Integrand size = 11, antiderivative size = 130 \[ \int \frac {(a+b x)^{10}}{x^{19}} \, dx=-\frac {a^{10}}{18 x^{18}}-\frac {10 a^9 b}{17 x^{17}}-\frac {45 a^8 b^2}{16 x^{16}}-\frac {8 a^7 b^3}{x^{15}}-\frac {15 a^6 b^4}{x^{14}}-\frac {252 a^5 b^5}{13 x^{13}}-\frac {35 a^4 b^6}{2 x^{12}}-\frac {120 a^3 b^7}{11 x^{11}}-\frac {9 a^2 b^8}{2 x^{10}}-\frac {10 a b^9}{9 x^9}-\frac {b^{10}}{8 x^8} \] Output:
-1/18*a^10/x^18-10/17*a^9*b/x^17-45/16*a^8*b^2/x^16-8*a^7*b^3/x^15-15*a^6* b^4/x^14-252/13*a^5*b^5/x^13-35/2*a^4*b^6/x^12-120/11*a^3*b^7/x^11-9/2*a^2 *b^8/x^10-10/9*a*b^9/x^9-1/8*b^10/x^8
Time = 0.00 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10}}{x^{19}} \, dx=-\frac {a^{10}}{18 x^{18}}-\frac {10 a^9 b}{17 x^{17}}-\frac {45 a^8 b^2}{16 x^{16}}-\frac {8 a^7 b^3}{x^{15}}-\frac {15 a^6 b^4}{x^{14}}-\frac {252 a^5 b^5}{13 x^{13}}-\frac {35 a^4 b^6}{2 x^{12}}-\frac {120 a^3 b^7}{11 x^{11}}-\frac {9 a^2 b^8}{2 x^{10}}-\frac {10 a b^9}{9 x^9}-\frac {b^{10}}{8 x^8} \] Input:
Integrate[(a + b*x)^10/x^19,x]
Output:
-1/18*a^10/x^18 - (10*a^9*b)/(17*x^17) - (45*a^8*b^2)/(16*x^16) - (8*a^7*b ^3)/x^15 - (15*a^6*b^4)/x^14 - (252*a^5*b^5)/(13*x^13) - (35*a^4*b^6)/(2*x ^12) - (120*a^3*b^7)/(11*x^11) - (9*a^2*b^8)/(2*x^10) - (10*a*b^9)/(9*x^9) - b^10/(8*x^8)
Time = 0.22 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {53, 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.
\(\displaystyle \int \frac {(a+b x)^{10}}{x^{19}} \, dx\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \int \left (\frac {a^{10}}{x^{19}}+\frac {10 a^9 b}{x^{18}}+\frac {45 a^8 b^2}{x^{17}}+\frac {120 a^7 b^3}{x^{16}}+\frac {210 a^6 b^4}{x^{15}}+\frac {252 a^5 b^5}{x^{14}}+\frac {210 a^4 b^6}{x^{13}}+\frac {120 a^3 b^7}{x^{12}}+\frac {45 a^2 b^8}{x^{11}}+\frac {10 a b^9}{x^{10}}+\frac {b^{10}}{x^9}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^{10}}{18 x^{18}}-\frac {10 a^9 b}{17 x^{17}}-\frac {45 a^8 b^2}{16 x^{16}}-\frac {8 a^7 b^3}{x^{15}}-\frac {15 a^6 b^4}{x^{14}}-\frac {252 a^5 b^5}{13 x^{13}}-\frac {35 a^4 b^6}{2 x^{12}}-\frac {120 a^3 b^7}{11 x^{11}}-\frac {9 a^2 b^8}{2 x^{10}}-\frac {10 a b^9}{9 x^9}-\frac {b^{10}}{8 x^8}\) |
Input:
Int[(a + b*x)^10/x^19,x]
Output:
-1/18*a^10/x^18 - (10*a^9*b)/(17*x^17) - (45*a^8*b^2)/(16*x^16) - (8*a^7*b ^3)/x^15 - (15*a^6*b^4)/x^14 - (252*a^5*b^5)/(13*x^13) - (35*a^4*b^6)/(2*x ^12) - (120*a^3*b^7)/(11*x^11) - (9*a^2*b^8)/(2*x^10) - (10*a*b^9)/(9*x^9) - b^10/(8*x^8)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Time = 0.05 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.86
method | result | size |
norman | \(\frac {-\frac {1}{18} a^{10}-\frac {10}{17} a^{9} b x -\frac {45}{16} a^{8} b^{2} x^{2}-8 a^{7} b^{3} x^{3}-15 a^{6} b^{4} x^{4}-\frac {252}{13} a^{5} b^{5} x^{5}-\frac {35}{2} a^{4} b^{6} x^{6}-\frac {120}{11} a^{3} b^{7} x^{7}-\frac {9}{2} a^{2} b^{8} x^{8}-\frac {10}{9} a \,b^{9} x^{9}-\frac {1}{8} b^{10} x^{10}}{x^{18}}\) | \(112\) |
risch | \(\frac {-\frac {1}{18} a^{10}-\frac {10}{17} a^{9} b x -\frac {45}{16} a^{8} b^{2} x^{2}-8 a^{7} b^{3} x^{3}-15 a^{6} b^{4} x^{4}-\frac {252}{13} a^{5} b^{5} x^{5}-\frac {35}{2} a^{4} b^{6} x^{6}-\frac {120}{11} a^{3} b^{7} x^{7}-\frac {9}{2} a^{2} b^{8} x^{8}-\frac {10}{9} a \,b^{9} x^{9}-\frac {1}{8} b^{10} x^{10}}{x^{18}}\) | \(112\) |
gosper | \(-\frac {43758 b^{10} x^{10}+388960 a \,b^{9} x^{9}+1575288 a^{2} b^{8} x^{8}+3818880 a^{3} b^{7} x^{7}+6126120 a^{4} b^{6} x^{6}+6785856 a^{5} b^{5} x^{5}+5250960 a^{6} b^{4} x^{4}+2800512 a^{7} b^{3} x^{3}+984555 a^{8} b^{2} x^{2}+205920 a^{9} b x +19448 a^{10}}{350064 x^{18}}\) | \(113\) |
default | \(-\frac {a^{10}}{18 x^{18}}-\frac {10 a^{9} b}{17 x^{17}}-\frac {45 a^{8} b^{2}}{16 x^{16}}-\frac {8 a^{7} b^{3}}{x^{15}}-\frac {15 a^{6} b^{4}}{x^{14}}-\frac {252 a^{5} b^{5}}{13 x^{13}}-\frac {35 a^{4} b^{6}}{2 x^{12}}-\frac {120 a^{3} b^{7}}{11 x^{11}}-\frac {9 a^{2} b^{8}}{2 x^{10}}-\frac {10 a \,b^{9}}{9 x^{9}}-\frac {b^{10}}{8 x^{8}}\) | \(113\) |
parallelrisch | \(\frac {-43758 b^{10} x^{10}-388960 a \,b^{9} x^{9}-1575288 a^{2} b^{8} x^{8}-3818880 a^{3} b^{7} x^{7}-6126120 a^{4} b^{6} x^{6}-6785856 a^{5} b^{5} x^{5}-5250960 a^{6} b^{4} x^{4}-2800512 a^{7} b^{3} x^{3}-984555 a^{8} b^{2} x^{2}-205920 a^{9} b x -19448 a^{10}}{350064 x^{18}}\) | \(113\) |
orering | \(-\frac {43758 b^{10} x^{10}+388960 a \,b^{9} x^{9}+1575288 a^{2} b^{8} x^{8}+3818880 a^{3} b^{7} x^{7}+6126120 a^{4} b^{6} x^{6}+6785856 a^{5} b^{5} x^{5}+5250960 a^{6} b^{4} x^{4}+2800512 a^{7} b^{3} x^{3}+984555 a^{8} b^{2} x^{2}+205920 a^{9} b x +19448 a^{10}}{350064 x^{18}}\) | \(113\) |
Input:
int((b*x+a)^10/x^19,x,method=_RETURNVERBOSE)
Output:
1/x^18*(-1/18*a^10-10/17*a^9*b*x-45/16*a^8*b^2*x^2-8*a^7*b^3*x^3-15*a^6*b^ 4*x^4-252/13*a^5*b^5*x^5-35/2*a^4*b^6*x^6-120/11*a^3*b^7*x^7-9/2*a^2*b^8*x ^8-10/9*a*b^9*x^9-1/8*b^10*x^10)
Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^{10}}{x^{19}} \, dx=-\frac {43758 \, b^{10} x^{10} + 388960 \, a b^{9} x^{9} + 1575288 \, a^{2} b^{8} x^{8} + 3818880 \, a^{3} b^{7} x^{7} + 6126120 \, a^{4} b^{6} x^{6} + 6785856 \, a^{5} b^{5} x^{5} + 5250960 \, a^{6} b^{4} x^{4} + 2800512 \, a^{7} b^{3} x^{3} + 984555 \, a^{8} b^{2} x^{2} + 205920 \, a^{9} b x + 19448 \, a^{10}}{350064 \, x^{18}} \] Input:
integrate((b*x+a)^10/x^19,x, algorithm="fricas")
Output:
-1/350064*(43758*b^10*x^10 + 388960*a*b^9*x^9 + 1575288*a^2*b^8*x^8 + 3818 880*a^3*b^7*x^7 + 6126120*a^4*b^6*x^6 + 6785856*a^5*b^5*x^5 + 5250960*a^6* b^4*x^4 + 2800512*a^7*b^3*x^3 + 984555*a^8*b^2*x^2 + 205920*a^9*b*x + 1944 8*a^10)/x^18
Time = 0.58 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^{10}}{x^{19}} \, dx=\frac {- 19448 a^{10} - 205920 a^{9} b x - 984555 a^{8} b^{2} x^{2} - 2800512 a^{7} b^{3} x^{3} - 5250960 a^{6} b^{4} x^{4} - 6785856 a^{5} b^{5} x^{5} - 6126120 a^{4} b^{6} x^{6} - 3818880 a^{3} b^{7} x^{7} - 1575288 a^{2} b^{8} x^{8} - 388960 a b^{9} x^{9} - 43758 b^{10} x^{10}}{350064 x^{18}} \] Input:
integrate((b*x+a)**10/x**19,x)
Output:
(-19448*a**10 - 205920*a**9*b*x - 984555*a**8*b**2*x**2 - 2800512*a**7*b** 3*x**3 - 5250960*a**6*b**4*x**4 - 6785856*a**5*b**5*x**5 - 6126120*a**4*b* *6*x**6 - 3818880*a**3*b**7*x**7 - 1575288*a**2*b**8*x**8 - 388960*a*b**9* x**9 - 43758*b**10*x**10)/(350064*x**18)
Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^{10}}{x^{19}} \, dx=-\frac {43758 \, b^{10} x^{10} + 388960 \, a b^{9} x^{9} + 1575288 \, a^{2} b^{8} x^{8} + 3818880 \, a^{3} b^{7} x^{7} + 6126120 \, a^{4} b^{6} x^{6} + 6785856 \, a^{5} b^{5} x^{5} + 5250960 \, a^{6} b^{4} x^{4} + 2800512 \, a^{7} b^{3} x^{3} + 984555 \, a^{8} b^{2} x^{2} + 205920 \, a^{9} b x + 19448 \, a^{10}}{350064 \, x^{18}} \] Input:
integrate((b*x+a)^10/x^19,x, algorithm="maxima")
Output:
-1/350064*(43758*b^10*x^10 + 388960*a*b^9*x^9 + 1575288*a^2*b^8*x^8 + 3818 880*a^3*b^7*x^7 + 6126120*a^4*b^6*x^6 + 6785856*a^5*b^5*x^5 + 5250960*a^6* b^4*x^4 + 2800512*a^7*b^3*x^3 + 984555*a^8*b^2*x^2 + 205920*a^9*b*x + 1944 8*a^10)/x^18
Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^{10}}{x^{19}} \, dx=-\frac {43758 \, b^{10} x^{10} + 388960 \, a b^{9} x^{9} + 1575288 \, a^{2} b^{8} x^{8} + 3818880 \, a^{3} b^{7} x^{7} + 6126120 \, a^{4} b^{6} x^{6} + 6785856 \, a^{5} b^{5} x^{5} + 5250960 \, a^{6} b^{4} x^{4} + 2800512 \, a^{7} b^{3} x^{3} + 984555 \, a^{8} b^{2} x^{2} + 205920 \, a^{9} b x + 19448 \, a^{10}}{350064 \, x^{18}} \] Input:
integrate((b*x+a)^10/x^19,x, algorithm="giac")
Output:
-1/350064*(43758*b^10*x^10 + 388960*a*b^9*x^9 + 1575288*a^2*b^8*x^8 + 3818 880*a^3*b^7*x^7 + 6126120*a^4*b^6*x^6 + 6785856*a^5*b^5*x^5 + 5250960*a^6* b^4*x^4 + 2800512*a^7*b^3*x^3 + 984555*a^8*b^2*x^2 + 205920*a^9*b*x + 1944 8*a^10)/x^18
Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^{10}}{x^{19}} \, dx=-\frac {\frac {a^{10}}{18}+\frac {10\,a^9\,b\,x}{17}+\frac {45\,a^8\,b^2\,x^2}{16}+8\,a^7\,b^3\,x^3+15\,a^6\,b^4\,x^4+\frac {252\,a^5\,b^5\,x^5}{13}+\frac {35\,a^4\,b^6\,x^6}{2}+\frac {120\,a^3\,b^7\,x^7}{11}+\frac {9\,a^2\,b^8\,x^8}{2}+\frac {10\,a\,b^9\,x^9}{9}+\frac {b^{10}\,x^{10}}{8}}{x^{18}} \] Input:
int((a + b*x)^10/x^19,x)
Output:
-(a^10/18 + (b^10*x^10)/8 + (10*a*b^9*x^9)/9 + (45*a^8*b^2*x^2)/16 + 8*a^7 *b^3*x^3 + 15*a^6*b^4*x^4 + (252*a^5*b^5*x^5)/13 + (35*a^4*b^6*x^6)/2 + (1 20*a^3*b^7*x^7)/11 + (9*a^2*b^8*x^8)/2 + (10*a^9*b*x)/17)/x^18
Time = 0.16 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^{10}}{x^{19}} \, dx=\frac {-43758 b^{10} x^{10}-388960 a \,b^{9} x^{9}-1575288 a^{2} b^{8} x^{8}-3818880 a^{3} b^{7} x^{7}-6126120 a^{4} b^{6} x^{6}-6785856 a^{5} b^{5} x^{5}-5250960 a^{6} b^{4} x^{4}-2800512 a^{7} b^{3} x^{3}-984555 a^{8} b^{2} x^{2}-205920 a^{9} b x -19448 a^{10}}{350064 x^{18}} \] Input:
int((b*x+a)^10/x^19,x)
Output:
( - 19448*a**10 - 205920*a**9*b*x - 984555*a**8*b**2*x**2 - 2800512*a**7*b **3*x**3 - 5250960*a**6*b**4*x**4 - 6785856*a**5*b**5*x**5 - 6126120*a**4* b**6*x**6 - 3818880*a**3*b**7*x**7 - 1575288*a**2*b**8*x**8 - 388960*a*b** 9*x**9 - 43758*b**10*x**10)/(350064*x**18)