\(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx\) [61]

Optimal result

Integrand size = 20, antiderivative size = 235 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx=-\frac {(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac {(7 b B d+4 A b e-11 a B e) (a+b x)^7}{110 e (b d-a e)^2 (d+e x)^{10}}+\frac {b (7 b B d+4 A b e-11 a B e) (a+b x)^7}{330 e (b d-a e)^3 (d+e x)^9}+\frac {b^2 (7 b B d+4 A b e-11 a B e) (a+b x)^7}{1320 e (b d-a e)^4 (d+e x)^8}+\frac {b^3 (7 b B d+4 A b e-11 a B e) (a+b x)^7}{9240 e (b d-a e)^5 (d+e x)^7} \] Output:

-1/11*(-A*e+B*d)*(b*x+a)^7/e/(-a*e+b*d)/(e*x+d)^11+1/110*(4*A*b*e-11*B*a*e 
+7*B*b*d)*(b*x+a)^7/e/(-a*e+b*d)^2/(e*x+d)^10+1/330*b*(4*A*b*e-11*B*a*e+7* 
B*b*d)*(b*x+a)^7/e/(-a*e+b*d)^3/(e*x+d)^9+1/1320*b^2*(4*A*b*e-11*B*a*e+7*B 
*b*d)*(b*x+a)^7/e/(-a*e+b*d)^4/(e*x+d)^8+1/9240*b^3*(4*A*b*e-11*B*a*e+7*B* 
b*d)*(b*x+a)^7/e/(-a*e+b*d)^5/(e*x+d)^7
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(605\) vs. \(2(235)=470\).

Time = 0.16 (sec) , antiderivative size = 605, normalized size of antiderivative = 2.57 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx=-\frac {84 a^6 e^6 (10 A e+B (d+11 e x))+56 a^5 b e^5 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+35 a^4 b^2 e^4 \left (8 A e \left (d^2+11 d e x+55 e^2 x^2\right )+3 B \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )\right )+20 a^3 b^3 e^3 \left (7 A e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+4 B \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )+10 a^2 b^4 e^2 \left (6 A e \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 B \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )\right )+4 a b^5 e \left (5 A e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+6 B \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )\right )+b^6 \left (4 A e \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )+7 B \left (d^7+11 d^6 e x+55 d^5 e^2 x^2+165 d^4 e^3 x^3+330 d^3 e^4 x^4+462 d^2 e^5 x^5+462 d e^6 x^6+330 e^7 x^7\right )\right )}{9240 e^8 (d+e x)^{11}} \] Input:

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^12,x]
 

Output:

-1/9240*(84*a^6*e^6*(10*A*e + B*(d + 11*e*x)) + 56*a^5*b*e^5*(9*A*e*(d + 1 
1*e*x) + 2*B*(d^2 + 11*d*e*x + 55*e^2*x^2)) + 35*a^4*b^2*e^4*(8*A*e*(d^2 + 
 11*d*e*x + 55*e^2*x^2) + 3*B*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x 
^3)) + 20*a^3*b^3*e^3*(7*A*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^ 
3) + 4*B*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4) 
) + 10*a^2*b^4*e^2*(6*A*e*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x 
^3 + 330*e^4*x^4) + 5*B*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x 
^3 + 330*d*e^4*x^4 + 462*e^5*x^5)) + 4*a*b^5*e*(5*A*e*(d^5 + 11*d^4*e*x + 
55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5) + 6*B*(d^6 
 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d 
*e^5*x^5 + 462*e^6*x^6)) + b^6*(4*A*e*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 
 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x^6) + 7*B*(d 
^7 + 11*d^6*e*x + 55*d^5*e^2*x^2 + 165*d^4*e^3*x^3 + 330*d^3*e^4*x^4 + 462 
*d^2*e^5*x^5 + 462*d*e^6*x^6 + 330*e^7*x^7)))/(e^8*(d + e*x)^11)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {87, 55, 55, 55, 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.

\(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-11 a B e+4 A b e+7 b B d) \int \frac {(a+b x)^6}{(d+e x)^{11}}dx}{11 e (b d-a e)}-\frac {(a+b x)^7 (B d-A e)}{11 e (d+e x)^{11} (b d-a e)}\)

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 48

\(\displaystyle \frac {\left (\frac {(a+b x)^7}{10 (d+e x)^{10} (b d-a e)}+\frac {3 b \left (\frac {(a+b x)^7}{9 (d+e x)^9 (b d-a e)}+\frac {2 b \left (\frac {b (a+b x)^7}{56 (d+e x)^7 (b d-a e)^2}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)}\right )}{9 (b d-a e)}\right )}{10 (b d-a e)}\right ) (-11 a B e+4 A b e+7 b B d)}{11 e (b d-a e)}-\frac {(a+b x)^7 (B d-A e)}{11 e (d+e x)^{11} (b d-a e)}\)

Input:

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^12,x]
 

Output:

-1/11*((B*d - A*e)*(a + b*x)^7)/(e*(b*d - a*e)*(d + e*x)^11) + ((7*b*B*d + 
 4*A*b*e - 11*a*B*e)*((a + b*x)^7/(10*(b*d - a*e)*(d + e*x)^10) + (3*b*((a 
 + b*x)^7/(9*(b*d - a*e)*(d + e*x)^9) + (2*b*((a + b*x)^7/(8*(b*d - a*e)*( 
d + e*x)^8) + (b*(a + b*x)^7)/(56*(b*d - a*e)^2*(d + e*x)^7)))/(9*(b*d - a 
*e))))/(10*(b*d - a*e))))/(11*e*(b*d - a*e))
 

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 55

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] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 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. \(788\) vs. \(2(225)=450\).

Time = 0.24 (sec) , antiderivative size = 789, normalized size of antiderivative = 3.36

method	result	size
risch	\(\frac {-\frac {b^{6} B \,x^{7}}{4 e}-\frac {b^{5} \left (4 A b e +24 B a e +7 B b d \right ) x^{6}}{20 e^{2}}-\frac {b^{4} \left (20 A a b \,e^{2}+4 A \,b^{2} d e +50 B \,a^{2} e^{2}+24 B a b d e +7 b^{2} B \,d^{2}\right ) x^{5}}{20 e^{3}}-\frac {b^{3} \left (60 A \,a^{2} b \,e^{3}+20 A a \,b^{2} d \,e^{2}+4 A \,b^{3} d^{2} e +80 B \,a^{3} e^{3}+50 B \,a^{2} b d \,e^{2}+24 B a \,b^{2} d^{2} e +7 b^{3} B \,d^{3}\right ) x^{4}}{28 e^{4}}-\frac {b^{2} \left (140 A \,a^{3} b \,e^{4}+60 A \,a^{2} b^{2} d \,e^{3}+20 A a \,b^{3} d^{2} e^{2}+4 A \,b^{4} d^{3} e +105 B \,a^{4} e^{4}+80 B \,a^{3} b d \,e^{3}+50 B \,a^{2} b^{2} d^{2} e^{2}+24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) x^{3}}{56 e^{5}}-\frac {b \left (280 A \,a^{4} b \,e^{5}+140 A \,a^{3} b^{2} d \,e^{4}+60 A \,a^{2} b^{3} d^{2} e^{3}+20 A a \,b^{4} d^{3} e^{2}+4 A \,b^{5} d^{4} e +112 B \,a^{5} e^{5}+105 B \,a^{4} b d \,e^{4}+80 B \,a^{3} b^{2} d^{2} e^{3}+50 B \,a^{2} b^{3} d^{3} e^{2}+24 B a \,b^{4} d^{4} e +7 B \,b^{5} d^{5}\right ) x^{2}}{168 e^{6}}-\frac {\left (504 A \,a^{5} b \,e^{6}+280 A \,a^{4} b^{2} d \,e^{5}+140 A \,a^{3} b^{3} d^{2} e^{4}+60 A \,a^{2} b^{4} d^{3} e^{3}+20 A a \,b^{5} d^{4} e^{2}+4 A \,b^{6} d^{5} e +84 B \,a^{6} e^{6}+112 B \,a^{5} b d \,e^{5}+105 B \,a^{4} b^{2} d^{2} e^{4}+80 B \,a^{3} b^{3} d^{3} e^{3}+50 B \,a^{2} b^{4} d^{4} e^{2}+24 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}\right ) x}{840 e^{7}}-\frac {840 a^{6} A \,e^{7}+504 A \,a^{5} b d \,e^{6}+280 A \,a^{4} b^{2} d^{2} e^{5}+140 A \,a^{3} b^{3} d^{3} e^{4}+60 A \,a^{2} b^{4} d^{4} e^{3}+20 A a \,b^{5} d^{5} e^{2}+4 A \,b^{6} d^{6} e +84 B \,a^{6} d \,e^{6}+112 B \,a^{5} b \,d^{2} e^{5}+105 B \,a^{4} b^{2} d^{3} e^{4}+80 B \,a^{3} b^{3} d^{4} e^{3}+50 B \,a^{2} b^{4} d^{5} e^{2}+24 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{9240 e^{8}}}{\left (e x +d \right )^{11}}\)	\(789\)
default	\(-\frac {b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right )}{2 e^{8} \left (e x +d \right )^{6}}-\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right )}{7 e^{8} \left (e x +d \right )^{7}}-\frac {5 b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{8 e^{8} \left (e x +d \right )^{8}}-\frac {b^{5} \left (A b e +6 B a e -7 B b d \right )}{5 e^{8} \left (e x +d \right )^{5}}-\frac {a^{6} A \,e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{11 e^{8} \left (e x +d \right )^{11}}-\frac {b^{6} B}{4 e^{8} \left (e x +d \right )^{4}}-\frac {b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{3 e^{8} \left (e x +d \right )^{9}}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{10 e^{8} \left (e x +d \right )^{10}}\)	\(814\)
norman	\(\frac {-\frac {b^{6} B \,x^{7}}{4 e}-\frac {\left (4 A \,b^{6} e^{4}+24 B a \,b^{5} e^{4}+7 b^{6} B d \,e^{3}\right ) x^{6}}{20 e^{5}}-\frac {\left (20 A a \,b^{5} e^{5}+4 A \,b^{6} d \,e^{4}+50 B \,a^{2} b^{4} e^{5}+24 B a \,b^{5} d \,e^{4}+7 b^{6} B \,d^{2} e^{3}\right ) x^{5}}{20 e^{6}}-\frac {\left (60 A \,a^{2} b^{4} e^{6}+20 A a \,b^{5} d \,e^{5}+4 A \,b^{6} d^{2} e^{4}+80 B \,a^{3} b^{3} e^{6}+50 B \,a^{2} b^{4} d \,e^{5}+24 B a \,b^{5} d^{2} e^{4}+7 b^{6} B \,d^{3} e^{3}\right ) x^{4}}{28 e^{7}}-\frac {\left (140 A \,a^{3} b^{3} e^{7}+60 A \,a^{2} b^{4} d \,e^{6}+20 A a \,b^{5} d^{2} e^{5}+4 A \,b^{6} d^{3} e^{4}+105 B \,a^{4} b^{2} e^{7}+80 B \,a^{3} b^{3} d \,e^{6}+50 B \,a^{2} b^{4} d^{2} e^{5}+24 B a \,b^{5} d^{3} e^{4}+7 b^{6} B \,d^{4} e^{3}\right ) x^{3}}{56 e^{8}}-\frac {\left (280 A \,a^{4} b^{2} e^{8}+140 A \,a^{3} b^{3} d \,e^{7}+60 A \,a^{2} b^{4} d^{2} e^{6}+20 A a \,b^{5} d^{3} e^{5}+4 A \,b^{6} d^{4} e^{4}+112 B \,a^{5} b \,e^{8}+105 B \,a^{4} b^{2} d \,e^{7}+80 B \,a^{3} b^{3} d^{2} e^{6}+50 B \,a^{2} b^{4} d^{3} e^{5}+24 B a \,b^{5} d^{4} e^{4}+7 b^{6} B \,d^{5} e^{3}\right ) x^{2}}{168 e^{9}}-\frac {\left (504 A \,a^{5} b \,e^{9}+280 A \,a^{4} b^{2} d \,e^{8}+140 A \,a^{3} b^{3} d^{2} e^{7}+60 A \,a^{2} b^{4} d^{3} e^{6}+20 A a \,b^{5} d^{4} e^{5}+4 A \,b^{6} d^{5} e^{4}+84 B \,a^{6} e^{9}+112 B \,a^{5} b d \,e^{8}+105 B \,a^{4} b^{2} d^{2} e^{7}+80 B \,a^{3} b^{3} d^{3} e^{6}+50 B \,a^{2} b^{4} d^{4} e^{5}+24 B a \,b^{5} d^{5} e^{4}+7 b^{6} B \,d^{6} e^{3}\right ) x}{840 e^{10}}-\frac {840 a^{6} A \,e^{10}+504 A \,a^{5} b d \,e^{9}+280 A \,a^{4} b^{2} d^{2} e^{8}+140 A \,a^{3} b^{3} d^{3} e^{7}+60 A \,a^{2} b^{4} d^{4} e^{6}+20 A a \,b^{5} d^{5} e^{5}+4 A \,b^{6} d^{6} e^{4}+84 B \,a^{6} d \,e^{9}+112 B \,a^{5} b \,d^{2} e^{8}+105 B \,a^{4} b^{2} d^{3} e^{7}+80 B \,a^{3} b^{3} d^{4} e^{6}+50 B \,a^{2} b^{4} d^{5} e^{5}+24 B a \,b^{5} d^{6} e^{4}+7 B \,b^{6} d^{7} e^{3}}{9240 e^{11}}}{\left (e x +d \right )^{11}}\)	\(858\)
gosper	\(-\frac {2310 B \,x^{7} b^{6} e^{7}+1848 A \,x^{6} b^{6} e^{7}+11088 B \,x^{6} a \,b^{5} e^{7}+3234 B \,x^{6} b^{6} d \,e^{6}+9240 A \,x^{5} a \,b^{5} e^{7}+1848 A \,x^{5} b^{6} d \,e^{6}+23100 B \,x^{5} a^{2} b^{4} e^{7}+11088 B \,x^{5} a \,b^{5} d \,e^{6}+3234 B \,x^{5} b^{6} d^{2} e^{5}+19800 A \,x^{4} a^{2} b^{4} e^{7}+6600 A \,x^{4} a \,b^{5} d \,e^{6}+1320 A \,x^{4} b^{6} d^{2} e^{5}+26400 B \,x^{4} a^{3} b^{3} e^{7}+16500 B \,x^{4} a^{2} b^{4} d \,e^{6}+7920 B \,x^{4} a \,b^{5} d^{2} e^{5}+2310 B \,x^{4} b^{6} d^{3} e^{4}+23100 A \,x^{3} a^{3} b^{3} e^{7}+9900 A \,x^{3} a^{2} b^{4} d \,e^{6}+3300 A \,x^{3} a \,b^{5} d^{2} e^{5}+660 A \,x^{3} b^{6} d^{3} e^{4}+17325 B \,x^{3} a^{4} b^{2} e^{7}+13200 B \,x^{3} a^{3} b^{3} d \,e^{6}+8250 B \,x^{3} a^{2} b^{4} d^{2} e^{5}+3960 B \,x^{3} a \,b^{5} d^{3} e^{4}+1155 B \,x^{3} b^{6} d^{4} e^{3}+15400 A \,x^{2} a^{4} b^{2} e^{7}+7700 A \,x^{2} a^{3} b^{3} d \,e^{6}+3300 A \,x^{2} a^{2} b^{4} d^{2} e^{5}+1100 A \,x^{2} a \,b^{5} d^{3} e^{4}+220 A \,x^{2} b^{6} d^{4} e^{3}+6160 B \,x^{2} a^{5} b \,e^{7}+5775 B \,x^{2} a^{4} b^{2} d \,e^{6}+4400 B \,x^{2} a^{3} b^{3} d^{2} e^{5}+2750 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+1320 B \,x^{2} a \,b^{5} d^{4} e^{3}+385 B \,x^{2} b^{6} d^{5} e^{2}+5544 A x \,a^{5} b \,e^{7}+3080 A x \,a^{4} b^{2} d \,e^{6}+1540 A x \,a^{3} b^{3} d^{2} e^{5}+660 A x \,a^{2} b^{4} d^{3} e^{4}+220 A x a \,b^{5} d^{4} e^{3}+44 A x \,b^{6} d^{5} e^{2}+924 B x \,a^{6} e^{7}+1232 B x \,a^{5} b d \,e^{6}+1155 B x \,a^{4} b^{2} d^{2} e^{5}+880 B x \,a^{3} b^{3} d^{3} e^{4}+550 B x \,a^{2} b^{4} d^{4} e^{3}+264 B x a \,b^{5} d^{5} e^{2}+77 B x \,b^{6} d^{6} e +840 a^{6} A \,e^{7}+504 A \,a^{5} b d \,e^{6}+280 A \,a^{4} b^{2} d^{2} e^{5}+140 A \,a^{3} b^{3} d^{3} e^{4}+60 A \,a^{2} b^{4} d^{4} e^{3}+20 A a \,b^{5} d^{5} e^{2}+4 A \,b^{6} d^{6} e +84 B \,a^{6} d \,e^{6}+112 B \,a^{5} b \,d^{2} e^{5}+105 B \,a^{4} b^{2} d^{3} e^{4}+80 B \,a^{3} b^{3} d^{4} e^{3}+50 B \,a^{2} b^{4} d^{5} e^{2}+24 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{9240 e^{8} \left (e x +d \right )^{11}}\)	\(913\)
orering	\(-\frac {2310 B \,x^{7} b^{6} e^{7}+1848 A \,x^{6} b^{6} e^{7}+11088 B \,x^{6} a \,b^{5} e^{7}+3234 B \,x^{6} b^{6} d \,e^{6}+9240 A \,x^{5} a \,b^{5} e^{7}+1848 A \,x^{5} b^{6} d \,e^{6}+23100 B \,x^{5} a^{2} b^{4} e^{7}+11088 B \,x^{5} a \,b^{5} d \,e^{6}+3234 B \,x^{5} b^{6} d^{2} e^{5}+19800 A \,x^{4} a^{2} b^{4} e^{7}+6600 A \,x^{4} a \,b^{5} d \,e^{6}+1320 A \,x^{4} b^{6} d^{2} e^{5}+26400 B \,x^{4} a^{3} b^{3} e^{7}+16500 B \,x^{4} a^{2} b^{4} d \,e^{6}+7920 B \,x^{4} a \,b^{5} d^{2} e^{5}+2310 B \,x^{4} b^{6} d^{3} e^{4}+23100 A \,x^{3} a^{3} b^{3} e^{7}+9900 A \,x^{3} a^{2} b^{4} d \,e^{6}+3300 A \,x^{3} a \,b^{5} d^{2} e^{5}+660 A \,x^{3} b^{6} d^{3} e^{4}+17325 B \,x^{3} a^{4} b^{2} e^{7}+13200 B \,x^{3} a^{3} b^{3} d \,e^{6}+8250 B \,x^{3} a^{2} b^{4} d^{2} e^{5}+3960 B \,x^{3} a \,b^{5} d^{3} e^{4}+1155 B \,x^{3} b^{6} d^{4} e^{3}+15400 A \,x^{2} a^{4} b^{2} e^{7}+7700 A \,x^{2} a^{3} b^{3} d \,e^{6}+3300 A \,x^{2} a^{2} b^{4} d^{2} e^{5}+1100 A \,x^{2} a \,b^{5} d^{3} e^{4}+220 A \,x^{2} b^{6} d^{4} e^{3}+6160 B \,x^{2} a^{5} b \,e^{7}+5775 B \,x^{2} a^{4} b^{2} d \,e^{6}+4400 B \,x^{2} a^{3} b^{3} d^{2} e^{5}+2750 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+1320 B \,x^{2} a \,b^{5} d^{4} e^{3}+385 B \,x^{2} b^{6} d^{5} e^{2}+5544 A x \,a^{5} b \,e^{7}+3080 A x \,a^{4} b^{2} d \,e^{6}+1540 A x \,a^{3} b^{3} d^{2} e^{5}+660 A x \,a^{2} b^{4} d^{3} e^{4}+220 A x a \,b^{5} d^{4} e^{3}+44 A x \,b^{6} d^{5} e^{2}+924 B x \,a^{6} e^{7}+1232 B x \,a^{5} b d \,e^{6}+1155 B x \,a^{4} b^{2} d^{2} e^{5}+880 B x \,a^{3} b^{3} d^{3} e^{4}+550 B x \,a^{2} b^{4} d^{4} e^{3}+264 B x a \,b^{5} d^{5} e^{2}+77 B x \,b^{6} d^{6} e +840 a^{6} A \,e^{7}+504 A \,a^{5} b d \,e^{6}+280 A \,a^{4} b^{2} d^{2} e^{5}+140 A \,a^{3} b^{3} d^{3} e^{4}+60 A \,a^{2} b^{4} d^{4} e^{3}+20 A a \,b^{5} d^{5} e^{2}+4 A \,b^{6} d^{6} e +84 B \,a^{6} d \,e^{6}+112 B \,a^{5} b \,d^{2} e^{5}+105 B \,a^{4} b^{2} d^{3} e^{4}+80 B \,a^{3} b^{3} d^{4} e^{3}+50 B \,a^{2} b^{4} d^{5} e^{2}+24 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{9240 e^{8} \left (e x +d \right )^{11}}\)	\(913\)
parallelrisch	\(-\frac {2310 B \,b^{6} x^{7} e^{10}+1848 A \,b^{6} e^{10} x^{6}+11088 B a \,b^{5} e^{10} x^{6}+3234 B \,b^{6} d \,e^{9} x^{6}+9240 A a \,b^{5} e^{10} x^{5}+1848 A \,b^{6} d \,e^{9} x^{5}+23100 B \,a^{2} b^{4} e^{10} x^{5}+11088 B a \,b^{5} d \,e^{9} x^{5}+3234 B \,b^{6} d^{2} e^{8} x^{5}+19800 A \,a^{2} b^{4} e^{10} x^{4}+6600 A a \,b^{5} d \,e^{9} x^{4}+1320 A \,b^{6} d^{2} e^{8} x^{4}+26400 B \,a^{3} b^{3} e^{10} x^{4}+16500 B \,a^{2} b^{4} d \,e^{9} x^{4}+7920 B a \,b^{5} d^{2} e^{8} x^{4}+2310 B \,b^{6} d^{3} e^{7} x^{4}+23100 A \,a^{3} b^{3} e^{10} x^{3}+9900 A \,a^{2} b^{4} d \,e^{9} x^{3}+3300 A a \,b^{5} d^{2} e^{8} x^{3}+660 A \,b^{6} d^{3} e^{7} x^{3}+17325 B \,a^{4} b^{2} e^{10} x^{3}+13200 B \,a^{3} b^{3} d \,e^{9} x^{3}+8250 B \,a^{2} b^{4} d^{2} e^{8} x^{3}+3960 B a \,b^{5} d^{3} e^{7} x^{3}+1155 B \,b^{6} d^{4} e^{6} x^{3}+15400 A \,a^{4} b^{2} e^{10} x^{2}+7700 A \,a^{3} b^{3} d \,e^{9} x^{2}+3300 A \,a^{2} b^{4} d^{2} e^{8} x^{2}+1100 A a \,b^{5} d^{3} e^{7} x^{2}+220 A \,b^{6} d^{4} e^{6} x^{2}+6160 B \,a^{5} b \,e^{10} x^{2}+5775 B \,a^{4} b^{2} d \,e^{9} x^{2}+4400 B \,a^{3} b^{3} d^{2} e^{8} x^{2}+2750 B \,a^{2} b^{4} d^{3} e^{7} x^{2}+1320 B a \,b^{5} d^{4} e^{6} x^{2}+385 B \,b^{6} d^{5} e^{5} x^{2}+5544 A \,a^{5} b \,e^{10} x +3080 A \,a^{4} b^{2} d \,e^{9} x +1540 A \,a^{3} b^{3} d^{2} e^{8} x +660 A \,a^{2} b^{4} d^{3} e^{7} x +220 A a \,b^{5} d^{4} e^{6} x +44 A \,b^{6} d^{5} e^{5} x +924 B \,a^{6} e^{10} x +1232 B \,a^{5} b d \,e^{9} x +1155 B \,a^{4} b^{2} d^{2} e^{8} x +880 B \,a^{3} b^{3} d^{3} e^{7} x +550 B \,a^{2} b^{4} d^{4} e^{6} x +264 B a \,b^{5} d^{5} e^{5} x +77 B \,b^{6} d^{6} e^{4} x +840 a^{6} A \,e^{10}+504 A \,a^{5} b d \,e^{9}+280 A \,a^{4} b^{2} d^{2} e^{8}+140 A \,a^{3} b^{3} d^{3} e^{7}+60 A \,a^{2} b^{4} d^{4} e^{6}+20 A a \,b^{5} d^{5} e^{5}+4 A \,b^{6} d^{6} e^{4}+84 B \,a^{6} d \,e^{9}+112 B \,a^{5} b \,d^{2} e^{8}+105 B \,a^{4} b^{2} d^{3} e^{7}+80 B \,a^{3} b^{3} d^{4} e^{6}+50 B \,a^{2} b^{4} d^{5} e^{5}+24 B a \,b^{5} d^{6} e^{4}+7 B \,b^{6} d^{7} e^{3}}{9240 e^{11} \left (e x +d \right )^{11}}\)	\(922\)

Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^12,x,method=_RETURNVERBOSE)
 

Output:

(-1/4*b^6*B/e*x^7-1/20*b^5/e^2*(4*A*b*e+24*B*a*e+7*B*b*d)*x^6-1/20*b^4/e^3 
*(20*A*a*b*e^2+4*A*b^2*d*e+50*B*a^2*e^2+24*B*a*b*d*e+7*B*b^2*d^2)*x^5-1/28 
*b^3/e^4*(60*A*a^2*b*e^3+20*A*a*b^2*d*e^2+4*A*b^3*d^2*e+80*B*a^3*e^3+50*B* 
a^2*b*d*e^2+24*B*a*b^2*d^2*e+7*B*b^3*d^3)*x^4-1/56*b^2/e^5*(140*A*a^3*b*e^ 
4+60*A*a^2*b^2*d*e^3+20*A*a*b^3*d^2*e^2+4*A*b^4*d^3*e+105*B*a^4*e^4+80*B*a 
^3*b*d*e^3+50*B*a^2*b^2*d^2*e^2+24*B*a*b^3*d^3*e+7*B*b^4*d^4)*x^3-1/168*b/ 
e^6*(280*A*a^4*b*e^5+140*A*a^3*b^2*d*e^4+60*A*a^2*b^3*d^2*e^3+20*A*a*b^4*d 
^3*e^2+4*A*b^5*d^4*e+112*B*a^5*e^5+105*B*a^4*b*d*e^4+80*B*a^3*b^2*d^2*e^3+ 
50*B*a^2*b^3*d^3*e^2+24*B*a*b^4*d^4*e+7*B*b^5*d^5)*x^2-1/840/e^7*(504*A*a^ 
5*b*e^6+280*A*a^4*b^2*d*e^5+140*A*a^3*b^3*d^2*e^4+60*A*a^2*b^4*d^3*e^3+20* 
A*a*b^5*d^4*e^2+4*A*b^6*d^5*e+84*B*a^6*e^6+112*B*a^5*b*d*e^5+105*B*a^4*b^2 
*d^2*e^4+80*B*a^3*b^3*d^3*e^3+50*B*a^2*b^4*d^4*e^2+24*B*a*b^5*d^5*e+7*B*b^ 
6*d^6)*x-1/9240/e^8*(840*A*a^6*e^7+504*A*a^5*b*d*e^6+280*A*a^4*b^2*d^2*e^5 
+140*A*a^3*b^3*d^3*e^4+60*A*a^2*b^4*d^4*e^3+20*A*a*b^5*d^5*e^2+4*A*b^6*d^6 
*e+84*B*a^6*d*e^6+112*B*a^5*b*d^2*e^5+105*B*a^4*b^2*d^3*e^4+80*B*a^3*b^3*d 
^4*e^3+50*B*a^2*b^4*d^5*e^2+24*B*a*b^5*d^6*e+7*B*b^6*d^7))/(e*x+d)^11
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 883 vs. \(2 (225) = 450\).

Time = 0.09 (sec) , antiderivative size = 883, normalized size of antiderivative = 3.76 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^12,x, algorithm="fricas")
 

Output:

-1/9240*(2310*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 840*A*a^6*e^7 + 4*(6*B*a*b^5 + 
 A*b^6)*d^6*e + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 20*(4*B*a^3*b^3 + 3 
*A*a^2*b^4)*d^4*e^3 + 35*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 56*(2*B*a^5 
*b + 5*A*a^4*b^2)*d^2*e^5 + 84*(B*a^6 + 6*A*a^5*b)*d*e^6 + 462*(7*B*b^6*d* 
e^6 + 4*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 462*(7*B*b^6*d^2*e^5 + 4*(6*B*a*b^5 
 + A*b^6)*d*e^6 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 330*(7*B*b^6*d^3 
*e^4 + 4*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 
+ 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 165*(7*B*b^6*d^4*e^3 + 4*(6*B* 
a*b^5 + A*b^6)*d^3*e^4 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 20*(4*B*a^ 
3*b^3 + 3*A*a^2*b^4)*d*e^6 + 35*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 55* 
(7*B*b^6*d^5*e^2 + 4*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 10*(5*B*a^2*b^4 + 2*A*a 
*b^5)*d^3*e^4 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 35*(3*B*a^4*b^2 + 
 4*A*a^3*b^3)*d*e^6 + 56*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 11*(7*B*b^6* 
d^6*e + 4*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e 
^3 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 35*(3*B*a^4*b^2 + 4*A*a^3*b^ 
3)*d^2*e^5 + 56*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 84*(B*a^6 + 6*A*a^5*b)*e 
^7)*x)/(e^19*x^11 + 11*d*e^18*x^10 + 55*d^2*e^17*x^9 + 165*d^3*e^16*x^8 + 
330*d^4*e^15*x^7 + 462*d^5*e^14*x^6 + 462*d^6*e^13*x^5 + 330*d^7*e^12*x^4 
+ 165*d^8*e^11*x^3 + 55*d^9*e^10*x^2 + 11*d^10*e^9*x + d^11*e^8)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**6*(B*x+A)/(e*x+d)**12,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 883 vs. \(2 (225) = 450\).

Time = 0.13 (sec) , antiderivative size = 883, normalized size of antiderivative = 3.76 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^12,x, algorithm="maxima")
 

Output:

-1/9240*(2310*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 840*A*a^6*e^7 + 4*(6*B*a*b^5 + 
 A*b^6)*d^6*e + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 20*(4*B*a^3*b^3 + 3 
*A*a^2*b^4)*d^4*e^3 + 35*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 56*(2*B*a^5 
*b + 5*A*a^4*b^2)*d^2*e^5 + 84*(B*a^6 + 6*A*a^5*b)*d*e^6 + 462*(7*B*b^6*d* 
e^6 + 4*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 462*(7*B*b^6*d^2*e^5 + 4*(6*B*a*b^5 
 + A*b^6)*d*e^6 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 330*(7*B*b^6*d^3 
*e^4 + 4*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 
+ 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 165*(7*B*b^6*d^4*e^3 + 4*(6*B* 
a*b^5 + A*b^6)*d^3*e^4 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 20*(4*B*a^ 
3*b^3 + 3*A*a^2*b^4)*d*e^6 + 35*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 55* 
(7*B*b^6*d^5*e^2 + 4*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 10*(5*B*a^2*b^4 + 2*A*a 
*b^5)*d^3*e^4 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 35*(3*B*a^4*b^2 + 
 4*A*a^3*b^3)*d*e^6 + 56*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 11*(7*B*b^6* 
d^6*e + 4*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e 
^3 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 35*(3*B*a^4*b^2 + 4*A*a^3*b^ 
3)*d^2*e^5 + 56*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 84*(B*a^6 + 6*A*a^5*b)*e 
^7)*x)/(e^19*x^11 + 11*d*e^18*x^10 + 55*d^2*e^17*x^9 + 165*d^3*e^16*x^8 + 
330*d^4*e^15*x^7 + 462*d^5*e^14*x^6 + 462*d^6*e^13*x^5 + 330*d^7*e^12*x^4 
+ 165*d^8*e^11*x^3 + 55*d^9*e^10*x^2 + 11*d^10*e^9*x + d^11*e^8)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (225) = 450\).

Time = 0.13 (sec) , antiderivative size = 912, normalized size of antiderivative = 3.88 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^12,x, algorithm="giac")
 

Output:

-1/9240*(2310*B*b^6*e^7*x^7 + 3234*B*b^6*d*e^6*x^6 + 11088*B*a*b^5*e^7*x^6 
 + 1848*A*b^6*e^7*x^6 + 3234*B*b^6*d^2*e^5*x^5 + 11088*B*a*b^5*d*e^6*x^5 + 
 1848*A*b^6*d*e^6*x^5 + 23100*B*a^2*b^4*e^7*x^5 + 9240*A*a*b^5*e^7*x^5 + 2 
310*B*b^6*d^3*e^4*x^4 + 7920*B*a*b^5*d^2*e^5*x^4 + 1320*A*b^6*d^2*e^5*x^4 
+ 16500*B*a^2*b^4*d*e^6*x^4 + 6600*A*a*b^5*d*e^6*x^4 + 26400*B*a^3*b^3*e^7 
*x^4 + 19800*A*a^2*b^4*e^7*x^4 + 1155*B*b^6*d^4*e^3*x^3 + 3960*B*a*b^5*d^3 
*e^4*x^3 + 660*A*b^6*d^3*e^4*x^3 + 8250*B*a^2*b^4*d^2*e^5*x^3 + 3300*A*a*b 
^5*d^2*e^5*x^3 + 13200*B*a^3*b^3*d*e^6*x^3 + 9900*A*a^2*b^4*d*e^6*x^3 + 17 
325*B*a^4*b^2*e^7*x^3 + 23100*A*a^3*b^3*e^7*x^3 + 385*B*b^6*d^5*e^2*x^2 + 
1320*B*a*b^5*d^4*e^3*x^2 + 220*A*b^6*d^4*e^3*x^2 + 2750*B*a^2*b^4*d^3*e^4* 
x^2 + 1100*A*a*b^5*d^3*e^4*x^2 + 4400*B*a^3*b^3*d^2*e^5*x^2 + 3300*A*a^2*b 
^4*d^2*e^5*x^2 + 5775*B*a^4*b^2*d*e^6*x^2 + 7700*A*a^3*b^3*d*e^6*x^2 + 616 
0*B*a^5*b*e^7*x^2 + 15400*A*a^4*b^2*e^7*x^2 + 77*B*b^6*d^6*e*x + 264*B*a*b 
^5*d^5*e^2*x + 44*A*b^6*d^5*e^2*x + 550*B*a^2*b^4*d^4*e^3*x + 220*A*a*b^5* 
d^4*e^3*x + 880*B*a^3*b^3*d^3*e^4*x + 660*A*a^2*b^4*d^3*e^4*x + 1155*B*a^4 
*b^2*d^2*e^5*x + 1540*A*a^3*b^3*d^2*e^5*x + 1232*B*a^5*b*d*e^6*x + 3080*A* 
a^4*b^2*d*e^6*x + 924*B*a^6*e^7*x + 5544*A*a^5*b*e^7*x + 7*B*b^6*d^7 + 24* 
B*a*b^5*d^6*e + 4*A*b^6*d^6*e + 50*B*a^2*b^4*d^5*e^2 + 20*A*a*b^5*d^5*e^2 
+ 80*B*a^3*b^3*d^4*e^3 + 60*A*a^2*b^4*d^4*e^3 + 105*B*a^4*b^2*d^3*e^4 + 14 
0*A*a^3*b^3*d^3*e^4 + 112*B*a^5*b*d^2*e^5 + 280*A*a^4*b^2*d^2*e^5 + 84*...
 

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 899, normalized size of antiderivative = 3.83 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx=-\frac {\frac {84\,B\,a^6\,d\,e^6+840\,A\,a^6\,e^7+112\,B\,a^5\,b\,d^2\,e^5+504\,A\,a^5\,b\,d\,e^6+105\,B\,a^4\,b^2\,d^3\,e^4+280\,A\,a^4\,b^2\,d^2\,e^5+80\,B\,a^3\,b^3\,d^4\,e^3+140\,A\,a^3\,b^3\,d^3\,e^4+50\,B\,a^2\,b^4\,d^5\,e^2+60\,A\,a^2\,b^4\,d^4\,e^3+24\,B\,a\,b^5\,d^6\,e+20\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+4\,A\,b^6\,d^6\,e}{9240\,e^8}+\frac {x\,\left (84\,B\,a^6\,e^6+112\,B\,a^5\,b\,d\,e^5+504\,A\,a^5\,b\,e^6+105\,B\,a^4\,b^2\,d^2\,e^4+280\,A\,a^4\,b^2\,d\,e^5+80\,B\,a^3\,b^3\,d^3\,e^3+140\,A\,a^3\,b^3\,d^2\,e^4+50\,B\,a^2\,b^4\,d^4\,e^2+60\,A\,a^2\,b^4\,d^3\,e^3+24\,B\,a\,b^5\,d^5\,e+20\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+4\,A\,b^6\,d^5\,e\right )}{840\,e^7}+\frac {b^3\,x^4\,\left (80\,B\,a^3\,e^3+50\,B\,a^2\,b\,d\,e^2+60\,A\,a^2\,b\,e^3+24\,B\,a\,b^2\,d^2\,e+20\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+4\,A\,b^3\,d^2\,e\right )}{28\,e^4}+\frac {b^5\,x^6\,\left (4\,A\,b\,e+24\,B\,a\,e+7\,B\,b\,d\right )}{20\,e^2}+\frac {b\,x^2\,\left (112\,B\,a^5\,e^5+105\,B\,a^4\,b\,d\,e^4+280\,A\,a^4\,b\,e^5+80\,B\,a^3\,b^2\,d^2\,e^3+140\,A\,a^3\,b^2\,d\,e^4+50\,B\,a^2\,b^3\,d^3\,e^2+60\,A\,a^2\,b^3\,d^2\,e^3+24\,B\,a\,b^4\,d^4\,e+20\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+4\,A\,b^5\,d^4\,e\right )}{168\,e^6}+\frac {b^2\,x^3\,\left (105\,B\,a^4\,e^4+80\,B\,a^3\,b\,d\,e^3+140\,A\,a^3\,b\,e^4+50\,B\,a^2\,b^2\,d^2\,e^2+60\,A\,a^2\,b^2\,d\,e^3+24\,B\,a\,b^3\,d^3\,e+20\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+4\,A\,b^4\,d^3\,e\right )}{56\,e^5}+\frac {b^4\,x^5\,\left (50\,B\,a^2\,e^2+24\,B\,a\,b\,d\,e+20\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+4\,A\,b^2\,d\,e\right )}{20\,e^3}+\frac {B\,b^6\,x^7}{4\,e}}{d^{11}+11\,d^{10}\,e\,x+55\,d^9\,e^2\,x^2+165\,d^8\,e^3\,x^3+330\,d^7\,e^4\,x^4+462\,d^6\,e^5\,x^5+462\,d^5\,e^6\,x^6+330\,d^4\,e^7\,x^7+165\,d^3\,e^8\,x^8+55\,d^2\,e^9\,x^9+11\,d\,e^{10}\,x^{10}+e^{11}\,x^{11}} \] Input:

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^12,x)
 

Output:

-((840*A*a^6*e^7 + 7*B*b^6*d^7 + 4*A*b^6*d^6*e + 84*B*a^6*d*e^6 + 20*A*a*b 
^5*d^5*e^2 + 112*B*a^5*b*d^2*e^5 + 60*A*a^2*b^4*d^4*e^3 + 140*A*a^3*b^3*d^ 
3*e^4 + 280*A*a^4*b^2*d^2*e^5 + 50*B*a^2*b^4*d^5*e^2 + 80*B*a^3*b^3*d^4*e^ 
3 + 105*B*a^4*b^2*d^3*e^4 + 504*A*a^5*b*d*e^6 + 24*B*a*b^5*d^6*e)/(9240*e^ 
8) + (x*(84*B*a^6*e^6 + 7*B*b^6*d^6 + 504*A*a^5*b*e^6 + 4*A*b^6*d^5*e + 20 
*A*a*b^5*d^4*e^2 + 280*A*a^4*b^2*d*e^5 + 60*A*a^2*b^4*d^3*e^3 + 140*A*a^3* 
b^3*d^2*e^4 + 50*B*a^2*b^4*d^4*e^2 + 80*B*a^3*b^3*d^3*e^3 + 105*B*a^4*b^2* 
d^2*e^4 + 24*B*a*b^5*d^5*e + 112*B*a^5*b*d*e^5))/(840*e^7) + (b^3*x^4*(80* 
B*a^3*e^3 + 7*B*b^3*d^3 + 60*A*a^2*b*e^3 + 4*A*b^3*d^2*e + 20*A*a*b^2*d*e^ 
2 + 24*B*a*b^2*d^2*e + 50*B*a^2*b*d*e^2))/(28*e^4) + (b^5*x^6*(4*A*b*e + 2 
4*B*a*e + 7*B*b*d))/(20*e^2) + (b*x^2*(112*B*a^5*e^5 + 7*B*b^5*d^5 + 280*A 
*a^4*b*e^5 + 4*A*b^5*d^4*e + 20*A*a*b^4*d^3*e^2 + 140*A*a^3*b^2*d*e^4 + 60 
*A*a^2*b^3*d^2*e^3 + 50*B*a^2*b^3*d^3*e^2 + 80*B*a^3*b^2*d^2*e^3 + 24*B*a* 
b^4*d^4*e + 105*B*a^4*b*d*e^4))/(168*e^6) + (b^2*x^3*(105*B*a^4*e^4 + 7*B* 
b^4*d^4 + 140*A*a^3*b*e^4 + 4*A*b^4*d^3*e + 20*A*a*b^3*d^2*e^2 + 60*A*a^2* 
b^2*d*e^3 + 50*B*a^2*b^2*d^2*e^2 + 24*B*a*b^3*d^3*e + 80*B*a^3*b*d*e^3))/( 
56*e^5) + (b^4*x^5*(50*B*a^2*e^2 + 7*B*b^2*d^2 + 20*A*a*b*e^2 + 4*A*b^2*d* 
e + 24*B*a*b*d*e))/(20*e^3) + (B*b^6*x^7)/(4*e))/(d^11 + e^11*x^11 + 11*d* 
e^10*x^10 + 55*d^9*e^2*x^2 + 165*d^8*e^3*x^3 + 330*d^7*e^4*x^4 + 462*d^6*e 
^5*x^5 + 462*d^5*e^6*x^6 + 330*d^4*e^7*x^7 + 165*d^3*e^8*x^8 + 55*d^2*e...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 607, normalized size of antiderivative = 2.58 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx=\frac {-330 b^{7} e^{7} x^{7}-1848 a \,b^{6} e^{7} x^{6}-462 b^{7} d \,e^{6} x^{6}-4620 a^{2} b^{5} e^{7} x^{5}-1848 a \,b^{6} d \,e^{6} x^{5}-462 b^{7} d^{2} e^{5} x^{5}-6600 a^{3} b^{4} e^{7} x^{4}-3300 a^{2} b^{5} d \,e^{6} x^{4}-1320 a \,b^{6} d^{2} e^{5} x^{4}-330 b^{7} d^{3} e^{4} x^{4}-5775 a^{4} b^{3} e^{7} x^{3}-3300 a^{3} b^{4} d \,e^{6} x^{3}-1650 a^{2} b^{5} d^{2} e^{5} x^{3}-660 a \,b^{6} d^{3} e^{4} x^{3}-165 b^{7} d^{4} e^{3} x^{3}-3080 a^{5} b^{2} e^{7} x^{2}-1925 a^{4} b^{3} d \,e^{6} x^{2}-1100 a^{3} b^{4} d^{2} e^{5} x^{2}-550 a^{2} b^{5} d^{3} e^{4} x^{2}-220 a \,b^{6} d^{4} e^{3} x^{2}-55 b^{7} d^{5} e^{2} x^{2}-924 a^{6} b \,e^{7} x -616 a^{5} b^{2} d \,e^{6} x -385 a^{4} b^{3} d^{2} e^{5} x -220 a^{3} b^{4} d^{3} e^{4} x -110 a^{2} b^{5} d^{4} e^{3} x -44 a \,b^{6} d^{5} e^{2} x -11 b^{7} d^{6} e x -120 a^{7} e^{7}-84 a^{6} b d \,e^{6}-56 a^{5} b^{2} d^{2} e^{5}-35 a^{4} b^{3} d^{3} e^{4}-20 a^{3} b^{4} d^{4} e^{3}-10 a^{2} b^{5} d^{5} e^{2}-4 a \,b^{6} d^{6} e -b^{7} d^{7}}{1320 e^{8} \left (e^{11} x^{11}+11 d \,e^{10} x^{10}+55 d^{2} e^{9} x^{9}+165 d^{3} e^{8} x^{8}+330 d^{4} e^{7} x^{7}+462 d^{5} e^{6} x^{6}+462 d^{6} e^{5} x^{5}+330 d^{7} e^{4} x^{4}+165 d^{8} e^{3} x^{3}+55 d^{9} e^{2} x^{2}+11 d^{10} e x +d^{11}\right )} \] Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^12,x)
 

Output:

( - 120*a**7*e**7 - 84*a**6*b*d*e**6 - 924*a**6*b*e**7*x - 56*a**5*b**2*d* 
*2*e**5 - 616*a**5*b**2*d*e**6*x - 3080*a**5*b**2*e**7*x**2 - 35*a**4*b**3 
*d**3*e**4 - 385*a**4*b**3*d**2*e**5*x - 1925*a**4*b**3*d*e**6*x**2 - 5775 
*a**4*b**3*e**7*x**3 - 20*a**3*b**4*d**4*e**3 - 220*a**3*b**4*d**3*e**4*x 
- 1100*a**3*b**4*d**2*e**5*x**2 - 3300*a**3*b**4*d*e**6*x**3 - 6600*a**3*b 
**4*e**7*x**4 - 10*a**2*b**5*d**5*e**2 - 110*a**2*b**5*d**4*e**3*x - 550*a 
**2*b**5*d**3*e**4*x**2 - 1650*a**2*b**5*d**2*e**5*x**3 - 3300*a**2*b**5*d 
*e**6*x**4 - 4620*a**2*b**5*e**7*x**5 - 4*a*b**6*d**6*e - 44*a*b**6*d**5*e 
**2*x - 220*a*b**6*d**4*e**3*x**2 - 660*a*b**6*d**3*e**4*x**3 - 1320*a*b** 
6*d**2*e**5*x**4 - 1848*a*b**6*d*e**6*x**5 - 1848*a*b**6*e**7*x**6 - b**7* 
d**7 - 11*b**7*d**6*e*x - 55*b**7*d**5*e**2*x**2 - 165*b**7*d**4*e**3*x**3 
 - 330*b**7*d**3*e**4*x**4 - 462*b**7*d**2*e**5*x**5 - 462*b**7*d*e**6*x** 
6 - 330*b**7*e**7*x**7)/(1320*e**8*(d**11 + 11*d**10*e*x + 55*d**9*e**2*x* 
*2 + 165*d**8*e**3*x**3 + 330*d**7*e**4*x**4 + 462*d**6*e**5*x**5 + 462*d* 
*5*e**6*x**6 + 330*d**4*e**7*x**7 + 165*d**3*e**8*x**8 + 55*d**2*e**9*x**9 
 + 11*d*e**10*x**10 + e**11*x**11))