Integrand size = 16, antiderivative size = 188 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{18}} \, dx=-\frac {A (a+b x)^{11}}{17 a x^{17}}+\frac {(6 A b-17 a B) (a+b x)^{11}}{272 a^2 x^{16}}-\frac {b (6 A b-17 a B) (a+b x)^{11}}{816 a^3 x^{15}}+\frac {b^2 (6 A b-17 a B) (a+b x)^{11}}{2856 a^4 x^{14}}-\frac {b^3 (6 A b-17 a B) (a+b x)^{11}}{12376 a^5 x^{13}}+\frac {b^4 (6 A b-17 a B) (a+b x)^{11}}{74256 a^6 x^{12}}-\frac {b^5 (6 A b-17 a B) (a+b x)^{11}}{816816 a^7 x^{11}} \]
-1/17*A*(b*x+a)^11/a/x^17+1/272*(6*A*b-17*B*a)*(b*x+a)^11/a^2/x^16-1/816*b *(6*A*b-17*B*a)*(b*x+a)^11/a^3/x^15+1/2856*b^2*(6*A*b-17*B*a)*(b*x+a)^11/a ^4/x^14-1/12376*b^3*(6*A*b-17*B*a)*(b*x+a)^11/a^5/x^13+1/74256*b^4*(6*A*b- 17*B*a)*(b*x+a)^11/a^6/x^12-1/816816*b^5*(6*A*b-17*B*a)*(b*x+a)^11/a^7/x^1 1
Time = 0.04 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.18 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{18}} \, dx=-\frac {b^{10} (6 A+7 B x)}{42 x^7}-\frac {5 a b^9 (7 A+8 B x)}{28 x^8}-\frac {5 a^2 b^8 (8 A+9 B x)}{8 x^9}-\frac {4 a^3 b^7 (9 A+10 B x)}{3 x^{10}}-\frac {21 a^4 b^6 (10 A+11 B x)}{11 x^{11}}-\frac {21 a^5 b^5 (11 A+12 B x)}{11 x^{12}}-\frac {35 a^6 b^4 (12 A+13 B x)}{26 x^{13}}-\frac {60 a^7 b^3 (13 A+14 B x)}{91 x^{14}}-\frac {3 a^8 b^2 (14 A+15 B x)}{14 x^{15}}-\frac {a^9 b (15 A+16 B x)}{24 x^{16}}-\frac {a^{10} (16 A+17 B x)}{272 x^{17}} \]
-1/42*(b^10*(6*A + 7*B*x))/x^7 - (5*a*b^9*(7*A + 8*B*x))/(28*x^8) - (5*a^2 *b^8*(8*A + 9*B*x))/(8*x^9) - (4*a^3*b^7*(9*A + 10*B*x))/(3*x^10) - (21*a^ 4*b^6*(10*A + 11*B*x))/(11*x^11) - (21*a^5*b^5*(11*A + 12*B*x))/(11*x^12) - (35*a^6*b^4*(12*A + 13*B*x))/(26*x^13) - (60*a^7*b^3*(13*A + 14*B*x))/(9 1*x^14) - (3*a^8*b^2*(14*A + 15*B*x))/(14*x^15) - (a^9*b*(15*A + 16*B*x))/ (24*x^16) - (a^10*(16*A + 17*B*x))/(272*x^17)
Time = 0.23 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {87, 55, 55, 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.
| \(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{x^{18}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {(6 A b-17 a B) \int \frac {(a+b x)^{10}}{x^{17}}dx}{17 a}-\frac {A (a+b x)^{11}}{17 a x^{17}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(6 A b-17 a B) \left (-\frac {5 b \int \frac {(a+b x)^{10}}{x^{16}}dx}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\right )}{17 a}-\frac {A (a+b x)^{11}}{17 a x^{17}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(6 A b-17 a B) \left (-\frac {5 b \left (-\frac {4 b \int \frac {(a+b x)^{10}}{x^{15}}dx}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\right )}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\right )}{17 a}-\frac {A (a+b x)^{11}}{17 a x^{17}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(6 A b-17 a B) \left (-\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \int \frac {(a+b x)^{10}}{x^{14}}dx}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\right )}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\right )}{17 a}-\frac {A (a+b x)^{11}}{17 a x^{17}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(6 A b-17 a B) \left (-\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \int \frac {(a+b x)^{10}}{x^{13}}dx}{13 a}-\frac {(a+b x)^{11}}{13 a x^{13}}\right )}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\right )}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\right )}{17 a}-\frac {A (a+b x)^{11}}{17 a x^{17}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(6 A b-17 a B) \left (-\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \left (-\frac {b \int \frac {(a+b x)^{10}}{x^{12}}dx}{12 a}-\frac {(a+b x)^{11}}{12 a x^{12}}\right )}{13 a}-\frac {(a+b x)^{11}}{13 a x^{13}}\right )}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\right )}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\right )}{17 a}-\frac {A (a+b x)^{11}}{17 a x^{17}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {\left (-\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \left (\frac {b (a+b x)^{11}}{132 a^2 x^{11}}-\frac {(a+b x)^{11}}{12 a x^{12}}\right )}{13 a}-\frac {(a+b x)^{11}}{13 a x^{13}}\right )}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\right )}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\right ) (6 A b-17 a B)}{17 a}-\frac {A (a+b x)^{11}}{17 a x^{17}}\) |
-1/17*(A*(a + b*x)^11)/(a*x^17) - ((6*A*b - 17*a*B)*(-1/16*(a + b*x)^11/(a *x^16) - (5*b*(-1/15*(a + b*x)^11/(a*x^15) - (4*b*(-1/14*(a + b*x)^11/(a*x ^14) - (3*b*(-1/13*(a + b*x)^11/(a*x^13) - (2*b*(-1/12*(a + b*x)^11/(a*x^1 2) + (b*(a + b*x)^11)/(132*a^2*x^11)))/(13*a)))/(14*a)))/(15*a)))/(16*a))) /(17*a)
3.2.65.3.1 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]))))
Time = 0.41 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.11
| method | result | size |
| default | \(-\frac {b^{10} B}{6 x^{6}}-\frac {15 a^{7} b^{2} \left (8 A b +3 B a \right )}{14 x^{14}}-\frac {a^{9} \left (10 A b +B a \right )}{16 x^{16}}-\frac {b^{9} \left (A b +10 B a \right )}{7 x^{7}}-\frac {5 a \,b^{8} \left (2 A b +9 B a \right )}{8 x^{8}}-\frac {3 a^{3} b^{6} \left (4 A b +7 B a \right )}{x^{10}}-\frac {30 a^{6} b^{3} \left (7 A b +4 B a \right )}{13 x^{13}}-\frac {a^{10} A}{17 x^{17}}-\frac {7 a^{5} b^{4} \left (6 A b +5 B a \right )}{2 x^{12}}-\frac {a^{8} b \left (9 A b +2 B a \right )}{3 x^{15}}-\frac {5 a^{2} b^{7} \left (3 A b +8 B a \right )}{3 x^{9}}-\frac {42 a^{4} b^{5} \left (5 A b +6 B a \right )}{11 x^{11}}\) | \(208\) |
| norman | \(\frac {-\frac {a^{10} A}{17}+\left (-\frac {5}{8} a^{9} b A -\frac {1}{16} a^{10} B \right ) x +\left (-3 a^{8} b^{2} A -\frac {2}{3} a^{9} b B \right ) x^{2}+\left (-\frac {60}{7} a^{7} b^{3} A -\frac {45}{14} a^{8} b^{2} B \right ) x^{3}+\left (-\frac {210}{13} a^{6} b^{4} A -\frac {120}{13} a^{7} b^{3} B \right ) x^{4}+\left (-21 a^{5} b^{5} A -\frac {35}{2} a^{6} b^{4} B \right ) x^{5}+\left (-\frac {210}{11} a^{4} b^{6} A -\frac {252}{11} a^{5} b^{5} B \right ) x^{6}+\left (-12 a^{3} b^{7} A -21 a^{4} b^{6} B \right ) x^{7}+\left (-5 a^{2} b^{8} A -\frac {40}{3} a^{3} b^{7} B \right ) x^{8}+\left (-\frac {5}{4} a \,b^{9} A -\frac {45}{8} a^{2} b^{8} B \right ) x^{9}+\left (-\frac {1}{7} b^{10} A -\frac {10}{7} a \,b^{9} B \right ) x^{10}-\frac {b^{10} B \,x^{11}}{6}}{x^{17}}\) | \(235\) |
| risch | \(\frac {-\frac {a^{10} A}{17}+\left (-\frac {5}{8} a^{9} b A -\frac {1}{16} a^{10} B \right ) x +\left (-3 a^{8} b^{2} A -\frac {2}{3} a^{9} b B \right ) x^{2}+\left (-\frac {60}{7} a^{7} b^{3} A -\frac {45}{14} a^{8} b^{2} B \right ) x^{3}+\left (-\frac {210}{13} a^{6} b^{4} A -\frac {120}{13} a^{7} b^{3} B \right ) x^{4}+\left (-21 a^{5} b^{5} A -\frac {35}{2} a^{6} b^{4} B \right ) x^{5}+\left (-\frac {210}{11} a^{4} b^{6} A -\frac {252}{11} a^{5} b^{5} B \right ) x^{6}+\left (-12 a^{3} b^{7} A -21 a^{4} b^{6} B \right ) x^{7}+\left (-5 a^{2} b^{8} A -\frac {40}{3} a^{3} b^{7} B \right ) x^{8}+\left (-\frac {5}{4} a \,b^{9} A -\frac {45}{8} a^{2} b^{8} B \right ) x^{9}+\left (-\frac {1}{7} b^{10} A -\frac {10}{7} a \,b^{9} B \right ) x^{10}-\frac {b^{10} B \,x^{11}}{6}}{x^{17}}\) | \(235\) |
| gosper | \(-\frac {136136 b^{10} B \,x^{11}+116688 A \,b^{10} x^{10}+1166880 B a \,b^{9} x^{10}+1021020 a A \,b^{9} x^{9}+4594590 B \,a^{2} b^{8} x^{9}+4084080 a^{2} A \,b^{8} x^{8}+10890880 B \,a^{3} b^{7} x^{8}+9801792 a^{3} A \,b^{7} x^{7}+17153136 B \,a^{4} b^{6} x^{7}+15593760 a^{4} A \,b^{6} x^{6}+18712512 B \,a^{5} b^{5} x^{6}+17153136 a^{5} A \,b^{5} x^{5}+14294280 B \,a^{6} b^{4} x^{5}+13194720 a^{6} A \,b^{4} x^{4}+7539840 B \,a^{7} b^{3} x^{4}+7001280 a^{7} A \,b^{3} x^{3}+2625480 B \,a^{8} b^{2} x^{3}+2450448 a^{8} A \,b^{2} x^{2}+544544 B \,a^{9} b \,x^{2}+510510 a^{9} A b x +51051 a^{10} B x +48048 a^{10} A}{816816 x^{17}}\) | \(244\) |
| parallelrisch | \(-\frac {136136 b^{10} B \,x^{11}+116688 A \,b^{10} x^{10}+1166880 B a \,b^{9} x^{10}+1021020 a A \,b^{9} x^{9}+4594590 B \,a^{2} b^{8} x^{9}+4084080 a^{2} A \,b^{8} x^{8}+10890880 B \,a^{3} b^{7} x^{8}+9801792 a^{3} A \,b^{7} x^{7}+17153136 B \,a^{4} b^{6} x^{7}+15593760 a^{4} A \,b^{6} x^{6}+18712512 B \,a^{5} b^{5} x^{6}+17153136 a^{5} A \,b^{5} x^{5}+14294280 B \,a^{6} b^{4} x^{5}+13194720 a^{6} A \,b^{4} x^{4}+7539840 B \,a^{7} b^{3} x^{4}+7001280 a^{7} A \,b^{3} x^{3}+2625480 B \,a^{8} b^{2} x^{3}+2450448 a^{8} A \,b^{2} x^{2}+544544 B \,a^{9} b \,x^{2}+510510 a^{9} A b x +51051 a^{10} B x +48048 a^{10} A}{816816 x^{17}}\) | \(244\) |
-1/6*b^10*B/x^6-15/14*a^7*b^2*(8*A*b+3*B*a)/x^14-1/16*a^9*(10*A*b+B*a)/x^1 6-1/7*b^9*(A*b+10*B*a)/x^7-5/8*a*b^8*(2*A*b+9*B*a)/x^8-3*a^3*b^6*(4*A*b+7* B*a)/x^10-30/13*a^6*b^3*(7*A*b+4*B*a)/x^13-1/17*a^10*A/x^17-7/2*a^5*b^4*(6 *A*b+5*B*a)/x^12-1/3*a^8*b*(9*A*b+2*B*a)/x^15-5/3*a^2*b^7*(3*A*b+8*B*a)/x^ 9-42/11*a^4*b^5*(5*A*b+6*B*a)/x^11
Time = 0.22 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.29 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{18}} \, dx=-\frac {136136 \, B b^{10} x^{11} + 48048 \, A a^{10} + 116688 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 510510 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 1361360 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 2450448 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 3118752 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 2858856 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 1884960 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 875160 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 272272 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 51051 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{816816 \, x^{17}} \]
-1/816816*(136136*B*b^10*x^11 + 48048*A*a^10 + 116688*(10*B*a*b^9 + A*b^10 )*x^10 + 510510*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 1361360*(8*B*a^3*b^7 + 3*A *a^2*b^8)*x^8 + 2450448*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 3118752*(6*B*a^5 *b^5 + 5*A*a^4*b^6)*x^6 + 2858856*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 188496 0*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 875160*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 272272*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 51051*(B*a^10 + 10*A*a^9*b)*x)/x ^17
Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{18}} \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.29 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{18}} \, dx=-\frac {136136 \, B b^{10} x^{11} + 48048 \, A a^{10} + 116688 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 510510 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 1361360 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 2450448 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 3118752 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 2858856 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 1884960 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 875160 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 272272 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 51051 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{816816 \, x^{17}} \]
-1/816816*(136136*B*b^10*x^11 + 48048*A*a^10 + 116688*(10*B*a*b^9 + A*b^10 )*x^10 + 510510*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 1361360*(8*B*a^3*b^7 + 3*A *a^2*b^8)*x^8 + 2450448*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 3118752*(6*B*a^5 *b^5 + 5*A*a^4*b^6)*x^6 + 2858856*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 188496 0*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 875160*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 272272*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 51051*(B*a^10 + 10*A*a^9*b)*x)/x ^17
Time = 0.27 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.29 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{18}} \, dx=-\frac {136136 \, B b^{10} x^{11} + 1166880 \, B a b^{9} x^{10} + 116688 \, A b^{10} x^{10} + 4594590 \, B a^{2} b^{8} x^{9} + 1021020 \, A a b^{9} x^{9} + 10890880 \, B a^{3} b^{7} x^{8} + 4084080 \, A a^{2} b^{8} x^{8} + 17153136 \, B a^{4} b^{6} x^{7} + 9801792 \, A a^{3} b^{7} x^{7} + 18712512 \, B a^{5} b^{5} x^{6} + 15593760 \, A a^{4} b^{6} x^{6} + 14294280 \, B a^{6} b^{4} x^{5} + 17153136 \, A a^{5} b^{5} x^{5} + 7539840 \, B a^{7} b^{3} x^{4} + 13194720 \, A a^{6} b^{4} x^{4} + 2625480 \, B a^{8} b^{2} x^{3} + 7001280 \, A a^{7} b^{3} x^{3} + 544544 \, B a^{9} b x^{2} + 2450448 \, A a^{8} b^{2} x^{2} + 51051 \, B a^{10} x + 510510 \, A a^{9} b x + 48048 \, A a^{10}}{816816 \, x^{17}} \]
-1/816816*(136136*B*b^10*x^11 + 1166880*B*a*b^9*x^10 + 116688*A*b^10*x^10 + 4594590*B*a^2*b^8*x^9 + 1021020*A*a*b^9*x^9 + 10890880*B*a^3*b^7*x^8 + 4 084080*A*a^2*b^8*x^8 + 17153136*B*a^4*b^6*x^7 + 9801792*A*a^3*b^7*x^7 + 18 712512*B*a^5*b^5*x^6 + 15593760*A*a^4*b^6*x^6 + 14294280*B*a^6*b^4*x^5 + 1 7153136*A*a^5*b^5*x^5 + 7539840*B*a^7*b^3*x^4 + 13194720*A*a^6*b^4*x^4 + 2 625480*B*a^8*b^2*x^3 + 7001280*A*a^7*b^3*x^3 + 544544*B*a^9*b*x^2 + 245044 8*A*a^8*b^2*x^2 + 51051*B*a^10*x + 510510*A*a^9*b*x + 48048*A*a^10)/x^17
Time = 0.14 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.25 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{18}} \, dx=-\frac {x\,\left (\frac {B\,a^{10}}{16}+\frac {5\,A\,b\,a^9}{8}\right )+\frac {A\,a^{10}}{17}+x^2\,\left (\frac {2\,B\,a^9\,b}{3}+3\,A\,a^8\,b^2\right )+x^9\,\left (\frac {45\,B\,a^2\,b^8}{8}+\frac {5\,A\,a\,b^9}{4}\right )+x^{10}\,\left (\frac {A\,b^{10}}{7}+\frac {10\,B\,a\,b^9}{7}\right )+x^7\,\left (21\,B\,a^4\,b^6+12\,A\,a^3\,b^7\right )+x^8\,\left (\frac {40\,B\,a^3\,b^7}{3}+5\,A\,a^2\,b^8\right )+x^5\,\left (\frac {35\,B\,a^6\,b^4}{2}+21\,A\,a^5\,b^5\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{14}+\frac {60\,A\,a^7\,b^3}{7}\right )+x^4\,\left (\frac {120\,B\,a^7\,b^3}{13}+\frac {210\,A\,a^6\,b^4}{13}\right )+x^6\,\left (\frac {252\,B\,a^5\,b^5}{11}+\frac {210\,A\,a^4\,b^6}{11}\right )+\frac {B\,b^{10}\,x^{11}}{6}}{x^{17}} \]
-(x*((B*a^10)/16 + (5*A*a^9*b)/8) + (A*a^10)/17 + x^2*(3*A*a^8*b^2 + (2*B* a^9*b)/3) + x^9*((45*B*a^2*b^8)/8 + (5*A*a*b^9)/4) + x^10*((A*b^10)/7 + (1 0*B*a*b^9)/7) + x^7*(12*A*a^3*b^7 + 21*B*a^4*b^6) + x^8*(5*A*a^2*b^8 + (40 *B*a^3*b^7)/3) + x^5*(21*A*a^5*b^5 + (35*B*a^6*b^4)/2) + x^3*((60*A*a^7*b^ 3)/7 + (45*B*a^8*b^2)/14) + x^4*((210*A*a^6*b^4)/13 + (120*B*a^7*b^3)/13) + x^6*((210*A*a^4*b^6)/11 + (252*B*a^5*b^5)/11) + (B*b^10*x^11)/6)/x^17