Integrand size = 20, antiderivative size = 185 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{11}} \, dx=-\frac {(B d-A e) (a+b x)^7}{10 e (b d-a e) (d+e x)^{10}}+\frac {(7 b B d+3 A b e-10 a B e) (a+b x)^7}{90 e (b d-a e)^2 (d+e x)^9}+\frac {b (7 b B d+3 A b e-10 a B e) (a+b x)^7}{360 e (b d-a e)^3 (d+e x)^8}+\frac {b^2 (7 b B d+3 A b e-10 a B e) (a+b x)^7}{2520 e (b d-a e)^4 (d+e x)^7} \]
-1/10*(-A*e+B*d)*(b*x+a)^7/e/(-a*e+b*d)/(e*x+d)^10+1/90*(3*A*b*e-10*B*a*e+ 7*B*b*d)*(b*x+a)^7/e/(-a*e+b*d)^2/(e*x+d)^9+1/360*b*(3*A*b*e-10*B*a*e+7*B* b*d)*(b*x+a)^7/e/(-a*e+b*d)^3/(e*x+d)^8+1/2520*b^2*(3*A*b*e-10*B*a*e+7*B*b *d)*(b*x+a)^7/e/(-a*e+b*d)^4/(e*x+d)^7
Leaf count is larger than twice the leaf count of optimal. \(602\) vs. \(2(185)=370\).
Time = 0.17 (sec) , antiderivative size = 602, normalized size of antiderivative = 3.25 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{11}} \, dx=-\frac {28 a^6 e^6 (9 A e+B (d+10 e x))+42 a^5 b e^5 \left (4 A e (d+10 e x)+B \left (d^2+10 d e x+45 e^2 x^2\right )\right )+15 a^4 b^2 e^4 \left (7 A e \left (d^2+10 d e x+45 e^2 x^2\right )+3 B \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )\right )+20 a^3 b^3 e^3 \left (3 A e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+2 B \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )\right )+30 a^2 b^4 e^2 \left (A e \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+B \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )\right )+6 a b^5 e \left (2 A e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+3 B \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )+b^6 \left (3 A e \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )+7 B \left (d^7+10 d^6 e x+45 d^5 e^2 x^2+120 d^4 e^3 x^3+210 d^3 e^4 x^4+252 d^2 e^5 x^5+210 d e^6 x^6+120 e^7 x^7\right )\right )}{2520 e^8 (d+e x)^{10}} \]
-1/2520*(28*a^6*e^6*(9*A*e + B*(d + 10*e*x)) + 42*a^5*b*e^5*(4*A*e*(d + 10 *e*x) + B*(d^2 + 10*d*e*x + 45*e^2*x^2)) + 15*a^4*b^2*e^4*(7*A*e*(d^2 + 10 *d*e*x + 45*e^2*x^2) + 3*B*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) ) + 20*a^3*b^3*e^3*(3*A*e*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 2*B*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4)) + 30*a^2*b^4*e^2*(A*e*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) + B*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 21 0*d*e^4*x^4 + 252*e^5*x^5)) + 6*a*b^5*e*(2*A*e*(d^5 + 10*d^4*e*x + 45*d^3* e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5) + 3*B*(d^6 + 10*d ^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^ 5 + 210*e^6*x^6)) + b^6*(3*A*e*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^ 3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6) + 7*B*(d^7 + 10 *d^6*e*x + 45*d^5*e^2*x^2 + 120*d^4*e^3*x^3 + 210*d^3*e^4*x^4 + 252*d^2*e^ 5*x^5 + 210*d*e^6*x^6 + 120*e^7*x^7)))/(e^8*(d + e*x)^10)
Time = 0.25 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {87, 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.
| \(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{11}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-10 a B e+3 A b e+7 b B d) \int \frac {(a+b x)^6}{(d+e x)^{10}}dx}{10 e (b d-a e)}-\frac {(a+b x)^7 (B d-A e)}{10 e (d+e x)^{10} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-10 a B e+3 A b e+7 b B d) \left (\frac {2 b \int \frac {(a+b x)^6}{(d+e x)^9}dx}{9 (b d-a e)}+\frac {(a+b x)^7}{9 (d+e x)^9 (b d-a e)}\right )}{10 e (b d-a e)}-\frac {(a+b x)^7 (B d-A e)}{10 e (d+e x)^{10} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-10 a B e+3 A b e+7 b B d) \left (\frac {2 b \left (\frac {b \int \frac {(a+b x)^6}{(d+e x)^8}dx}{8 (b d-a e)}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)}\right )}{9 (b d-a e)}+\frac {(a+b x)^7}{9 (d+e x)^9 (b d-a e)}\right )}{10 e (b d-a e)}-\frac {(a+b x)^7 (B d-A e)}{10 e (d+e x)^{10} (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\left (\frac {(a+b x)^7}{9 (d+e x)^9 (b d-a e)}+\frac {2 b \left (\frac {b (a+b x)^7}{56 (d+e x)^7 (b d-a e)^2}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)}\right )}{9 (b d-a e)}\right ) (-10 a B e+3 A b e+7 b B d)}{10 e (b d-a e)}-\frac {(a+b x)^7 (B d-A e)}{10 e (d+e x)^{10} (b d-a e)}\) |
-1/10*((B*d - A*e)*(a + b*x)^7)/(e*(b*d - a*e)*(d + e*x)^10) + ((7*b*B*d + 3*A*b*e - 10*a*B*e)*((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*e*(b*d - a*e))
3.11.70.3.1 Defintions of rubi rules used
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]
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])
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]))))
Leaf count of result is larger than twice the leaf count of optimal. \(788\) vs. \(2(177)=354\).
Time = 0.75 (sec) , antiderivative size = 789, normalized size of antiderivative = 4.26
| method | result | size |
| risch | \(\frac {-\frac {b^{6} B \,x^{7}}{3 e}-\frac {b^{5} \left (3 A b e +18 B a e +7 B b d \right ) x^{6}}{12 e^{2}}-\frac {b^{4} \left (12 A a b \,e^{2}+3 A \,b^{2} d e +30 B \,a^{2} e^{2}+18 B a b d e +7 b^{2} B \,d^{2}\right ) x^{5}}{10 e^{3}}-\frac {b^{3} \left (30 A \,a^{2} b \,e^{3}+12 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +40 B \,a^{3} e^{3}+30 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e +7 b^{3} B \,d^{3}\right ) x^{4}}{12 e^{4}}-\frac {b^{2} \left (60 A \,a^{3} b \,e^{4}+30 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}+3 A \,b^{4} d^{3} e +45 B \,a^{4} e^{4}+40 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}+18 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) x^{3}}{21 e^{5}}-\frac {b \left (105 A \,a^{4} b \,e^{5}+60 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}+12 A a \,b^{4} d^{3} e^{2}+3 A \,b^{5} d^{4} e +42 B \,a^{5} e^{5}+45 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}+30 B \,a^{2} b^{3} d^{3} e^{2}+18 B a \,b^{4} d^{4} e +7 B \,b^{5} d^{5}\right ) x^{2}}{56 e^{6}}-\frac {\left (168 A \,a^{5} b \,e^{6}+105 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}+30 A \,a^{2} b^{4} d^{3} e^{3}+12 A a \,b^{5} d^{4} e^{2}+3 A \,b^{6} d^{5} e +28 B \,a^{6} e^{6}+42 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}+40 B \,a^{3} b^{3} d^{3} e^{3}+30 B \,a^{2} b^{4} d^{4} e^{2}+18 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}\right ) x}{252 e^{7}}-\frac {252 A \,a^{6} e^{7}+168 A \,a^{5} b d \,e^{6}+105 A \,a^{4} b^{2} d^{2} e^{5}+60 A \,a^{3} b^{3} d^{3} e^{4}+30 A \,a^{2} b^{4} d^{4} e^{3}+12 A a \,b^{5} d^{5} e^{2}+3 A \,b^{6} d^{6} e +28 B \,a^{6} d \,e^{6}+42 B \,a^{5} b \,d^{2} e^{5}+45 B \,a^{4} b^{2} d^{3} e^{4}+40 B \,a^{3} b^{3} d^{4} e^{3}+30 B \,a^{2} b^{4} d^{5} e^{2}+18 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{2520 e^{8}}}{\left (e x +d \right )^{10}}\) | \(789\) |
| default | \(-\frac {A \,a^{6} e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{10 e^{8} \left (e x +d \right )^{10}}-\frac {3 b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{8 e^{8} \left (e x +d \right )^{8}}-\frac {5 b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{7 e^{8} \left (e x +d \right )^{7}}-\frac {3 b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right )}{5 e^{8} \left (e x +d \right )^{5}}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{9 e^{8} \left (e x +d \right )^{9}}-\frac {b^{6} B}{3 e^{8} \left (e x +d \right )^{3}}-\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right )}{6 e^{8} \left (e x +d \right )^{6}}-\frac {b^{5} \left (A b e +6 B a e -7 B b d \right )}{4 e^{8} \left (e x +d \right )^{4}}\) | \(814\) |
| norman | \(\frac {-\frac {b^{6} B \,x^{7}}{3 e}-\frac {\left (3 A \,b^{6} e^{3}+18 B a \,b^{5} e^{3}+7 b^{6} B d \,e^{2}\right ) x^{6}}{12 e^{4}}-\frac {\left (12 A a \,b^{5} e^{4}+3 A \,b^{6} d \,e^{3}+30 B \,a^{2} b^{4} e^{4}+18 B a \,b^{5} d \,e^{3}+7 b^{6} B \,d^{2} e^{2}\right ) x^{5}}{10 e^{5}}-\frac {\left (30 A \,a^{2} b^{4} e^{5}+12 A a \,b^{5} d \,e^{4}+3 A \,b^{6} d^{2} e^{3}+40 B \,a^{3} b^{3} e^{5}+30 B \,a^{2} b^{4} d \,e^{4}+18 B a \,b^{5} d^{2} e^{3}+7 b^{6} B \,d^{3} e^{2}\right ) x^{4}}{12 e^{6}}-\frac {\left (60 A \,a^{3} b^{3} e^{6}+30 A \,a^{2} b^{4} d \,e^{5}+12 A a \,b^{5} d^{2} e^{4}+3 A \,b^{6} d^{3} e^{3}+45 B \,a^{4} b^{2} e^{6}+40 B \,a^{3} b^{3} d \,e^{5}+30 B \,a^{2} b^{4} d^{2} e^{4}+18 B a \,b^{5} d^{3} e^{3}+7 b^{6} B \,d^{4} e^{2}\right ) x^{3}}{21 e^{7}}-\frac {\left (105 A \,a^{4} b^{2} e^{7}+60 A \,a^{3} b^{3} d \,e^{6}+30 A \,a^{2} b^{4} d^{2} e^{5}+12 A a \,b^{5} d^{3} e^{4}+3 A \,b^{6} d^{4} e^{3}+42 B \,a^{5} b \,e^{7}+45 B \,a^{4} b^{2} d \,e^{6}+40 B \,a^{3} b^{3} d^{2} e^{5}+30 B \,a^{2} b^{4} d^{3} e^{4}+18 B a \,b^{5} d^{4} e^{3}+7 b^{6} B \,d^{5} e^{2}\right ) x^{2}}{56 e^{8}}-\frac {\left (168 A \,a^{5} b \,e^{8}+105 A \,a^{4} b^{2} d \,e^{7}+60 A \,a^{3} b^{3} d^{2} e^{6}+30 A \,a^{2} b^{4} d^{3} e^{5}+12 A a \,b^{5} d^{4} e^{4}+3 A \,b^{6} d^{5} e^{3}+28 B \,a^{6} e^{8}+42 B \,a^{5} b d \,e^{7}+45 B \,a^{4} b^{2} d^{2} e^{6}+40 B \,a^{3} b^{3} d^{3} e^{5}+30 B \,a^{2} b^{4} d^{4} e^{4}+18 B a \,b^{5} d^{5} e^{3}+7 b^{6} B \,d^{6} e^{2}\right ) x}{252 e^{9}}-\frac {252 A \,a^{6} e^{9}+168 A \,a^{5} b d \,e^{8}+105 A \,a^{4} b^{2} d^{2} e^{7}+60 A \,a^{3} b^{3} d^{3} e^{6}+30 A \,a^{2} b^{4} d^{4} e^{5}+12 A a \,b^{5} d^{5} e^{4}+3 A \,b^{6} d^{6} e^{3}+28 B \,a^{6} d \,e^{8}+42 B \,a^{5} b \,d^{2} e^{7}+45 B \,a^{4} b^{2} d^{3} e^{6}+40 B \,a^{3} b^{3} d^{4} e^{5}+30 B \,a^{2} b^{4} d^{5} e^{4}+18 B a \,b^{5} d^{6} e^{3}+7 B \,b^{6} d^{7} e^{2}}{2520 e^{10}}}{\left (e x +d \right )^{10}}\) | \(858\) |
| gosper | \(-\frac {840 B \,x^{7} b^{6} e^{7}+630 A \,x^{6} b^{6} e^{7}+3780 B \,x^{6} a \,b^{5} e^{7}+1470 B \,x^{6} b^{6} d \,e^{6}+3024 A \,x^{5} a \,b^{5} e^{7}+756 A \,x^{5} b^{6} d \,e^{6}+7560 B \,x^{5} a^{2} b^{4} e^{7}+4536 B \,x^{5} a \,b^{5} d \,e^{6}+1764 B \,x^{5} b^{6} d^{2} e^{5}+6300 A \,x^{4} a^{2} b^{4} e^{7}+2520 A \,x^{4} a \,b^{5} d \,e^{6}+630 A \,x^{4} b^{6} d^{2} e^{5}+8400 B \,x^{4} a^{3} b^{3} e^{7}+6300 B \,x^{4} a^{2} b^{4} d \,e^{6}+3780 B \,x^{4} a \,b^{5} d^{2} e^{5}+1470 B \,x^{4} b^{6} d^{3} e^{4}+7200 A \,x^{3} a^{3} b^{3} e^{7}+3600 A \,x^{3} a^{2} b^{4} d \,e^{6}+1440 A \,x^{3} a \,b^{5} d^{2} e^{5}+360 A \,x^{3} b^{6} d^{3} e^{4}+5400 B \,x^{3} a^{4} b^{2} e^{7}+4800 B \,x^{3} a^{3} b^{3} d \,e^{6}+3600 B \,x^{3} a^{2} b^{4} d^{2} e^{5}+2160 B \,x^{3} a \,b^{5} d^{3} e^{4}+840 B \,x^{3} b^{6} d^{4} e^{3}+4725 A \,x^{2} a^{4} b^{2} e^{7}+2700 A \,x^{2} a^{3} b^{3} d \,e^{6}+1350 A \,x^{2} a^{2} b^{4} d^{2} e^{5}+540 A \,x^{2} a \,b^{5} d^{3} e^{4}+135 A \,x^{2} b^{6} d^{4} e^{3}+1890 B \,x^{2} a^{5} b \,e^{7}+2025 B \,x^{2} a^{4} b^{2} d \,e^{6}+1800 B \,x^{2} a^{3} b^{3} d^{2} e^{5}+1350 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+810 B \,x^{2} a \,b^{5} d^{4} e^{3}+315 B \,x^{2} b^{6} d^{5} e^{2}+1680 A x \,a^{5} b \,e^{7}+1050 A x \,a^{4} b^{2} d \,e^{6}+600 A x \,a^{3} b^{3} d^{2} e^{5}+300 A x \,a^{2} b^{4} d^{3} e^{4}+120 A x a \,b^{5} d^{4} e^{3}+30 A x \,b^{6} d^{5} e^{2}+280 B x \,a^{6} e^{7}+420 B x \,a^{5} b d \,e^{6}+450 B x \,a^{4} b^{2} d^{2} e^{5}+400 B x \,a^{3} b^{3} d^{3} e^{4}+300 B x \,a^{2} b^{4} d^{4} e^{3}+180 B x a \,b^{5} d^{5} e^{2}+70 B x \,b^{6} d^{6} e +252 A \,a^{6} e^{7}+168 A \,a^{5} b d \,e^{6}+105 A \,a^{4} b^{2} d^{2} e^{5}+60 A \,a^{3} b^{3} d^{3} e^{4}+30 A \,a^{2} b^{4} d^{4} e^{3}+12 A a \,b^{5} d^{5} e^{2}+3 A \,b^{6} d^{6} e +28 B \,a^{6} d \,e^{6}+42 B \,a^{5} b \,d^{2} e^{5}+45 B \,a^{4} b^{2} d^{3} e^{4}+40 B \,a^{3} b^{3} d^{4} e^{3}+30 B \,a^{2} b^{4} d^{5} e^{2}+18 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{2520 e^{8} \left (e x +d \right )^{10}}\) | \(913\) |
| parallelrisch | \(-\frac {840 B \,b^{6} x^{7} e^{9}+630 A \,b^{6} e^{9} x^{6}+3780 B a \,b^{5} e^{9} x^{6}+1470 B \,b^{6} d \,e^{8} x^{6}+3024 A a \,b^{5} e^{9} x^{5}+756 A \,b^{6} d \,e^{8} x^{5}+7560 B \,a^{2} b^{4} e^{9} x^{5}+4536 B a \,b^{5} d \,e^{8} x^{5}+1764 B \,b^{6} d^{2} e^{7} x^{5}+6300 A \,a^{2} b^{4} e^{9} x^{4}+2520 A a \,b^{5} d \,e^{8} x^{4}+630 A \,b^{6} d^{2} e^{7} x^{4}+8400 B \,a^{3} b^{3} e^{9} x^{4}+6300 B \,a^{2} b^{4} d \,e^{8} x^{4}+3780 B a \,b^{5} d^{2} e^{7} x^{4}+1470 B \,b^{6} d^{3} e^{6} x^{4}+7200 A \,a^{3} b^{3} e^{9} x^{3}+3600 A \,a^{2} b^{4} d \,e^{8} x^{3}+1440 A a \,b^{5} d^{2} e^{7} x^{3}+360 A \,b^{6} d^{3} e^{6} x^{3}+5400 B \,a^{4} b^{2} e^{9} x^{3}+4800 B \,a^{3} b^{3} d \,e^{8} x^{3}+3600 B \,a^{2} b^{4} d^{2} e^{7} x^{3}+2160 B a \,b^{5} d^{3} e^{6} x^{3}+840 B \,b^{6} d^{4} e^{5} x^{3}+4725 A \,a^{4} b^{2} e^{9} x^{2}+2700 A \,a^{3} b^{3} d \,e^{8} x^{2}+1350 A \,a^{2} b^{4} d^{2} e^{7} x^{2}+540 A a \,b^{5} d^{3} e^{6} x^{2}+135 A \,b^{6} d^{4} e^{5} x^{2}+1890 B \,a^{5} b \,e^{9} x^{2}+2025 B \,a^{4} b^{2} d \,e^{8} x^{2}+1800 B \,a^{3} b^{3} d^{2} e^{7} x^{2}+1350 B \,a^{2} b^{4} d^{3} e^{6} x^{2}+810 B a \,b^{5} d^{4} e^{5} x^{2}+315 B \,b^{6} d^{5} e^{4} x^{2}+1680 A \,a^{5} b \,e^{9} x +1050 A \,a^{4} b^{2} d \,e^{8} x +600 A \,a^{3} b^{3} d^{2} e^{7} x +300 A \,a^{2} b^{4} d^{3} e^{6} x +120 A a \,b^{5} d^{4} e^{5} x +30 A \,b^{6} d^{5} e^{4} x +280 B \,a^{6} e^{9} x +420 B \,a^{5} b d \,e^{8} x +450 B \,a^{4} b^{2} d^{2} e^{7} x +400 B \,a^{3} b^{3} d^{3} e^{6} x +300 B \,a^{2} b^{4} d^{4} e^{5} x +180 B a \,b^{5} d^{5} e^{4} x +70 B \,b^{6} d^{6} e^{3} x +252 A \,a^{6} e^{9}+168 A \,a^{5} b d \,e^{8}+105 A \,a^{4} b^{2} d^{2} e^{7}+60 A \,a^{3} b^{3} d^{3} e^{6}+30 A \,a^{2} b^{4} d^{4} e^{5}+12 A a \,b^{5} d^{5} e^{4}+3 A \,b^{6} d^{6} e^{3}+28 B \,a^{6} d \,e^{8}+42 B \,a^{5} b \,d^{2} e^{7}+45 B \,a^{4} b^{2} d^{3} e^{6}+40 B \,a^{3} b^{3} d^{4} e^{5}+30 B \,a^{2} b^{4} d^{5} e^{4}+18 B a \,b^{5} d^{6} e^{3}+7 B \,b^{6} d^{7} e^{2}}{2520 e^{10} \left (e x +d \right )^{10}}\) | \(922\) |
(-1/3*b^6*B/e*x^7-1/12*b^5/e^2*(3*A*b*e+18*B*a*e+7*B*b*d)*x^6-1/10*b^4/e^3 *(12*A*a*b*e^2+3*A*b^2*d*e+30*B*a^2*e^2+18*B*a*b*d*e+7*B*b^2*d^2)*x^5-1/12 *b^3/e^4*(30*A*a^2*b*e^3+12*A*a*b^2*d*e^2+3*A*b^3*d^2*e+40*B*a^3*e^3+30*B* a^2*b*d*e^2+18*B*a*b^2*d^2*e+7*B*b^3*d^3)*x^4-1/21*b^2/e^5*(60*A*a^3*b*e^4 +30*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2+3*A*b^4*d^3*e+45*B*a^4*e^4+40*B*a^3 *b*d*e^3+30*B*a^2*b^2*d^2*e^2+18*B*a*b^3*d^3*e+7*B*b^4*d^4)*x^3-1/56*b/e^6 *(105*A*a^4*b*e^5+60*A*a^3*b^2*d*e^4+30*A*a^2*b^3*d^2*e^3+12*A*a*b^4*d^3*e ^2+3*A*b^5*d^4*e+42*B*a^5*e^5+45*B*a^4*b*d*e^4+40*B*a^3*b^2*d^2*e^3+30*B*a ^2*b^3*d^3*e^2+18*B*a*b^4*d^4*e+7*B*b^5*d^5)*x^2-1/252/e^7*(168*A*a^5*b*e^ 6+105*A*a^4*b^2*d*e^5+60*A*a^3*b^3*d^2*e^4+30*A*a^2*b^4*d^3*e^3+12*A*a*b^5 *d^4*e^2+3*A*b^6*d^5*e+28*B*a^6*e^6+42*B*a^5*b*d*e^5+45*B*a^4*b^2*d^2*e^4+ 40*B*a^3*b^3*d^3*e^3+30*B*a^2*b^4*d^4*e^2+18*B*a*b^5*d^5*e+7*B*b^6*d^6)*x- 1/2520/e^8*(252*A*a^6*e^7+168*A*a^5*b*d*e^6+105*A*a^4*b^2*d^2*e^5+60*A*a^3 *b^3*d^3*e^4+30*A*a^2*b^4*d^4*e^3+12*A*a*b^5*d^5*e^2+3*A*b^6*d^6*e+28*B*a^ 6*d*e^6+42*B*a^5*b*d^2*e^5+45*B*a^4*b^2*d^3*e^4+40*B*a^3*b^3*d^4*e^3+30*B* a^2*b^4*d^5*e^2+18*B*a*b^5*d^6*e+7*B*b^6*d^7))/(e*x+d)^10
Leaf count of result is larger than twice the leaf count of optimal. 872 vs. \(2 (177) = 354\).
Time = 0.24 (sec) , antiderivative size = 872, normalized size of antiderivative = 4.71 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{11}} \, dx=-\frac {840 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 252 \, A a^{6} e^{7} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 21 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 28 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 210 \, {\left (7 \, B b^{6} d e^{6} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 252 \, {\left (7 \, B b^{6} d^{2} e^{5} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 210 \, {\left (7 \, B b^{6} d^{3} e^{4} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 120 \, {\left (7 \, B b^{6} d^{4} e^{3} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 45 \, {\left (7 \, B b^{6} d^{5} e^{2} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 21 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 10 \, {\left (7 \, B b^{6} d^{6} e + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 21 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 28 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{2520 \, {\left (e^{18} x^{10} + 10 \, d e^{17} x^{9} + 45 \, d^{2} e^{16} x^{8} + 120 \, d^{3} e^{15} x^{7} + 210 \, d^{4} e^{14} x^{6} + 252 \, d^{5} e^{13} x^{5} + 210 \, d^{6} e^{12} x^{4} + 120 \, d^{7} e^{11} x^{3} + 45 \, d^{8} e^{10} x^{2} + 10 \, d^{9} e^{9} x + d^{10} e^{8}\right )}} \]
-1/2520*(840*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 252*A*a^6*e^7 + 3*(6*B*a*b^5 + A*b^6)*d^6*e + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 10*(4*B*a^3*b^3 + 3*A *a^2*b^4)*d^4*e^3 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 21*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + 28*(B*a^6 + 6*A*a^5*b)*d*e^6 + 210*(7*B*b^6*d*e^ 6 + 3*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 252*(7*B*b^6*d^2*e^5 + 3*(6*B*a*b^5 + A*b^6)*d*e^6 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 210*(7*B*b^6*d^3*e^ 4 + 3*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 10 *(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 120*(7*B*b^6*d^4*e^3 + 3*(6*B*a*b^ 5 + A*b^6)*d^3*e^4 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 45*(7*B* b^6*d^5*e^2 + 3*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)* d^3*e^4 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 15*(3*B*a^4*b^2 + 4*A*a ^3*b^3)*d*e^6 + 21*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 10*(7*B*b^6*d^6*e + 3*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 10 *(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2* e^5 + 21*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 28*(B*a^6 + 6*A*a^5*b)*e^7)*x)/ (e^18*x^10 + 10*d*e^17*x^9 + 45*d^2*e^16*x^8 + 120*d^3*e^15*x^7 + 210*d^4* e^14*x^6 + 252*d^5*e^13*x^5 + 210*d^6*e^12*x^4 + 120*d^7*e^11*x^3 + 45*d^8 *e^10*x^2 + 10*d^9*e^9*x + d^10*e^8)
Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{11}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 872 vs. \(2 (177) = 354\).
Time = 0.24 (sec) , antiderivative size = 872, normalized size of antiderivative = 4.71 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{11}} \, dx=-\frac {840 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 252 \, A a^{6} e^{7} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 21 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 28 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 210 \, {\left (7 \, B b^{6} d e^{6} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 252 \, {\left (7 \, B b^{6} d^{2} e^{5} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 210 \, {\left (7 \, B b^{6} d^{3} e^{4} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 120 \, {\left (7 \, B b^{6} d^{4} e^{3} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 45 \, {\left (7 \, B b^{6} d^{5} e^{2} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 21 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 10 \, {\left (7 \, B b^{6} d^{6} e + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 21 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 28 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{2520 \, {\left (e^{18} x^{10} + 10 \, d e^{17} x^{9} + 45 \, d^{2} e^{16} x^{8} + 120 \, d^{3} e^{15} x^{7} + 210 \, d^{4} e^{14} x^{6} + 252 \, d^{5} e^{13} x^{5} + 210 \, d^{6} e^{12} x^{4} + 120 \, d^{7} e^{11} x^{3} + 45 \, d^{8} e^{10} x^{2} + 10 \, d^{9} e^{9} x + d^{10} e^{8}\right )}} \]
-1/2520*(840*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 252*A*a^6*e^7 + 3*(6*B*a*b^5 + A*b^6)*d^6*e + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 10*(4*B*a^3*b^3 + 3*A *a^2*b^4)*d^4*e^3 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 21*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + 28*(B*a^6 + 6*A*a^5*b)*d*e^6 + 210*(7*B*b^6*d*e^ 6 + 3*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 252*(7*B*b^6*d^2*e^5 + 3*(6*B*a*b^5 + A*b^6)*d*e^6 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 210*(7*B*b^6*d^3*e^ 4 + 3*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 10 *(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 120*(7*B*b^6*d^4*e^3 + 3*(6*B*a*b^ 5 + A*b^6)*d^3*e^4 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 45*(7*B* b^6*d^5*e^2 + 3*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)* d^3*e^4 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 15*(3*B*a^4*b^2 + 4*A*a ^3*b^3)*d*e^6 + 21*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 10*(7*B*b^6*d^6*e + 3*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 10 *(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2* e^5 + 21*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 28*(B*a^6 + 6*A*a^5*b)*e^7)*x)/ (e^18*x^10 + 10*d*e^17*x^9 + 45*d^2*e^16*x^8 + 120*d^3*e^15*x^7 + 210*d^4* e^14*x^6 + 252*d^5*e^13*x^5 + 210*d^6*e^12*x^4 + 120*d^7*e^11*x^3 + 45*d^8 *e^10*x^2 + 10*d^9*e^9*x + d^10*e^8)
Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (177) = 354\).
Time = 0.31 (sec) , antiderivative size = 912, normalized size of antiderivative = 4.93 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{11}} \, dx=-\frac {840 \, B b^{6} e^{7} x^{7} + 1470 \, B b^{6} d e^{6} x^{6} + 3780 \, B a b^{5} e^{7} x^{6} + 630 \, A b^{6} e^{7} x^{6} + 1764 \, B b^{6} d^{2} e^{5} x^{5} + 4536 \, B a b^{5} d e^{6} x^{5} + 756 \, A b^{6} d e^{6} x^{5} + 7560 \, B a^{2} b^{4} e^{7} x^{5} + 3024 \, A a b^{5} e^{7} x^{5} + 1470 \, B b^{6} d^{3} e^{4} x^{4} + 3780 \, B a b^{5} d^{2} e^{5} x^{4} + 630 \, A b^{6} d^{2} e^{5} x^{4} + 6300 \, B a^{2} b^{4} d e^{6} x^{4} + 2520 \, A a b^{5} d e^{6} x^{4} + 8400 \, B a^{3} b^{3} e^{7} x^{4} + 6300 \, A a^{2} b^{4} e^{7} x^{4} + 840 \, B b^{6} d^{4} e^{3} x^{3} + 2160 \, B a b^{5} d^{3} e^{4} x^{3} + 360 \, A b^{6} d^{3} e^{4} x^{3} + 3600 \, B a^{2} b^{4} d^{2} e^{5} x^{3} + 1440 \, A a b^{5} d^{2} e^{5} x^{3} + 4800 \, B a^{3} b^{3} d e^{6} x^{3} + 3600 \, A a^{2} b^{4} d e^{6} x^{3} + 5400 \, B a^{4} b^{2} e^{7} x^{3} + 7200 \, A a^{3} b^{3} e^{7} x^{3} + 315 \, B b^{6} d^{5} e^{2} x^{2} + 810 \, B a b^{5} d^{4} e^{3} x^{2} + 135 \, A b^{6} d^{4} e^{3} x^{2} + 1350 \, B a^{2} b^{4} d^{3} e^{4} x^{2} + 540 \, A a b^{5} d^{3} e^{4} x^{2} + 1800 \, B a^{3} b^{3} d^{2} e^{5} x^{2} + 1350 \, A a^{2} b^{4} d^{2} e^{5} x^{2} + 2025 \, B a^{4} b^{2} d e^{6} x^{2} + 2700 \, A a^{3} b^{3} d e^{6} x^{2} + 1890 \, B a^{5} b e^{7} x^{2} + 4725 \, A a^{4} b^{2} e^{7} x^{2} + 70 \, B b^{6} d^{6} e x + 180 \, B a b^{5} d^{5} e^{2} x + 30 \, A b^{6} d^{5} e^{2} x + 300 \, B a^{2} b^{4} d^{4} e^{3} x + 120 \, A a b^{5} d^{4} e^{3} x + 400 \, B a^{3} b^{3} d^{3} e^{4} x + 300 \, A a^{2} b^{4} d^{3} e^{4} x + 450 \, B a^{4} b^{2} d^{2} e^{5} x + 600 \, A a^{3} b^{3} d^{2} e^{5} x + 420 \, B a^{5} b d e^{6} x + 1050 \, A a^{4} b^{2} d e^{6} x + 280 \, B a^{6} e^{7} x + 1680 \, A a^{5} b e^{7} x + 7 \, B b^{6} d^{7} + 18 \, B a b^{5} d^{6} e + 3 \, A b^{6} d^{6} e + 30 \, B a^{2} b^{4} d^{5} e^{2} + 12 \, A a b^{5} d^{5} e^{2} + 40 \, B a^{3} b^{3} d^{4} e^{3} + 30 \, A a^{2} b^{4} d^{4} e^{3} + 45 \, B a^{4} b^{2} d^{3} e^{4} + 60 \, A a^{3} b^{3} d^{3} e^{4} + 42 \, B a^{5} b d^{2} e^{5} + 105 \, A a^{4} b^{2} d^{2} e^{5} + 28 \, B a^{6} d e^{6} + 168 \, A a^{5} b d e^{6} + 252 \, A a^{6} e^{7}}{2520 \, {\left (e x + d\right )}^{10} e^{8}} \]
-1/2520*(840*B*b^6*e^7*x^7 + 1470*B*b^6*d*e^6*x^6 + 3780*B*a*b^5*e^7*x^6 + 630*A*b^6*e^7*x^6 + 1764*B*b^6*d^2*e^5*x^5 + 4536*B*a*b^5*d*e^6*x^5 + 756 *A*b^6*d*e^6*x^5 + 7560*B*a^2*b^4*e^7*x^5 + 3024*A*a*b^5*e^7*x^5 + 1470*B* b^6*d^3*e^4*x^4 + 3780*B*a*b^5*d^2*e^5*x^4 + 630*A*b^6*d^2*e^5*x^4 + 6300* B*a^2*b^4*d*e^6*x^4 + 2520*A*a*b^5*d*e^6*x^4 + 8400*B*a^3*b^3*e^7*x^4 + 63 00*A*a^2*b^4*e^7*x^4 + 840*B*b^6*d^4*e^3*x^3 + 2160*B*a*b^5*d^3*e^4*x^3 + 360*A*b^6*d^3*e^4*x^3 + 3600*B*a^2*b^4*d^2*e^5*x^3 + 1440*A*a*b^5*d^2*e^5* x^3 + 4800*B*a^3*b^3*d*e^6*x^3 + 3600*A*a^2*b^4*d*e^6*x^3 + 5400*B*a^4*b^2 *e^7*x^3 + 7200*A*a^3*b^3*e^7*x^3 + 315*B*b^6*d^5*e^2*x^2 + 810*B*a*b^5*d^ 4*e^3*x^2 + 135*A*b^6*d^4*e^3*x^2 + 1350*B*a^2*b^4*d^3*e^4*x^2 + 540*A*a*b ^5*d^3*e^4*x^2 + 1800*B*a^3*b^3*d^2*e^5*x^2 + 1350*A*a^2*b^4*d^2*e^5*x^2 + 2025*B*a^4*b^2*d*e^6*x^2 + 2700*A*a^3*b^3*d*e^6*x^2 + 1890*B*a^5*b*e^7*x^ 2 + 4725*A*a^4*b^2*e^7*x^2 + 70*B*b^6*d^6*e*x + 180*B*a*b^5*d^5*e^2*x + 30 *A*b^6*d^5*e^2*x + 300*B*a^2*b^4*d^4*e^3*x + 120*A*a*b^5*d^4*e^3*x + 400*B *a^3*b^3*d^3*e^4*x + 300*A*a^2*b^4*d^3*e^4*x + 450*B*a^4*b^2*d^2*e^5*x + 6 00*A*a^3*b^3*d^2*e^5*x + 420*B*a^5*b*d*e^6*x + 1050*A*a^4*b^2*d*e^6*x + 28 0*B*a^6*e^7*x + 1680*A*a^5*b*e^7*x + 7*B*b^6*d^7 + 18*B*a*b^5*d^6*e + 3*A* b^6*d^6*e + 30*B*a^2*b^4*d^5*e^2 + 12*A*a*b^5*d^5*e^2 + 40*B*a^3*b^3*d^4*e ^3 + 30*A*a^2*b^4*d^4*e^3 + 45*B*a^4*b^2*d^3*e^4 + 60*A*a^3*b^3*d^3*e^4 + 42*B*a^5*b*d^2*e^5 + 105*A*a^4*b^2*d^2*e^5 + 28*B*a^6*d*e^6 + 168*A*a^5...
Time = 1.80 (sec) , antiderivative size = 888, normalized size of antiderivative = 4.80 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{11}} \, dx=-\frac {\frac {28\,B\,a^6\,d\,e^6+252\,A\,a^6\,e^7+42\,B\,a^5\,b\,d^2\,e^5+168\,A\,a^5\,b\,d\,e^6+45\,B\,a^4\,b^2\,d^3\,e^4+105\,A\,a^4\,b^2\,d^2\,e^5+40\,B\,a^3\,b^3\,d^4\,e^3+60\,A\,a^3\,b^3\,d^3\,e^4+30\,B\,a^2\,b^4\,d^5\,e^2+30\,A\,a^2\,b^4\,d^4\,e^3+18\,B\,a\,b^5\,d^6\,e+12\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+3\,A\,b^6\,d^6\,e}{2520\,e^8}+\frac {x\,\left (28\,B\,a^6\,e^6+42\,B\,a^5\,b\,d\,e^5+168\,A\,a^5\,b\,e^6+45\,B\,a^4\,b^2\,d^2\,e^4+105\,A\,a^4\,b^2\,d\,e^5+40\,B\,a^3\,b^3\,d^3\,e^3+60\,A\,a^3\,b^3\,d^2\,e^4+30\,B\,a^2\,b^4\,d^4\,e^2+30\,A\,a^2\,b^4\,d^3\,e^3+18\,B\,a\,b^5\,d^5\,e+12\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+3\,A\,b^6\,d^5\,e\right )}{252\,e^7}+\frac {b^3\,x^4\,\left (40\,B\,a^3\,e^3+30\,B\,a^2\,b\,d\,e^2+30\,A\,a^2\,b\,e^3+18\,B\,a\,b^2\,d^2\,e+12\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+3\,A\,b^3\,d^2\,e\right )}{12\,e^4}+\frac {b^5\,x^6\,\left (3\,A\,b\,e+18\,B\,a\,e+7\,B\,b\,d\right )}{12\,e^2}+\frac {b\,x^2\,\left (42\,B\,a^5\,e^5+45\,B\,a^4\,b\,d\,e^4+105\,A\,a^4\,b\,e^5+40\,B\,a^3\,b^2\,d^2\,e^3+60\,A\,a^3\,b^2\,d\,e^4+30\,B\,a^2\,b^3\,d^3\,e^2+30\,A\,a^2\,b^3\,d^2\,e^3+18\,B\,a\,b^4\,d^4\,e+12\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+3\,A\,b^5\,d^4\,e\right )}{56\,e^6}+\frac {b^2\,x^3\,\left (45\,B\,a^4\,e^4+40\,B\,a^3\,b\,d\,e^3+60\,A\,a^3\,b\,e^4+30\,B\,a^2\,b^2\,d^2\,e^2+30\,A\,a^2\,b^2\,d\,e^3+18\,B\,a\,b^3\,d^3\,e+12\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+3\,A\,b^4\,d^3\,e\right )}{21\,e^5}+\frac {b^4\,x^5\,\left (30\,B\,a^2\,e^2+18\,B\,a\,b\,d\,e+12\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+3\,A\,b^2\,d\,e\right )}{10\,e^3}+\frac {B\,b^6\,x^7}{3\,e}}{d^{10}+10\,d^9\,e\,x+45\,d^8\,e^2\,x^2+120\,d^7\,e^3\,x^3+210\,d^6\,e^4\,x^4+252\,d^5\,e^5\,x^5+210\,d^4\,e^6\,x^6+120\,d^3\,e^7\,x^7+45\,d^2\,e^8\,x^8+10\,d\,e^9\,x^9+e^{10}\,x^{10}} \]
-((252*A*a^6*e^7 + 7*B*b^6*d^7 + 3*A*b^6*d^6*e + 28*B*a^6*d*e^6 + 12*A*a*b ^5*d^5*e^2 + 42*B*a^5*b*d^2*e^5 + 30*A*a^2*b^4*d^4*e^3 + 60*A*a^3*b^3*d^3* e^4 + 105*A*a^4*b^2*d^2*e^5 + 30*B*a^2*b^4*d^5*e^2 + 40*B*a^3*b^3*d^4*e^3 + 45*B*a^4*b^2*d^3*e^4 + 168*A*a^5*b*d*e^6 + 18*B*a*b^5*d^6*e)/(2520*e^8) + (x*(28*B*a^6*e^6 + 7*B*b^6*d^6 + 168*A*a^5*b*e^6 + 3*A*b^6*d^5*e + 12*A* a*b^5*d^4*e^2 + 105*A*a^4*b^2*d*e^5 + 30*A*a^2*b^4*d^3*e^3 + 60*A*a^3*b^3* d^2*e^4 + 30*B*a^2*b^4*d^4*e^2 + 40*B*a^3*b^3*d^3*e^3 + 45*B*a^4*b^2*d^2*e ^4 + 18*B*a*b^5*d^5*e + 42*B*a^5*b*d*e^5))/(252*e^7) + (b^3*x^4*(40*B*a^3* e^3 + 7*B*b^3*d^3 + 30*A*a^2*b*e^3 + 3*A*b^3*d^2*e + 12*A*a*b^2*d*e^2 + 18 *B*a*b^2*d^2*e + 30*B*a^2*b*d*e^2))/(12*e^4) + (b^5*x^6*(3*A*b*e + 18*B*a* e + 7*B*b*d))/(12*e^2) + (b*x^2*(42*B*a^5*e^5 + 7*B*b^5*d^5 + 105*A*a^4*b* e^5 + 3*A*b^5*d^4*e + 12*A*a*b^4*d^3*e^2 + 60*A*a^3*b^2*d*e^4 + 30*A*a^2*b ^3*d^2*e^3 + 30*B*a^2*b^3*d^3*e^2 + 40*B*a^3*b^2*d^2*e^3 + 18*B*a*b^4*d^4* e + 45*B*a^4*b*d*e^4))/(56*e^6) + (b^2*x^3*(45*B*a^4*e^4 + 7*B*b^4*d^4 + 6 0*A*a^3*b*e^4 + 3*A*b^4*d^3*e + 12*A*a*b^3*d^2*e^2 + 30*A*a^2*b^2*d*e^3 + 30*B*a^2*b^2*d^2*e^2 + 18*B*a*b^3*d^3*e + 40*B*a^3*b*d*e^3))/(21*e^5) + (b ^4*x^5*(30*B*a^2*e^2 + 7*B*b^2*d^2 + 12*A*a*b*e^2 + 3*A*b^2*d*e + 18*B*a*b *d*e))/(10*e^3) + (B*b^6*x^7)/(3*e))/(d^10 + e^10*x^10 + 10*d*e^9*x^9 + 45 *d^8*e^2*x^2 + 120*d^7*e^3*x^3 + 210*d^6*e^4*x^4 + 252*d^5*e^5*x^5 + 210*d ^4*e^6*x^6 + 120*d^3*e^7*x^7 + 45*d^2*e^8*x^8 + 10*d^9*e*x)