Integrand size = 16, antiderivative size = 240 \[ \int x^8 (a+b x)^{10} (A+B x) \, dx=\frac {a^8 (A b-a B) (a+b x)^{11}}{11 b^{10}}-\frac {a^7 (8 A b-9 a B) (a+b x)^{12}}{12 b^{10}}+\frac {4 a^6 (7 A b-9 a B) (a+b x)^{13}}{13 b^{10}}-\frac {2 a^5 (2 A b-3 a B) (a+b x)^{14}}{b^{10}}+\frac {14 a^4 (5 A b-9 a B) (a+b x)^{15}}{15 b^{10}}-\frac {7 a^3 (4 A b-9 a B) (a+b x)^{16}}{8 b^{10}}+\frac {28 a^2 (A b-3 a B) (a+b x)^{17}}{17 b^{10}}-\frac {2 a (2 A b-9 a B) (a+b x)^{18}}{9 b^{10}}+\frac {(A b-9 a B) (a+b x)^{19}}{19 b^{10}}+\frac {B (a+b x)^{20}}{20 b^{10}} \]
1/11*a^8*(A*b-B*a)*(b*x+a)^11/b^10-1/12*a^7*(8*A*b-9*B*a)*(b*x+a)^12/b^10+ 4/13*a^6*(7*A*b-9*B*a)*(b*x+a)^13/b^10-2*a^5*(2*A*b-3*B*a)*(b*x+a)^14/b^10 +14/15*a^4*(5*A*b-9*B*a)*(b*x+a)^15/b^10-7/8*a^3*(4*A*b-9*B*a)*(b*x+a)^16/ b^10+28/17*a^2*(A*b-3*B*a)*(b*x+a)^17/b^10-2/9*a*(2*A*b-9*B*a)*(b*x+a)^18/ b^10+1/19*(A*b-9*B*a)*(b*x+a)^19/b^10+1/20*B*(b*x+a)^20/b^10
Time = 0.02 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.95 \[ \int x^8 (a+b x)^{10} (A+B x) \, dx=\frac {1}{9} a^{10} A x^9+\frac {1}{10} a^9 (10 A b+a B) x^{10}+\frac {5}{11} a^8 b (9 A b+2 a B) x^{11}+\frac {5}{4} a^7 b^2 (8 A b+3 a B) x^{12}+\frac {30}{13} a^6 b^3 (7 A b+4 a B) x^{13}+3 a^5 b^4 (6 A b+5 a B) x^{14}+\frac {14}{5} a^4 b^5 (5 A b+6 a B) x^{15}+\frac {15}{8} a^3 b^6 (4 A b+7 a B) x^{16}+\frac {15}{17} a^2 b^7 (3 A b+8 a B) x^{17}+\frac {5}{18} a b^8 (2 A b+9 a B) x^{18}+\frac {1}{19} b^9 (A b+10 a B) x^{19}+\frac {1}{20} b^{10} B x^{20} \]
(a^10*A*x^9)/9 + (a^9*(10*A*b + a*B)*x^10)/10 + (5*a^8*b*(9*A*b + 2*a*B)*x ^11)/11 + (5*a^7*b^2*(8*A*b + 3*a*B)*x^12)/4 + (30*a^6*b^3*(7*A*b + 4*a*B) *x^13)/13 + 3*a^5*b^4*(6*A*b + 5*a*B)*x^14 + (14*a^4*b^5*(5*A*b + 6*a*B)*x ^15)/5 + (15*a^3*b^6*(4*A*b + 7*a*B)*x^16)/8 + (15*a^2*b^7*(3*A*b + 8*a*B) *x^17)/17 + (5*a*b^8*(2*A*b + 9*a*B)*x^18)/18 + (b^9*(A*b + 10*a*B)*x^19)/ 19 + (b^10*B*x^20)/20
Time = 0.44 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {85, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^8 (a+b x)^{10} (A+B x) \, dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (-\frac {a^8 (a+b x)^{10} (a B-A b)}{b^9}+\frac {a^7 (a+b x)^{11} (9 a B-8 A b)}{b^9}-\frac {4 a^6 (a+b x)^{12} (9 a B-7 A b)}{b^9}+\frac {28 a^5 (a+b x)^{13} (3 a B-2 A b)}{b^9}-\frac {14 a^4 (a+b x)^{14} (9 a B-5 A b)}{b^9}+\frac {14 a^3 (a+b x)^{15} (9 a B-4 A b)}{b^9}-\frac {28 a^2 (a+b x)^{16} (3 a B-A b)}{b^9}+\frac {(a+b x)^{18} (A b-9 a B)}{b^9}+\frac {4 a (a+b x)^{17} (9 a B-2 A b)}{b^9}+\frac {B (a+b x)^{19}}{b^9}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^8 (a+b x)^{11} (A b-a B)}{11 b^{10}}-\frac {a^7 (a+b x)^{12} (8 A b-9 a B)}{12 b^{10}}+\frac {4 a^6 (a+b x)^{13} (7 A b-9 a B)}{13 b^{10}}-\frac {2 a^5 (a+b x)^{14} (2 A b-3 a B)}{b^{10}}+\frac {14 a^4 (a+b x)^{15} (5 A b-9 a B)}{15 b^{10}}-\frac {7 a^3 (a+b x)^{16} (4 A b-9 a B)}{8 b^{10}}+\frac {28 a^2 (a+b x)^{17} (A b-3 a B)}{17 b^{10}}+\frac {(a+b x)^{19} (A b-9 a B)}{19 b^{10}}-\frac {2 a (a+b x)^{18} (2 A b-9 a B)}{9 b^{10}}+\frac {B (a+b x)^{20}}{20 b^{10}}\) |
(a^8*(A*b - a*B)*(a + b*x)^11)/(11*b^10) - (a^7*(8*A*b - 9*a*B)*(a + b*x)^ 12)/(12*b^10) + (4*a^6*(7*A*b - 9*a*B)*(a + b*x)^13)/(13*b^10) - (2*a^5*(2 *A*b - 3*a*B)*(a + b*x)^14)/b^10 + (14*a^4*(5*A*b - 9*a*B)*(a + b*x)^15)/( 15*b^10) - (7*a^3*(4*A*b - 9*a*B)*(a + b*x)^16)/(8*b^10) + (28*a^2*(A*b - 3*a*B)*(a + b*x)^17)/(17*b^10) - (2*a*(2*A*b - 9*a*B)*(a + b*x)^18)/(9*b^1 0) + ((A*b - 9*a*B)*(a + b*x)^19)/(19*b^10) + (B*(a + b*x)^20)/(20*b^10)
3.2.39.3.1 Defintions of rubi rules used
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Time = 0.39 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.98
| method | result | size |
| norman | \(\frac {a^{10} A \,x^{9}}{9}+\left (a^{9} b A +\frac {1}{10} a^{10} B \right ) x^{10}+\left (\frac {45}{11} a^{8} b^{2} A +\frac {10}{11} a^{9} b B \right ) x^{11}+\left (10 a^{7} b^{3} A +\frac {15}{4} a^{8} b^{2} B \right ) x^{12}+\left (\frac {210}{13} a^{6} b^{4} A +\frac {120}{13} a^{7} b^{3} B \right ) x^{13}+\left (18 a^{5} b^{5} A +15 a^{6} b^{4} B \right ) x^{14}+\left (14 a^{4} b^{6} A +\frac {84}{5} a^{5} b^{5} B \right ) x^{15}+\left (\frac {15}{2} a^{3} b^{7} A +\frac {105}{8} a^{4} b^{6} B \right ) x^{16}+\left (\frac {45}{17} a^{2} b^{8} A +\frac {120}{17} a^{3} b^{7} B \right ) x^{17}+\left (\frac {5}{9} a \,b^{9} A +\frac {5}{2} a^{2} b^{8} B \right ) x^{18}+\left (\frac {1}{19} b^{10} A +\frac {10}{19} a \,b^{9} B \right ) x^{19}+\frac {b^{10} B \,x^{20}}{20}\) | \(235\) |
| default | \(\frac {b^{10} B \,x^{20}}{20}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{19}}{19}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{18}}{18}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{17}}{17}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{16}}{16}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{15}}{15}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{14}}{14}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{13}}{13}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{12}}{12}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{11}}{11}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{10}}{10}+\frac {a^{10} A \,x^{9}}{9}\) | \(244\) |
| gosper | \(\frac {1}{9} a^{10} A \,x^{9}+x^{10} a^{9} b A +\frac {1}{10} x^{10} a^{10} B +\frac {45}{11} x^{11} a^{8} b^{2} A +\frac {10}{11} x^{11} a^{9} b B +10 x^{12} a^{7} b^{3} A +\frac {15}{4} x^{12} a^{8} b^{2} B +\frac {210}{13} x^{13} a^{6} b^{4} A +\frac {120}{13} x^{13} a^{7} b^{3} B +18 A \,a^{5} b^{5} x^{14}+15 B \,a^{6} b^{4} x^{14}+14 x^{15} a^{4} b^{6} A +\frac {84}{5} x^{15} a^{5} b^{5} B +\frac {15}{2} x^{16} a^{3} b^{7} A +\frac {105}{8} x^{16} a^{4} b^{6} B +\frac {45}{17} x^{17} a^{2} b^{8} A +\frac {120}{17} x^{17} a^{3} b^{7} B +\frac {5}{9} x^{18} a \,b^{9} A +\frac {5}{2} x^{18} a^{2} b^{8} B +\frac {1}{19} x^{19} b^{10} A +\frac {10}{19} x^{19} a \,b^{9} B +\frac {1}{20} b^{10} B \,x^{20}\) | \(245\) |
| risch | \(\frac {1}{9} a^{10} A \,x^{9}+x^{10} a^{9} b A +\frac {1}{10} x^{10} a^{10} B +\frac {45}{11} x^{11} a^{8} b^{2} A +\frac {10}{11} x^{11} a^{9} b B +10 x^{12} a^{7} b^{3} A +\frac {15}{4} x^{12} a^{8} b^{2} B +\frac {210}{13} x^{13} a^{6} b^{4} A +\frac {120}{13} x^{13} a^{7} b^{3} B +18 A \,a^{5} b^{5} x^{14}+15 B \,a^{6} b^{4} x^{14}+14 x^{15} a^{4} b^{6} A +\frac {84}{5} x^{15} a^{5} b^{5} B +\frac {15}{2} x^{16} a^{3} b^{7} A +\frac {105}{8} x^{16} a^{4} b^{6} B +\frac {45}{17} x^{17} a^{2} b^{8} A +\frac {120}{17} x^{17} a^{3} b^{7} B +\frac {5}{9} x^{18} a \,b^{9} A +\frac {5}{2} x^{18} a^{2} b^{8} B +\frac {1}{19} x^{19} b^{10} A +\frac {10}{19} x^{19} a \,b^{9} B +\frac {1}{20} b^{10} B \,x^{20}\) | \(245\) |
| parallelrisch | \(\frac {1}{9} a^{10} A \,x^{9}+x^{10} a^{9} b A +\frac {1}{10} x^{10} a^{10} B +\frac {45}{11} x^{11} a^{8} b^{2} A +\frac {10}{11} x^{11} a^{9} b B +10 x^{12} a^{7} b^{3} A +\frac {15}{4} x^{12} a^{8} b^{2} B +\frac {210}{13} x^{13} a^{6} b^{4} A +\frac {120}{13} x^{13} a^{7} b^{3} B +18 A \,a^{5} b^{5} x^{14}+15 B \,a^{6} b^{4} x^{14}+14 x^{15} a^{4} b^{6} A +\frac {84}{5} x^{15} a^{5} b^{5} B +\frac {15}{2} x^{16} a^{3} b^{7} A +\frac {105}{8} x^{16} a^{4} b^{6} B +\frac {45}{17} x^{17} a^{2} b^{8} A +\frac {120}{17} x^{17} a^{3} b^{7} B +\frac {5}{9} x^{18} a \,b^{9} A +\frac {5}{2} x^{18} a^{2} b^{8} B +\frac {1}{19} x^{19} b^{10} A +\frac {10}{19} x^{19} a \,b^{9} B +\frac {1}{20} b^{10} B \,x^{20}\) | \(245\) |
1/9*a^10*A*x^9+(a^9*b*A+1/10*a^10*B)*x^10+(45/11*a^8*b^2*A+10/11*a^9*b*B)* x^11+(10*a^7*b^3*A+15/4*a^8*b^2*B)*x^12+(210/13*a^6*b^4*A+120/13*a^7*b^3*B )*x^13+(18*A*a^5*b^5+15*B*a^6*b^4)*x^14+(14*a^4*b^6*A+84/5*a^5*b^5*B)*x^15 +(15/2*a^3*b^7*A+105/8*a^4*b^6*B)*x^16+(45/17*a^2*b^8*A+120/17*a^3*b^7*B)* x^17+(5/9*a*b^9*A+5/2*a^2*b^8*B)*x^18+(1/19*b^10*A+10/19*a*b^9*B)*x^19+1/2 0*b^10*B*x^20
Time = 0.22 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.01 \[ \int x^8 (a+b x)^{10} (A+B x) \, dx=\frac {1}{20} \, B b^{10} x^{20} + \frac {1}{9} \, A a^{10} x^{9} + \frac {1}{19} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{19} + \frac {5}{18} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{18} + \frac {15}{17} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{17} + \frac {15}{8} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{16} + \frac {14}{5} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{15} + 3 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{14} + \frac {30}{13} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{13} + \frac {5}{4} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{12} + \frac {5}{11} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{11} + \frac {1}{10} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{10} \]
1/20*B*b^10*x^20 + 1/9*A*a^10*x^9 + 1/19*(10*B*a*b^9 + A*b^10)*x^19 + 5/18 *(9*B*a^2*b^8 + 2*A*a*b^9)*x^18 + 15/17*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^17 + 15/8*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^16 + 14/5*(6*B*a^5*b^5 + 5*A*a^4*b^6)* x^15 + 3*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^14 + 30/13*(4*B*a^7*b^3 + 7*A*a^6*b ^4)*x^13 + 5/4*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^12 + 5/11*(2*B*a^9*b + 9*A*a^ 8*b^2)*x^11 + 1/10*(B*a^10 + 10*A*a^9*b)*x^10
Time = 0.04 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.10 \[ \int x^8 (a+b x)^{10} (A+B x) \, dx=\frac {A a^{10} x^{9}}{9} + \frac {B b^{10} x^{20}}{20} + x^{19} \left (\frac {A b^{10}}{19} + \frac {10 B a b^{9}}{19}\right ) + x^{18} \cdot \left (\frac {5 A a b^{9}}{9} + \frac {5 B a^{2} b^{8}}{2}\right ) + x^{17} \cdot \left (\frac {45 A a^{2} b^{8}}{17} + \frac {120 B a^{3} b^{7}}{17}\right ) + x^{16} \cdot \left (\frac {15 A a^{3} b^{7}}{2} + \frac {105 B a^{4} b^{6}}{8}\right ) + x^{15} \cdot \left (14 A a^{4} b^{6} + \frac {84 B a^{5} b^{5}}{5}\right ) + x^{14} \cdot \left (18 A a^{5} b^{5} + 15 B a^{6} b^{4}\right ) + x^{13} \cdot \left (\frac {210 A a^{6} b^{4}}{13} + \frac {120 B a^{7} b^{3}}{13}\right ) + x^{12} \cdot \left (10 A a^{7} b^{3} + \frac {15 B a^{8} b^{2}}{4}\right ) + x^{11} \cdot \left (\frac {45 A a^{8} b^{2}}{11} + \frac {10 B a^{9} b}{11}\right ) + x^{10} \left (A a^{9} b + \frac {B a^{10}}{10}\right ) \]
A*a**10*x**9/9 + B*b**10*x**20/20 + x**19*(A*b**10/19 + 10*B*a*b**9/19) + x**18*(5*A*a*b**9/9 + 5*B*a**2*b**8/2) + x**17*(45*A*a**2*b**8/17 + 120*B* a**3*b**7/17) + x**16*(15*A*a**3*b**7/2 + 105*B*a**4*b**6/8) + x**15*(14*A *a**4*b**6 + 84*B*a**5*b**5/5) + x**14*(18*A*a**5*b**5 + 15*B*a**6*b**4) + x**13*(210*A*a**6*b**4/13 + 120*B*a**7*b**3/13) + x**12*(10*A*a**7*b**3 + 15*B*a**8*b**2/4) + x**11*(45*A*a**8*b**2/11 + 10*B*a**9*b/11) + x**10*(A *a**9*b + B*a**10/10)
Time = 0.20 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.01 \[ \int x^8 (a+b x)^{10} (A+B x) \, dx=\frac {1}{20} \, B b^{10} x^{20} + \frac {1}{9} \, A a^{10} x^{9} + \frac {1}{19} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{19} + \frac {5}{18} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{18} + \frac {15}{17} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{17} + \frac {15}{8} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{16} + \frac {14}{5} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{15} + 3 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{14} + \frac {30}{13} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{13} + \frac {5}{4} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{12} + \frac {5}{11} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{11} + \frac {1}{10} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{10} \]
1/20*B*b^10*x^20 + 1/9*A*a^10*x^9 + 1/19*(10*B*a*b^9 + A*b^10)*x^19 + 5/18 *(9*B*a^2*b^8 + 2*A*a*b^9)*x^18 + 15/17*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^17 + 15/8*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^16 + 14/5*(6*B*a^5*b^5 + 5*A*a^4*b^6)* x^15 + 3*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^14 + 30/13*(4*B*a^7*b^3 + 7*A*a^6*b ^4)*x^13 + 5/4*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^12 + 5/11*(2*B*a^9*b + 9*A*a^ 8*b^2)*x^11 + 1/10*(B*a^10 + 10*A*a^9*b)*x^10
Time = 0.29 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.02 \[ \int x^8 (a+b x)^{10} (A+B x) \, dx=\frac {1}{20} \, B b^{10} x^{20} + \frac {10}{19} \, B a b^{9} x^{19} + \frac {1}{19} \, A b^{10} x^{19} + \frac {5}{2} \, B a^{2} b^{8} x^{18} + \frac {5}{9} \, A a b^{9} x^{18} + \frac {120}{17} \, B a^{3} b^{7} x^{17} + \frac {45}{17} \, A a^{2} b^{8} x^{17} + \frac {105}{8} \, B a^{4} b^{6} x^{16} + \frac {15}{2} \, A a^{3} b^{7} x^{16} + \frac {84}{5} \, B a^{5} b^{5} x^{15} + 14 \, A a^{4} b^{6} x^{15} + 15 \, B a^{6} b^{4} x^{14} + 18 \, A a^{5} b^{5} x^{14} + \frac {120}{13} \, B a^{7} b^{3} x^{13} + \frac {210}{13} \, A a^{6} b^{4} x^{13} + \frac {15}{4} \, B a^{8} b^{2} x^{12} + 10 \, A a^{7} b^{3} x^{12} + \frac {10}{11} \, B a^{9} b x^{11} + \frac {45}{11} \, A a^{8} b^{2} x^{11} + \frac {1}{10} \, B a^{10} x^{10} + A a^{9} b x^{10} + \frac {1}{9} \, A a^{10} x^{9} \]
1/20*B*b^10*x^20 + 10/19*B*a*b^9*x^19 + 1/19*A*b^10*x^19 + 5/2*B*a^2*b^8*x ^18 + 5/9*A*a*b^9*x^18 + 120/17*B*a^3*b^7*x^17 + 45/17*A*a^2*b^8*x^17 + 10 5/8*B*a^4*b^6*x^16 + 15/2*A*a^3*b^7*x^16 + 84/5*B*a^5*b^5*x^15 + 14*A*a^4* b^6*x^15 + 15*B*a^6*b^4*x^14 + 18*A*a^5*b^5*x^14 + 120/13*B*a^7*b^3*x^13 + 210/13*A*a^6*b^4*x^13 + 15/4*B*a^8*b^2*x^12 + 10*A*a^7*b^3*x^12 + 10/11*B *a^9*b*x^11 + 45/11*A*a^8*b^2*x^11 + 1/10*B*a^10*x^10 + A*a^9*b*x^10 + 1/9 *A*a^10*x^9
Time = 0.09 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.88 \[ \int x^8 (a+b x)^{10} (A+B x) \, dx=x^{10}\,\left (\frac {B\,a^{10}}{10}+A\,b\,a^9\right )+x^{19}\,\left (\frac {A\,b^{10}}{19}+\frac {10\,B\,a\,b^9}{19}\right )+\frac {A\,a^{10}\,x^9}{9}+\frac {B\,b^{10}\,x^{20}}{20}+\frac {5\,a^7\,b^2\,x^{12}\,\left (8\,A\,b+3\,B\,a\right )}{4}+\frac {30\,a^6\,b^3\,x^{13}\,\left (7\,A\,b+4\,B\,a\right )}{13}+3\,a^5\,b^4\,x^{14}\,\left (6\,A\,b+5\,B\,a\right )+\frac {14\,a^4\,b^5\,x^{15}\,\left (5\,A\,b+6\,B\,a\right )}{5}+\frac {15\,a^3\,b^6\,x^{16}\,\left (4\,A\,b+7\,B\,a\right )}{8}+\frac {15\,a^2\,b^7\,x^{17}\,\left (3\,A\,b+8\,B\,a\right )}{17}+\frac {5\,a^8\,b\,x^{11}\,\left (9\,A\,b+2\,B\,a\right )}{11}+\frac {5\,a\,b^8\,x^{18}\,\left (2\,A\,b+9\,B\,a\right )}{18} \]
x^10*((B*a^10)/10 + A*a^9*b) + x^19*((A*b^10)/19 + (10*B*a*b^9)/19) + (A*a ^10*x^9)/9 + (B*b^10*x^20)/20 + (5*a^7*b^2*x^12*(8*A*b + 3*B*a))/4 + (30*a ^6*b^3*x^13*(7*A*b + 4*B*a))/13 + 3*a^5*b^4*x^14*(6*A*b + 5*B*a) + (14*a^4 *b^5*x^15*(5*A*b + 6*B*a))/5 + (15*a^3*b^6*x^16*(4*A*b + 7*B*a))/8 + (15*a ^2*b^7*x^17*(3*A*b + 8*B*a))/17 + (5*a^8*b*x^11*(9*A*b + 2*B*a))/11 + (5*a *b^8*x^18*(2*A*b + 9*B*a))/18