Integrand size = 16, antiderivative size = 229 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx=-\frac {a^{10} A}{18 x^{18}}-\frac {a^9 (10 A b+a B)}{17 x^{17}}-\frac {5 a^8 b (9 A b+2 a B)}{16 x^{16}}-\frac {a^7 b^2 (8 A b+3 a B)}{x^{15}}-\frac {15 a^6 b^3 (7 A b+4 a B)}{7 x^{14}}-\frac {42 a^5 b^4 (6 A b+5 a B)}{13 x^{13}}-\frac {7 a^4 b^5 (5 A b+6 a B)}{2 x^{12}}-\frac {30 a^3 b^6 (4 A b+7 a B)}{11 x^{11}}-\frac {3 a^2 b^7 (3 A b+8 a B)}{2 x^{10}}-\frac {5 a b^8 (2 A b+9 a B)}{9 x^9}-\frac {b^9 (A b+10 a B)}{8 x^8}-\frac {b^{10} B}{7 x^7} \]
-1/18*a^10*A/x^18-1/17*a^9*(10*A*b+B*a)/x^17-5/16*a^8*b*(9*A*b+2*B*a)/x^16 -a^7*b^2*(8*A*b+3*B*a)/x^15-15/7*a^6*b^3*(7*A*b+4*B*a)/x^14-42/13*a^5*b^4* (6*A*b+5*B*a)/x^13-7/2*a^4*b^5*(5*A*b+6*B*a)/x^12-30/11*a^3*b^6*(4*A*b+7*B *a)/x^11-3/2*a^2*b^7*(3*A*b+8*B*a)/x^10-5/9*a*b^8*(2*A*b+9*B*a)/x^9-1/8*b^ 9*(A*b+10*B*a)/x^8-1/7*b^10*B/x^7
Time = 0.05 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx=-\frac {b^{10} (7 A+8 B x)}{56 x^8}-\frac {5 a b^9 (8 A+9 B x)}{36 x^9}-\frac {a^2 b^8 (9 A+10 B x)}{2 x^{10}}-\frac {12 a^3 b^7 (10 A+11 B x)}{11 x^{11}}-\frac {35 a^4 b^6 (11 A+12 B x)}{22 x^{12}}-\frac {21 a^5 b^5 (12 A+13 B x)}{13 x^{13}}-\frac {15 a^6 b^4 (13 A+14 B x)}{13 x^{14}}-\frac {4 a^7 b^3 (14 A+15 B x)}{7 x^{15}}-\frac {3 a^8 b^2 (15 A+16 B x)}{16 x^{16}}-\frac {5 a^9 b (16 A+17 B x)}{136 x^{17}}-\frac {a^{10} (17 A+18 B x)}{306 x^{18}} \]
-1/56*(b^10*(7*A + 8*B*x))/x^8 - (5*a*b^9*(8*A + 9*B*x))/(36*x^9) - (a^2*b ^8*(9*A + 10*B*x))/(2*x^10) - (12*a^3*b^7*(10*A + 11*B*x))/(11*x^11) - (35 *a^4*b^6*(11*A + 12*B*x))/(22*x^12) - (21*a^5*b^5*(12*A + 13*B*x))/(13*x^1 3) - (15*a^6*b^4*(13*A + 14*B*x))/(13*x^14) - (4*a^7*b^3*(14*A + 15*B*x))/ (7*x^15) - (3*a^8*b^2*(15*A + 16*B*x))/(16*x^16) - (5*a^9*b*(16*A + 17*B*x ))/(136*x^17) - (a^10*(17*A + 18*B*x))/(306*x^18)
Time = 0.39 (sec) , antiderivative size = 229, 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.
| \(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (\frac {a^{10} A}{x^{19}}+\frac {a^9 (a B+10 A b)}{x^{18}}+\frac {5 a^8 b (2 a B+9 A b)}{x^{17}}+\frac {15 a^7 b^2 (3 a B+8 A b)}{x^{16}}+\frac {30 a^6 b^3 (4 a B+7 A b)}{x^{15}}+\frac {42 a^5 b^4 (5 a B+6 A b)}{x^{14}}+\frac {42 a^4 b^5 (6 a B+5 A b)}{x^{13}}+\frac {30 a^3 b^6 (7 a B+4 A b)}{x^{12}}+\frac {15 a^2 b^7 (8 a B+3 A b)}{x^{11}}+\frac {b^9 (10 a B+A b)}{x^9}+\frac {5 a b^8 (9 a B+2 A b)}{x^{10}}+\frac {b^{10} B}{x^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{10} A}{18 x^{18}}-\frac {a^9 (a B+10 A b)}{17 x^{17}}-\frac {5 a^8 b (2 a B+9 A b)}{16 x^{16}}-\frac {a^7 b^2 (3 a B+8 A b)}{x^{15}}-\frac {15 a^6 b^3 (4 a B+7 A b)}{7 x^{14}}-\frac {42 a^5 b^4 (5 a B+6 A b)}{13 x^{13}}-\frac {7 a^4 b^5 (6 a B+5 A b)}{2 x^{12}}-\frac {30 a^3 b^6 (7 a B+4 A b)}{11 x^{11}}-\frac {3 a^2 b^7 (8 a B+3 A b)}{2 x^{10}}-\frac {b^9 (10 a B+A b)}{8 x^8}-\frac {5 a b^8 (9 a B+2 A b)}{9 x^9}-\frac {b^{10} B}{7 x^7}\) |
-1/18*(a^10*A)/x^18 - (a^9*(10*A*b + a*B))/(17*x^17) - (5*a^8*b*(9*A*b + 2 *a*B))/(16*x^16) - (a^7*b^2*(8*A*b + 3*a*B))/x^15 - (15*a^6*b^3*(7*A*b + 4 *a*B))/(7*x^14) - (42*a^5*b^4*(6*A*b + 5*a*B))/(13*x^13) - (7*a^4*b^5*(5*A *b + 6*a*B))/(2*x^12) - (30*a^3*b^6*(4*A*b + 7*a*B))/(11*x^11) - (3*a^2*b^ 7*(3*A*b + 8*a*B))/(2*x^10) - (5*a*b^8*(2*A*b + 9*a*B))/(9*x^9) - (b^9*(A* b + 10*a*B))/(8*x^8) - (b^10*B)/(7*x^7)
3.2.66.3.1 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 = 0.40 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.91
| method | result | size |
| default | \(-\frac {a^{10} A}{18 x^{18}}-\frac {a^{9} \left (10 A b +B a \right )}{17 x^{17}}-\frac {5 a^{8} b \left (9 A b +2 B a \right )}{16 x^{16}}-\frac {a^{7} b^{2} \left (8 A b +3 B a \right )}{x^{15}}-\frac {15 a^{6} b^{3} \left (7 A b +4 B a \right )}{7 x^{14}}-\frac {42 a^{5} b^{4} \left (6 A b +5 B a \right )}{13 x^{13}}-\frac {7 a^{4} b^{5} \left (5 A b +6 B a \right )}{2 x^{12}}-\frac {30 a^{3} b^{6} \left (4 A b +7 B a \right )}{11 x^{11}}-\frac {3 a^{2} b^{7} \left (3 A b +8 B a \right )}{2 x^{10}}-\frac {5 a \,b^{8} \left (2 A b +9 B a \right )}{9 x^{9}}-\frac {b^{9} \left (A b +10 B a \right )}{8 x^{8}}-\frac {b^{10} B}{7 x^{7}}\) | \(208\) |
| norman | \(\frac {-\frac {a^{10} A}{18}+\left (-\frac {10}{17} a^{9} b A -\frac {1}{17} a^{10} B \right ) x +\left (-\frac {45}{16} a^{8} b^{2} A -\frac {5}{8} a^{9} b B \right ) x^{2}+\left (-8 a^{7} b^{3} A -3 a^{8} b^{2} B \right ) x^{3}+\left (-15 a^{6} b^{4} A -\frac {60}{7} a^{7} b^{3} B \right ) x^{4}+\left (-\frac {252}{13} a^{5} b^{5} A -\frac {210}{13} a^{6} b^{4} B \right ) x^{5}+\left (-\frac {35}{2} a^{4} b^{6} A -21 a^{5} b^{5} B \right ) x^{6}+\left (-\frac {120}{11} a^{3} b^{7} A -\frac {210}{11} a^{4} b^{6} B \right ) x^{7}+\left (-\frac {9}{2} a^{2} b^{8} A -12 a^{3} b^{7} B \right ) x^{8}+\left (-\frac {10}{9} a \,b^{9} A -5 a^{2} b^{8} B \right ) x^{9}+\left (-\frac {1}{8} b^{10} A -\frac {5}{4} a \,b^{9} B \right ) x^{10}-\frac {b^{10} B \,x^{11}}{7}}{x^{18}}\) | \(235\) |
| risch | \(\frac {-\frac {a^{10} A}{18}+\left (-\frac {10}{17} a^{9} b A -\frac {1}{17} a^{10} B \right ) x +\left (-\frac {45}{16} a^{8} b^{2} A -\frac {5}{8} a^{9} b B \right ) x^{2}+\left (-8 a^{7} b^{3} A -3 a^{8} b^{2} B \right ) x^{3}+\left (-15 a^{6} b^{4} A -\frac {60}{7} a^{7} b^{3} B \right ) x^{4}+\left (-\frac {252}{13} a^{5} b^{5} A -\frac {210}{13} a^{6} b^{4} B \right ) x^{5}+\left (-\frac {35}{2} a^{4} b^{6} A -21 a^{5} b^{5} B \right ) x^{6}+\left (-\frac {120}{11} a^{3} b^{7} A -\frac {210}{11} a^{4} b^{6} B \right ) x^{7}+\left (-\frac {9}{2} a^{2} b^{8} A -12 a^{3} b^{7} B \right ) x^{8}+\left (-\frac {10}{9} a \,b^{9} A -5 a^{2} b^{8} B \right ) x^{9}+\left (-\frac {1}{8} b^{10} A -\frac {5}{4} a \,b^{9} B \right ) x^{10}-\frac {b^{10} B \,x^{11}}{7}}{x^{18}}\) | \(235\) |
| gosper | \(-\frac {350064 b^{10} B \,x^{11}+306306 A \,b^{10} x^{10}+3063060 B a \,b^{9} x^{10}+2722720 a A \,b^{9} x^{9}+12252240 B \,a^{2} b^{8} x^{9}+11027016 a^{2} A \,b^{8} x^{8}+29405376 B \,a^{3} b^{7} x^{8}+26732160 a^{3} A \,b^{7} x^{7}+46781280 B \,a^{4} b^{6} x^{7}+42882840 a^{4} A \,b^{6} x^{6}+51459408 B \,a^{5} b^{5} x^{6}+47500992 a^{5} A \,b^{5} x^{5}+39584160 B \,a^{6} b^{4} x^{5}+36756720 a^{6} A \,b^{4} x^{4}+21003840 B \,a^{7} b^{3} x^{4}+19603584 a^{7} A \,b^{3} x^{3}+7351344 B \,a^{8} b^{2} x^{3}+6891885 a^{8} A \,b^{2} x^{2}+1531530 B \,a^{9} b \,x^{2}+1441440 a^{9} A b x +144144 a^{10} B x +136136 a^{10} A}{2450448 x^{18}}\) | \(244\) |
| parallelrisch | \(-\frac {350064 b^{10} B \,x^{11}+306306 A \,b^{10} x^{10}+3063060 B a \,b^{9} x^{10}+2722720 a A \,b^{9} x^{9}+12252240 B \,a^{2} b^{8} x^{9}+11027016 a^{2} A \,b^{8} x^{8}+29405376 B \,a^{3} b^{7} x^{8}+26732160 a^{3} A \,b^{7} x^{7}+46781280 B \,a^{4} b^{6} x^{7}+42882840 a^{4} A \,b^{6} x^{6}+51459408 B \,a^{5} b^{5} x^{6}+47500992 a^{5} A \,b^{5} x^{5}+39584160 B \,a^{6} b^{4} x^{5}+36756720 a^{6} A \,b^{4} x^{4}+21003840 B \,a^{7} b^{3} x^{4}+19603584 a^{7} A \,b^{3} x^{3}+7351344 B \,a^{8} b^{2} x^{3}+6891885 a^{8} A \,b^{2} x^{2}+1531530 B \,a^{9} b \,x^{2}+1441440 a^{9} A b x +144144 a^{10} B x +136136 a^{10} A}{2450448 x^{18}}\) | \(244\) |
-1/18*a^10*A/x^18-1/17*a^9*(10*A*b+B*a)/x^17-5/16*a^8*b*(9*A*b+2*B*a)/x^16 -a^7*b^2*(8*A*b+3*B*a)/x^15-15/7*a^6*b^3*(7*A*b+4*B*a)/x^14-42/13*a^5*b^4* (6*A*b+5*B*a)/x^13-7/2*a^4*b^5*(5*A*b+6*B*a)/x^12-30/11*a^3*b^6*(4*A*b+7*B *a)/x^11-3/2*a^2*b^7*(3*A*b+8*B*a)/x^10-5/9*a*b^8*(2*A*b+9*B*a)/x^9-1/8*b^ 9*(A*b+10*B*a)/x^8-1/7*b^10*B/x^7
Time = 0.22 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx=-\frac {350064 \, B b^{10} x^{11} + 136136 \, A a^{10} + 306306 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 1361360 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 3675672 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 6683040 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 8576568 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 7916832 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 5250960 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 2450448 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 765765 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 144144 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2450448 \, x^{18}} \]
-1/2450448*(350064*B*b^10*x^11 + 136136*A*a^10 + 306306*(10*B*a*b^9 + A*b^ 10)*x^10 + 1361360*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 3675672*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 6683040*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 8576568*(6*B* a^5*b^5 + 5*A*a^4*b^6)*x^6 + 7916832*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 525 0960*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 2450448*(3*B*a^8*b^2 + 8*A*a^7*b^3) *x^3 + 765765*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 144144*(B*a^10 + 10*A*a^9*b) *x)/x^18
Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx=-\frac {350064 \, B b^{10} x^{11} + 136136 \, A a^{10} + 306306 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 1361360 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 3675672 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 6683040 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 8576568 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 7916832 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 5250960 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 2450448 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 765765 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 144144 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2450448 \, x^{18}} \]
-1/2450448*(350064*B*b^10*x^11 + 136136*A*a^10 + 306306*(10*B*a*b^9 + A*b^ 10)*x^10 + 1361360*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 3675672*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 6683040*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 8576568*(6*B* a^5*b^5 + 5*A*a^4*b^6)*x^6 + 7916832*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 525 0960*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 2450448*(3*B*a^8*b^2 + 8*A*a^7*b^3) *x^3 + 765765*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 144144*(B*a^10 + 10*A*a^9*b) *x)/x^18
Time = 0.28 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx=-\frac {350064 \, B b^{10} x^{11} + 3063060 \, B a b^{9} x^{10} + 306306 \, A b^{10} x^{10} + 12252240 \, B a^{2} b^{8} x^{9} + 2722720 \, A a b^{9} x^{9} + 29405376 \, B a^{3} b^{7} x^{8} + 11027016 \, A a^{2} b^{8} x^{8} + 46781280 \, B a^{4} b^{6} x^{7} + 26732160 \, A a^{3} b^{7} x^{7} + 51459408 \, B a^{5} b^{5} x^{6} + 42882840 \, A a^{4} b^{6} x^{6} + 39584160 \, B a^{6} b^{4} x^{5} + 47500992 \, A a^{5} b^{5} x^{5} + 21003840 \, B a^{7} b^{3} x^{4} + 36756720 \, A a^{6} b^{4} x^{4} + 7351344 \, B a^{8} b^{2} x^{3} + 19603584 \, A a^{7} b^{3} x^{3} + 1531530 \, B a^{9} b x^{2} + 6891885 \, A a^{8} b^{2} x^{2} + 144144 \, B a^{10} x + 1441440 \, A a^{9} b x + 136136 \, A a^{10}}{2450448 \, x^{18}} \]
-1/2450448*(350064*B*b^10*x^11 + 3063060*B*a*b^9*x^10 + 306306*A*b^10*x^10 + 12252240*B*a^2*b^8*x^9 + 2722720*A*a*b^9*x^9 + 29405376*B*a^3*b^7*x^8 + 11027016*A*a^2*b^8*x^8 + 46781280*B*a^4*b^6*x^7 + 26732160*A*a^3*b^7*x^7 + 51459408*B*a^5*b^5*x^6 + 42882840*A*a^4*b^6*x^6 + 39584160*B*a^6*b^4*x^5 + 47500992*A*a^5*b^5*x^5 + 21003840*B*a^7*b^3*x^4 + 36756720*A*a^6*b^4*x^ 4 + 7351344*B*a^8*b^2*x^3 + 19603584*A*a^7*b^3*x^3 + 1531530*B*a^9*b*x^2 + 6891885*A*a^8*b^2*x^2 + 144144*B*a^10*x + 1441440*A*a^9*b*x + 136136*A*a^ 10)/x^18
Time = 0.42 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx=-\frac {x\,\left (\frac {B\,a^{10}}{17}+\frac {10\,A\,b\,a^9}{17}\right )+\frac {A\,a^{10}}{18}+x^9\,\left (5\,B\,a^2\,b^8+\frac {10\,A\,a\,b^9}{9}\right )+x^2\,\left (\frac {5\,B\,a^9\,b}{8}+\frac {45\,A\,a^8\,b^2}{16}\right )+x^{10}\,\left (\frac {A\,b^{10}}{8}+\frac {5\,B\,a\,b^9}{4}\right )+x^3\,\left (3\,B\,a^8\,b^2+8\,A\,a^7\,b^3\right )+x^8\,\left (12\,B\,a^3\,b^7+\frac {9\,A\,a^2\,b^8}{2}\right )+x^6\,\left (21\,B\,a^5\,b^5+\frac {35\,A\,a^4\,b^6}{2}\right )+x^4\,\left (\frac {60\,B\,a^7\,b^3}{7}+15\,A\,a^6\,b^4\right )+x^7\,\left (\frac {210\,B\,a^4\,b^6}{11}+\frac {120\,A\,a^3\,b^7}{11}\right )+x^5\,\left (\frac {210\,B\,a^6\,b^4}{13}+\frac {252\,A\,a^5\,b^5}{13}\right )+\frac {B\,b^{10}\,x^{11}}{7}}{x^{18}} \]
-(x*((B*a^10)/17 + (10*A*a^9*b)/17) + (A*a^10)/18 + x^9*(5*B*a^2*b^8 + (10 *A*a*b^9)/9) + x^2*((45*A*a^8*b^2)/16 + (5*B*a^9*b)/8) + x^10*((A*b^10)/8 + (5*B*a*b^9)/4) + x^3*(8*A*a^7*b^3 + 3*B*a^8*b^2) + x^8*((9*A*a^2*b^8)/2 + 12*B*a^3*b^7) + x^6*((35*A*a^4*b^6)/2 + 21*B*a^5*b^5) + x^4*(15*A*a^6*b^ 4 + (60*B*a^7*b^3)/7) + x^7*((120*A*a^3*b^7)/11 + (210*B*a^4*b^6)/11) + x^ 5*((252*A*a^5*b^5)/13 + (210*B*a^6*b^4)/13) + (B*b^10*x^11)/7)/x^18