+5 B \,a^{2} b^{4} d^{4} e^{2}+6 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}\right ) x}{7 e^{7}}-\frac {7 a^{6} A \,e^{7}+6 A \,a^{5} b d \,e^{6}+5 A \,a^{4} b^{2} d^{2} e^{5}+4 A \,a^{3} b^{3} d^{3} e^{4}+3 A \,a^{2} b^{4} d^{4} e^{3}+2 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+2 B \,a^{5} b \,d^{2} e^{5}+3 B \,a^{4} b^{2} d^{3} e^{4}+4 B \,a^{3} b^{3} d^{4} e^{3}+5 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{56 e^{8}}}{\left (e x +d \right )^{8}}\)	\(780\)
norman	\(\frac {-\frac {b^{6} B \,x^{7}}{e}-\frac {\left (A \,b^{6} e +6 B a \,b^{5} e +7 b^{6} B d \right ) x^{6}}{2 e^{2}}-\frac {\left (2 A a \,b^{5} e^{2}+A \,b^{6} d e +5 B \,a^{2} b^{4} e^{2}+6 B a \,b^{5} d e +7 b^{6} B \,d^{2}\right ) x^{5}}{e^{3}}-\frac {5 \left (3 A \,a^{2} b^{4} e^{3}+2 A a \,b^{5} d \,e^{2}+A \,b^{6} d^{2} e +4 B \,a^{3} b^{3} e^{3}+5 B \,a^{2} b^{4} d \,e^{2}+6 B a \,b^{5} d^{2} e +7 b^{6} B \,d^{3}\right ) x^{4}}{4 e^{4}}-\frac {\left (4 A \,a^{3} b^{3} e^{4}+3 A \,a^{2} b^{4} d \,e^{3}+2 A a \,b^{5} d^{2} e^{2}+A \,b^{6} d^{3} e +3 B \,a^{4} b^{2} e^{4}+4 B \,a^{3} b^{3} d \,e^{3}+5 B \,a^{2} b^{4} d^{2} e^{2}+6 B a \,b^{5} d^{3} e +7 b^{6} B \,d^{4}\right ) x^{3}}{e^{5}}-\frac {\left (5 A \,a^{4} b^{2} e^{5}+4 A \,a^{3} b^{3} d \,e^{4}+3 A \,a^{2} b^{4} d^{2} e^{3}+2 A a \,b^{5} d^{3} e^{2}+A \,b^{6} d^{4} e +2 B \,a^{5} b \,e^{5}+3 B \,a^{4} b^{2} d \,e^{4}+4 B \,a^{3} b^{3} d^{2} e^{3}+5 B \,a^{2} b^{4} d^{3} e^{2}+6 B a \,b^{5} d^{4} e +7 b^{6} B \,d^{5}\right ) x^{2}}{2 e^{6}}-\frac {\left (6 A \,a^{5} b \,e^{6}+5 A \,a^{4} b^{2} d \,e^{5}+4 A \,a^{3} b^{3} d^{2} e^{4}+3 A \,a^{2} b^{4} d^{3} e^{3}+2 A a \,b^{5} d^{4} e^{2}+A \,b^{6} d^{5} e +B \,a^{6} e^{6}+2 B \,a^{5} b d \,e^{5}+3 B \,a^{4} b^{2} d^{2} e^{4}+4 B \,a^{3} b^{3} d^{3} e^{3}+5 B \,a^{2} b^{4} d^{4} e^{2}+6 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}\right ) x}{7 e^{7}}-\frac {7 a^{6} A \,e^{7}+6 A \,a^{5} b d \,e^{6}+5 A \,a^{4} b^{2} d^{2} e^{5}+4 A \,a^{3} b^{3} d^{3} e^{4}+3 A \,a^{2} b^{4} d^{4} e^{3}+2 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+2 B \,a^{5} b \,d^{2} e^{5}+3 B \,a^{4} b^{2} d^{3} e^{4}+4 B \,a^{3} b^{3} d^{4} e^{3}+5 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{56 e^{8}}}{\left (e x +d \right )^{8}}\)	\(800\)
default	\(-\frac {b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{2 e^{8} \left (e x +d \right )^{6}}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{7 e^{8} \left (e x +d \right )^{7}}-\frac {a^{6} A \,e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{8 e^{8} \left (e x +d \right )^{8}}-\frac {b^{6} B}{e^{8} \left (e x +d \right )}-\frac {b^{5} \left (A b e +6 B a e -7 B b d \right )}{2 e^{8} \left (e x +d \right )^{2}}-\frac {b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{e^{8} \left (e x +d \right )^{5}}-\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right )}{4 e^{8} \left (e x +d \right )^{4}}-\frac {b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right )}{e^{8} \left (e x +d \right )^{3}}\)	\(814\)
gosper	\(-\frac {56 B \,x^{7} b^{6} e^{7}+28 A \,x^{6} b^{6} e^{7}+168 B \,x^{6} a \,b^{5} e^{7}+196 B \,x^{6} b^{6} d \,e^{6}+112 A \,x^{5} a \,b^{5} e^{7}+56 A \,x^{5} b^{6} d \,e^{6}+280 B \,x^{5} a^{2} b^{4} e^{7}+336 B \,x^{5} a \,b^{5} d \,e^{6}+392 B \,x^{5} b^{6} d^{2} e^{5}+210 A \,x^{4} a^{2} b^{4} e^{7}+140 A \,x^{4} a \,b^{5} d \,e^{6}+70 A \,x^{4} b^{6} d^{2} e^{5}+280 B \,x^{4} a^{3} b^{3} e^{7}+350 B \,x^{4} a^{2} b^{4} d \,e^{6}+420 B \,x^{4} a \,b^{5} d^{2} e^{5}+490 B \,x^{4} b^{6} d^{3} e^{4}+224 A \,x^{3} a^{3} b^{3} e^{7}+168 A \,x^{3} a^{2} b^{4} d \,e^{6}+112 A \,x^{3} a \,b^{5} d^{2} e^{5}+56 A \,x^{3} b^{6} d^{3} e^{4}+168 B \,x^{3} a^{4} b^{2} e^{7}+224 B \,x^{3} a^{3} b^{3} d \,e^{6}+280 B \,x^{3} a^{2} b^{4} d^{2} e^{5}+336 B \,x^{3} a \,b^{5} d^{3} e^{4}+392 B \,x^{3} b^{6} d^{4} e^{3}+140 A \,x^{2} a^{4} b^{2} e^{7}+112 A \,x^{2} a^{3} b^{3} d \,e^{6}+84 A \,x^{2} a^{2} b^{4} d^{2} e^{5}+56 A \,x^{2} a \,b^{5} d^{3} e^{4}+28 A \,x^{2} b^{6} d^{4} e^{3}+56 B \,x^{2} a^{5} b \,e^{7}+84 B \,x^{2} a^{4} b^{2} d \,e^{6}+112 B \,x^{2} a^{3} b^{3} d^{2} e^{5}+140 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+168 B \,x^{2} a \,b^{5} d^{4} e^{3}+196 B \,x^{2} b^{6} d^{5} e^{2}+48 A x \,a^{5} b \,e^{7}+40 A x \,a^{4} b^{2} d \,e^{6}+32 A x \,a^{3} b^{3} d^{2} e^{5}+24 A x \,a^{2} b^{4} d^{3} e^{4}+16 A x a \,b^{5} d^{4} e^{3}+8 A x \,b^{6} d^{5} e^{2}+8 B x \,a^{6} e^{7}+16 B x \,a^{5} b d \,e^{6}+24 B x \,a^{4} b^{2} d^{2} e^{5}+32 B x \,a^{3} b^{3} d^{3} e^{4}+40 B x \,a^{2} b^{4} d^{4} e^{3}+48 B x a \,b^{5} d^{5} e^{2}+56 B x \,b^{6} d^{6} e +7 a^{6} A \,e^{7}+6 A \,a^{5} b d \,e^{6}+5 A \,a^{4} b^{2} d^{2} e^{5}+4 A \,a^{3} b^{3} d^{3} e^{4}+3 A \,a^{2} b^{4} d^{4} e^{3}+2 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+2 B \,a^{5} b \,d^{2} e^{5}+3 B \,a^{4} b^{2} d^{3} e^{4}+4 B \,a^{3} b^{3} d^{4} e^{3}+5 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{56 \left (e x +d \right )^{8} e^{8}}\)	\(911\)
parallelrisch	\(-\frac {56 B \,x^{7} b^{6} e^{7}+28 A \,x^{6} b^{6} e^{7}+168 B \,x^{6} a \,b^{5} e^{7}+196 B \,x^{6} b^{6} d \,e^{6}+112 A \,x^{5} a \,b^{5} e^{7}+56 A \,x^{5} b^{6} d \,e^{6}+280 B \,x^{5} a^{2} b^{4} e^{7}+336 B \,x^{5} a \,b^{5} d \,e^{6}+392 B \,x^{5} b^{6} d^{2} e^{5}+210 A \,x^{4} a^{2} b^{4} e^{7}+140 A \,x^{4} a \,b^{5} d \,e^{6}+70 A \,x^{4} b^{6} d^{2} e^{5}+280 B \,x^{4} a^{3} b^{3} e^{7}+350 B \,x^{4} a^{2} b^{4} d \,e^{6}+420 B \,x^{4} a \,b^{5} d^{2} e^{5}+490 B \,x^{4} b^{6} d^{3} e^{4}+224 A \,x^{3} a^{3} b^{3} e^{7}+168 A \,x^{3} a^{2} b^{4} d \,e^{6}+112 A \,x^{3} a \,b^{5} d^{2} e^{5}+56 A \,x^{3} b^{6} d^{3} e^{4}+168 B \,x^{3} a^{4} b^{2} e^{7}+224 B \,x^{3} a^{3} b^{3} d \,e^{6}+280 B \,x^{3} a^{2} b^{4} d^{2} e^{5}+336 B \,x^{3} a \,b^{5} d^{3} e^{4}+392 B \,x^{3} b^{6} d^{4} e^{3}+140 A \,x^{2} a^{4} b^{2} e^{7}+112 A \,x^{2} a^{3} b^{3} d \,e^{6}+84 A \,x^{2} a^{2} b^{4} d^{2} e^{5}+56 A \,x^{2} a \,b^{5} d^{3} e^{4}+28 A \,x^{2} b^{6} d^{4} e^{3}+56 B \,x^{2} a^{5} b \,e^{7}+84 B \,x^{2} a^{4} b^{2} d \,e^{6}+112 B \,x^{2} a^{3} b^{3} d^{2} e^{5}+140 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+168 B \,x^{2} a \,b^{5} d^{4} e^{3}+196 B \,x^{2} b^{6} d^{5} e^{2}+48 A x \,a^{5} b \,e^{7}+40 A x \,a^{4} b^{2} d \,e^{6}+32 A x \,a^{3} b^{3} d^{2} e^{5}+24 A x \,a^{2} b^{4} d^{3} e^{4}+16 A x a \,b^{5} d^{4} e^{3}+8 A x \,b^{6} d^{5} e^{2}+8 B x \,a^{6} e^{7}+16 B x \,a^{5} b d \,e^{6}+24 B x \,a^{4} b^{2} d^{2} e^{5}+32 B x \,a^{3} b^{3} d^{3} e^{4}+40 B x \,a^{2} b^{4} d^{4} e^{3}+48 B x a \,b^{5} d^{5} e^{2}+56 B x \,b^{6} d^{6} e +7 a^{6} A \,e^{7}+6 A \,a^{5} b d \,e^{6}+5 A \,a^{4} b^{2} d^{2} e^{5}+4 A \,a^{3} b^{3} d^{3} e^{4}+3 A \,a^{2} b^{4} d^{4} e^{3}+2 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+2 B \,a^{5} b \,d^{2} e^{5}+3 B \,a^{4} b^{2} d^{3} e^{4}+4 B \,a^{3} b^{3} d^{4} e^{3}+5 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{56 \left (e x +d \right )^{8} e^{8}}\)	\(911\)
orering	\(-\frac {56 B \,x^{7} b^{6} e^{7}+28 A \,x^{6} b^{6} e^{7}+168 B \,x^{6} a \,b^{5} e^{7}+196 B \,x^{6} b^{6} d \,e^{6}+112 A \,x^{5} a \,b^{5} e^{7}+56 A \,x^{5} b^{6} d \,e^{6}+280 B \,x^{5} a^{2} b^{4} e^{7}+336 B \,x^{5} a \,b^{5} d \,e^{6}+392 B \,x^{5} b^{6} d^{2} e^{5}+210 A \,x^{4} a^{2} b^{4} e^{7}+140 A \,x^{4} a \,b^{5} d \,e^{6}+70 A \,x^{4} b^{6} d^{2} e^{5}+280 B \,x^{4} a^{3} b^{3} e^{7}+350 B \,x^{4} a^{2} b^{4} d \,e^{6}+420 B \,x^{4} a \,b^{5} d^{2} e^{5}+490 B \,x^{4} b^{6} d^{3} e^{4}+224 A \,x^{3} a^{3} b^{3} e^{7}+168 A \,x^{3} a^{2} b^{4} d \,e^{6}+112 A \,x^{3} a \,b^{5} d^{2} e^{5}+56 A \,x^{3} b^{6} d^{3} e^{4}+168 B \,x^{3} a^{4} b^{2} e^{7}+224 B \,x^{3} a^{3} b^{3} d \,e^{6}+280 B \,x^{3} a^{2} b^{4} d^{2} e^{5}+336 B \,x^{3} a \,b^{5} d^{3} e^{4}+392 B \,x^{3} b^{6} d^{4} e^{3}+140 A \,x^{2} a^{4} b^{2} e^{7}+112 A \,x^{2} a^{3} b^{3} d \,e^{6}+84 A \,x^{2} a^{2} b^{4} d^{2} e^{5}+56 A \,x^{2} a \,b^{5} d^{3} e^{4}+28 A \,x^{2} b^{6} d^{4} e^{3}+56 B \,x^{2} a^{5} b \,e^{7}+84 B \,x^{2} a^{4} b^{2} d \,e^{6}+112 B \,x^{2} a^{3} b^{3} d^{2} e^{5}+140 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+168 B \,x^{2} a \,b^{5} d^{4} e^{3}+196 B \,x^{2} b^{6} d^{5} e^{2}+48 A x \,a^{5} b \,e^{7}+40 A x \,a^{4} b^{2} d \,e^{6}+32 A x \,a^{3} b^{3} d^{2} e^{5}+24 A x \,a^{2} b^{4} d^{3} e^{4}+16 A x a \,b^{5} d^{4} e^{3}+8 A x \,b^{6} d^{5} e^{2}+8 B x \,a^{6} e^{7}+16 B x \,a^{5} b d \,e^{6}+24 B x \,a^{4} b^{2} d^{2} e^{5}+32 B x \,a^{3} b^{3} d^{3} e^{4}+40 B x \,a^{2} b^{4} d^{4} e^{3}+48 B x a \,b^{5} d^{5} e^{2}+56 B x \,b^{6} d^{6} e +7 a^{6} A \,e^{7}+6 A \,a^{5} b d \,e^{6}+5 A \,a^{4} b^{2} d^{2} e^{5}+4 A \,a^{3} b^{3} d^{3} e^{4}+3 A \,a^{2} b^{4} d^{4} e^{3}+2 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+2 B \,a^{5} b \,d^{2} e^{5}+3 B \,a^{4} b^{2} d^{3} e^{4}+4 B \,a^{3} b^{3} d^{4} e^{3}+5 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{56 \left (e x +d \right )^{8} e^{8}}\)	\(911\)

Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^9,x,method=_RETURNVERBOSE)
 

Output:

(-b^6*B/e*x^7-1/2*b^5*(A*b*e+6*B*a*e+7*B*b*d)/e^2*x^6-b^4*(2*A*a*b*e^2+A*b 
^2*d*e+5*B*a^2*e^2+6*B*a*b*d*e+7*B*b^2*d^2)/e^3*x^5-5/4*b^3*(3*A*a^2*b*e^3 
+2*A*a*b^2*d*e^2+A*b^3*d^2*e+4*B*a^3*e^3+5*B*a^2*b*d*e^2+6*B*a*b^2*d^2*e+7 
*B*b^3*d^3)/e^4*x^4-b^2*(4*A*a^3*b*e^4+3*A*a^2*b^2*d*e^3+2*A*a*b^3*d^2*e^2 
+A*b^4*d^3*e+3*B*a^4*e^4+4*B*a^3*b*d*e^3+5*B*a^2*b^2*d^2*e^2+6*B*a*b^3*d^3 
*e+7*B*b^4*d^4)/e^5*x^3-1/2*b*(5*A*a^4*b*e^5+4*A*a^3*b^2*d*e^4+3*A*a^2*b^3 
*d^2*e^3+2*A*a*b^4*d^3*e^2+A*b^5*d^4*e+2*B*a^5*e^5+3*B*a^4*b*d*e^4+4*B*a^3 
*b^2*d^2*e^3+5*B*a^2*b^3*d^3*e^2+6*B*a*b^4*d^4*e+7*B*b^5*d^5)/e^6*x^2-1/7* 
(6*A*a^5*b*e^6+5*A*a^4*b^2*d*e^5+4*A*a^3*b^3*d^2*e^4+3*A*a^2*b^4*d^3*e^3+2 
*A*a*b^5*d^4*e^2+A*b^6*d^5*e+B*a^6*e^6+2*B*a^5*b*d*e^5+3*B*a^4*b^2*d^2*e^4 
+4*B*a^3*b^3*d^3*e^3+5*B*a^2*b^4*d^4*e^2+6*B*a*b^5*d^5*e+7*B*b^6*d^6)/e^7* 
x-1/56*(7*A*a^6*e^7+6*A*a^5*b*d*e^6+5*A*a^4*b^2*d^2*e^5+4*A*a^3*b^3*d^3*e^ 
4+3*A*a^2*b^4*d^4*e^3+2*A*a*b^5*d^5*e^2+A*b^6*d^6*e+B*a^6*d*e^6+2*B*a^5*b* 
d^2*e^5+3*B*a^4*b^2*d^3*e^4+4*B*a^3*b^3*d^4*e^3+5*B*a^2*b^4*d^5*e^2+6*B*a* 
b^5*d^6*e+7*B*b^6*d^7)/e^8)/(e*x+d)^8
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (82) = 164\).

Time = 0.09 (sec) , antiderivative size = 823, normalized size of antiderivative = 9.57 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^9,x, algorithm="fricas")
 

Output:

-1/56*(56*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 7*A*a^6*e^7 + (6*B*a*b^5 + A*b^6)* 
d^6*e + (5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^ 
4*e^3 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + (2*B*a^5*b + 5*A*a^4*b^2)*d^ 
2*e^5 + (B*a^6 + 6*A*a^5*b)*d*e^6 + 28*(7*B*b^6*d*e^6 + (6*B*a*b^5 + A*b^6 
)*e^7)*x^6 + 56*(7*B*b^6*d^2*e^5 + (6*B*a*b^5 + A*b^6)*d*e^6 + (5*B*a^2*b^ 
4 + 2*A*a*b^5)*e^7)*x^5 + 70*(7*B*b^6*d^3*e^4 + (6*B*a*b^5 + A*b^6)*d^2*e^ 
5 + (5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 
 + 56*(7*B*b^6*d^4*e^3 + (6*B*a*b^5 + A*b^6)*d^3*e^4 + (5*B*a^2*b^4 + 2*A* 
a*b^5)*d^2*e^5 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + (3*B*a^4*b^2 + 4*A*a^ 
3*b^3)*e^7)*x^3 + 28*(7*B*b^6*d^5*e^2 + (6*B*a*b^5 + A*b^6)*d^4*e^3 + (5*B 
*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + (3*B 
*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + (2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 8*(7* 
B*b^6*d^6*e + (6*B*a*b^5 + A*b^6)*d^5*e^2 + (5*B*a^2*b^4 + 2*A*a*b^5)*d^4* 
e^3 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d^ 
2*e^5 + (2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + (B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^1 
6*x^8 + 8*d*e^15*x^7 + 28*d^2*e^14*x^6 + 56*d^3*e^13*x^5 + 70*d^4*e^12*x^4 
 + 56*d^5*e^11*x^3 + 28*d^6*e^10*x^2 + 8*d^7*e^9*x + d^8*e^8)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**6*(B*x+A)/(e*x+d)**9,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (82) = 164\).

Time = 0.08 (sec) , antiderivative size = 823, normalized size of antiderivative = 9.57 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^9,x, algorithm="maxima")
 

Output:

-1/56*(56*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 7*A*a^6*e^7 + (6*B*a*b^5 + A*b^6)* 
d^6*e + (5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^ 
4*e^3 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + (2*B*a^5*b + 5*A*a^4*b^2)*d^ 
2*e^5 + (B*a^6 + 6*A*a^5*b)*d*e^6 + 28*(7*B*b^6*d*e^6 + (6*B*a*b^5 + A*b^6 
)*e^7)*x^6 + 56*(7*B*b^6*d^2*e^5 + (6*B*a*b^5 + A*b^6)*d*e^6 + (5*B*a^2*b^ 
4 + 2*A*a*b^5)*e^7)*x^5 + 70*(7*B*b^6*d^3*e^4 + (6*B*a*b^5 + A*b^6)*d^2*e^ 
5 + (5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 
 + 56*(7*B*b^6*d^4*e^3 + (6*B*a*b^5 + A*b^6)*d^3*e^4 + (5*B*a^2*b^4 + 2*A* 
a*b^5)*d^2*e^5 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + (3*B*a^4*b^2 + 4*A*a^ 
3*b^3)*e^7)*x^3 + 28*(7*B*b^6*d^5*e^2 + (6*B*a*b^5 + A*b^6)*d^4*e^3 + (5*B 
*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + (3*B 
*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + (2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 8*(7* 
B*b^6*d^6*e + (6*B*a*b^5 + A*b^6)*d^5*e^2 + (5*B*a^2*b^4 + 2*A*a*b^5)*d^4* 
e^3 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d^ 
2*e^5 + (2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + (B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^1 
6*x^8 + 8*d*e^15*x^7 + 28*d^2*e^14*x^6 + 56*d^3*e^13*x^5 + 70*d^4*e^12*x^4 
 + 56*d^5*e^11*x^3 + 28*d^6*e^10*x^2 + 8*d^7*e^9*x + d^8*e^8)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (82) = 164\).

Time = 0.13 (sec) , antiderivative size = 910, normalized size of antiderivative = 10.58 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^9,x, algorithm="giac")
 

Output:

-1/56*(56*B*b^6*e^7*x^7 + 196*B*b^6*d*e^6*x^6 + 168*B*a*b^5*e^7*x^6 + 28*A 
*b^6*e^7*x^6 + 392*B*b^6*d^2*e^5*x^5 + 336*B*a*b^5*d*e^6*x^5 + 56*A*b^6*d* 
e^6*x^5 + 280*B*a^2*b^4*e^7*x^5 + 112*A*a*b^5*e^7*x^5 + 490*B*b^6*d^3*e^4* 
x^4 + 420*B*a*b^5*d^2*e^5*x^4 + 70*A*b^6*d^2*e^5*x^4 + 350*B*a^2*b^4*d*e^6 
*x^4 + 140*A*a*b^5*d*e^6*x^4 + 280*B*a^3*b^3*e^7*x^4 + 210*A*a^2*b^4*e^7*x 
^4 + 392*B*b^6*d^4*e^3*x^3 + 336*B*a*b^5*d^3*e^4*x^3 + 56*A*b^6*d^3*e^4*x^ 
3 + 280*B*a^2*b^4*d^2*e^5*x^3 + 112*A*a*b^5*d^2*e^5*x^3 + 224*B*a^3*b^3*d* 
e^6*x^3 + 168*A*a^2*b^4*d*e^6*x^3 + 168*B*a^4*b^2*e^7*x^3 + 224*A*a^3*b^3* 
e^7*x^3 + 196*B*b^6*d^5*e^2*x^2 + 168*B*a*b^5*d^4*e^3*x^2 + 28*A*b^6*d^4*e 
^3*x^2 + 140*B*a^2*b^4*d^3*e^4*x^2 + 56*A*a*b^5*d^3*e^4*x^2 + 112*B*a^3*b^ 
3*d^2*e^5*x^2 + 84*A*a^2*b^4*d^2*e^5*x^2 + 84*B*a^4*b^2*d*e^6*x^2 + 112*A* 
a^3*b^3*d*e^6*x^2 + 56*B*a^5*b*e^7*x^2 + 140*A*a^4*b^2*e^7*x^2 + 56*B*b^6* 
d^6*e*x + 48*B*a*b^5*d^5*e^2*x + 8*A*b^6*d^5*e^2*x + 40*B*a^2*b^4*d^4*e^3* 
x + 16*A*a*b^5*d^4*e^3*x + 32*B*a^3*b^3*d^3*e^4*x + 24*A*a^2*b^4*d^3*e^4*x 
 + 24*B*a^4*b^2*d^2*e^5*x + 32*A*a^3*b^3*d^2*e^5*x + 16*B*a^5*b*d*e^6*x + 
40*A*a^4*b^2*d*e^6*x + 8*B*a^6*e^7*x + 48*A*a^5*b*e^7*x + 7*B*b^6*d^7 + 6* 
B*a*b^5*d^6*e + A*b^6*d^6*e + 5*B*a^2*b^4*d^5*e^2 + 2*A*a*b^5*d^5*e^2 + 4* 
B*a^3*b^3*d^4*e^3 + 3*A*a^2*b^4*d^4*e^3 + 3*B*a^4*b^2*d^3*e^4 + 4*A*a^3*b^ 
3*d^3*e^4 + 2*B*a^5*b*d^2*e^5 + 5*A*a^4*b^2*d^2*e^5 + B*a^6*d*e^6 + 6*A*a^ 
5*b*d*e^6 + 7*A*a^6*e^7)/((e*x + d)^8*e^8)
 

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 854, normalized size of antiderivative = 9.93 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx=-\frac {\frac {B\,a^6\,d\,e^6+7\,A\,a^6\,e^7+2\,B\,a^5\,b\,d^2\,e^5+6\,A\,a^5\,b\,d\,e^6+3\,B\,a^4\,b^2\,d^3\,e^4+5\,A\,a^4\,b^2\,d^2\,e^5+4\,B\,a^3\,b^3\,d^4\,e^3+4\,A\,a^3\,b^3\,d^3\,e^4+5\,B\,a^2\,b^4\,d^5\,e^2+3\,A\,a^2\,b^4\,d^4\,e^3+6\,B\,a\,b^5\,d^6\,e+2\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+A\,b^6\,d^6\,e}{56\,e^8}+\frac {x\,\left (B\,a^6\,e^6+2\,B\,a^5\,b\,d\,e^5+6\,A\,a^5\,b\,e^6+3\,B\,a^4\,b^2\,d^2\,e^4+5\,A\,a^4\,b^2\,d\,e^5+4\,B\,a^3\,b^3\,d^3\,e^3+4\,A\,a^3\,b^3\,d^2\,e^4+5\,B\,a^2\,b^4\,d^4\,e^2+3\,A\,a^2\,b^4\,d^3\,e^3+6\,B\,a\,b^5\,d^5\,e+2\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+A\,b^6\,d^5\,e\right )}{7\,e^7}+\frac {5\,b^3\,x^4\,\left (4\,B\,a^3\,e^3+5\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3+6\,B\,a\,b^2\,d^2\,e+2\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{4\,e^4}+\frac {b^5\,x^6\,\left (A\,b\,e+6\,B\,a\,e+7\,B\,b\,d\right )}{2\,e^2}+\frac {b\,x^2\,\left (2\,B\,a^5\,e^5+3\,B\,a^4\,b\,d\,e^4+5\,A\,a^4\,b\,e^5+4\,B\,a^3\,b^2\,d^2\,e^3+4\,A\,a^3\,b^2\,d\,e^4+5\,B\,a^2\,b^3\,d^3\,e^2+3\,A\,a^2\,b^3\,d^2\,e^3+6\,B\,a\,b^4\,d^4\,e+2\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+A\,b^5\,d^4\,e\right )}{2\,e^6}+\frac {b^2\,x^3\,\left (3\,B\,a^4\,e^4+4\,B\,a^3\,b\,d\,e^3+4\,A\,a^3\,b\,e^4+5\,B\,a^2\,b^2\,d^2\,e^2+3\,A\,a^2\,b^2\,d\,e^3+6\,B\,a\,b^3\,d^3\,e+2\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+A\,b^4\,d^3\,e\right )}{e^5}+\frac {b^4\,x^5\,\left (5\,B\,a^2\,e^2+6\,B\,a\,b\,d\,e+2\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+A\,b^2\,d\,e\right )}{e^3}+\frac {B\,b^6\,x^7}{e}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \] Input:

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^9,x)
 

Output:

-((7*A*a^6*e^7 + 7*B*b^6*d^7 + A*b^6*d^6*e + B*a^6*d*e^6 + 2*A*a*b^5*d^5*e 
^2 + 2*B*a^5*b*d^2*e^5 + 3*A*a^2*b^4*d^4*e^3 + 4*A*a^3*b^3*d^3*e^4 + 5*A*a 
^4*b^2*d^2*e^5 + 5*B*a^2*b^4*d^5*e^2 + 4*B*a^3*b^3*d^4*e^3 + 3*B*a^4*b^2*d 
^3*e^4 + 6*A*a^5*b*d*e^6 + 6*B*a*b^5*d^6*e)/(56*e^8) + (x*(B*a^6*e^6 + 7*B 
*b^6*d^6 + 6*A*a^5*b*e^6 + A*b^6*d^5*e + 2*A*a*b^5*d^4*e^2 + 5*A*a^4*b^2*d 
*e^5 + 3*A*a^2*b^4*d^3*e^3 + 4*A*a^3*b^3*d^2*e^4 + 5*B*a^2*b^4*d^4*e^2 + 4 
*B*a^3*b^3*d^3*e^3 + 3*B*a^4*b^2*d^2*e^4 + 6*B*a*b^5*d^5*e + 2*B*a^5*b*d*e 
^5))/(7*e^7) + (5*b^3*x^4*(4*B*a^3*e^3 + 7*B*b^3*d^3 + 3*A*a^2*b*e^3 + A*b 
^3*d^2*e + 2*A*a*b^2*d*e^2 + 6*B*a*b^2*d^2*e + 5*B*a^2*b*d*e^2))/(4*e^4) + 
 (b^5*x^6*(A*b*e + 6*B*a*e + 7*B*b*d))/(2*e^2) + (b*x^2*(2*B*a^5*e^5 + 7*B 
*b^5*d^5 + 5*A*a^4*b*e^5 + A*b^5*d^4*e + 2*A*a*b^4*d^3*e^2 + 4*A*a^3*b^2*d 
*e^4 + 3*A*a^2*b^3*d^2*e^3 + 5*B*a^2*b^3*d^3*e^2 + 4*B*a^3*b^2*d^2*e^3 + 6 
*B*a*b^4*d^4*e + 3*B*a^4*b*d*e^4))/(2*e^6) + (b^2*x^3*(3*B*a^4*e^4 + 7*B*b 
^4*d^4 + 4*A*a^3*b*e^4 + A*b^4*d^3*e + 2*A*a*b^3*d^2*e^2 + 3*A*a^2*b^2*d*e 
^3 + 5*B*a^2*b^2*d^2*e^2 + 6*B*a*b^3*d^3*e + 4*B*a^3*b*d*e^3))/e^5 + (b^4* 
x^5*(5*B*a^2*e^2 + 7*B*b^2*d^2 + 2*A*a*b*e^2 + A*b^2*d*e + 6*B*a*b*d*e))/e 
^3 + (B*b^6*x^7)/e)/(d^8 + e^8*x^8 + 8*d*e^7*x^7 + 28*d^6*e^2*x^2 + 56*d^5 
*e^3*x^3 + 70*d^4*e^4*x^4 + 56*d^3*e^5*x^5 + 28*d^2*e^6*x^6 + 8*d^7*e*x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 504, normalized size of antiderivative = 5.86 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx=\frac {b^{7} e^{7} x^{8}-28 a \,b^{6} d \,e^{6} x^{6}-56 a^{2} b^{5} d \,e^{6} x^{5}-56 a \,b^{6} d^{2} e^{5} x^{5}-70 a^{3} b^{4} d \,e^{6} x^{4}-70 a^{2} b^{5} d^{2} e^{5} x^{4}-70 a \,b^{6} d^{3} e^{4} x^{4}-56 a^{4} b^{3} d \,e^{6} x^{3}-56 a^{3} b^{4} d^{2} e^{5} x^{3}-56 a^{2} b^{5} d^{3} e^{4} x^{3}-56 a \,b^{6} d^{4} e^{3} x^{3}-28 a^{5} b^{2} d \,e^{6} x^{2}-28 a^{4} b^{3} d^{2} e^{5} x^{2}-28 a^{3} b^{4} d^{3} e^{4} x^{2}-28 a^{2} b^{5} d^{4} e^{3} x^{2}-28 a \,b^{6} d^{5} e^{2} x^{2}-8 a^{6} b d \,e^{6} x -8 a^{5} b^{2} d^{2} e^{5} x -8 a^{4} b^{3} d^{3} e^{4} x -8 a^{3} b^{4} d^{4} e^{3} x -8 a^{2} b^{5} d^{5} e^{2} x -8 a \,b^{6} d^{6} e x -a^{7} d \,e^{6}-a^{6} b \,d^{2} e^{5}-a^{5} b^{2} d^{3} e^{4}-a^{4} b^{3} d^{4} e^{3}-a^{3} b^{4} d^{5} e^{2}-a^{2} b^{5} d^{6} e -a \,b^{6} d^{7}}{8 d \,e^{7} \left (e^{8} x^{8}+8 d \,e^{7} x^{7}+28 d^{2} e^{6} x^{6}+56 d^{3} e^{5} x^{5}+70 d^{4} e^{4} x^{4}+56 d^{5} e^{3} x^{3}+28 d^{6} e^{2} x^{2}+8 d^{7} e x +d^{8}\right )} \] Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^9,x)
 

Output:

( - a**7*d*e**6 - a**6*b*d**2*e**5 - 8*a**6*b*d*e**6*x - a**5*b**2*d**3*e* 
*4 - 8*a**5*b**2*d**2*e**5*x - 28*a**5*b**2*d*e**6*x**2 - a**4*b**3*d**4*e 
**3 - 8*a**4*b**3*d**3*e**4*x - 28*a**4*b**3*d**2*e**5*x**2 - 56*a**4*b**3 
*d*e**6*x**3 - a**3*b**4*d**5*e**2 - 8*a**3*b**4*d**4*e**3*x - 28*a**3*b** 
4*d**3*e**4*x**2 - 56*a**3*b**4*d**2*e**5*x**3 - 70*a**3*b**4*d*e**6*x**4 
- a**2*b**5*d**6*e - 8*a**2*b**5*d**5*e**2*x - 28*a**2*b**5*d**4*e**3*x**2 
 - 56*a**2*b**5*d**3*e**4*x**3 - 70*a**2*b**5*d**2*e**5*x**4 - 56*a**2*b** 
5*d*e**6*x**5 - a*b**6*d**7 - 8*a*b**6*d**6*e*x - 28*a*b**6*d**5*e**2*x**2 
 - 56*a*b**6*d**4*e**3*x**3 - 70*a*b**6*d**3*e**4*x**4 - 56*a*b**6*d**2*e* 
*5*x**5 - 28*a*b**6*d*e**6*x**6 + b**7*e**7*x**8)/(8*d*e**7*(d**8 + 8*d**7 
*e*x + 28*d**6*e**2*x**2 + 56*d**5*e**3*x**3 + 70*d**4*e**4*x**4 + 56*d**3 
*e**5*x**5 + 28*d**2*e**6*x**6 + 8*d*e**7*x**7 + e**8*x**8))