Integrand size = 11, antiderivative size = 132 \[ \int x^{11} (a+b x)^{10} \, dx=\frac {a^{10} x^{12}}{12}+\frac {10}{13} a^9 b x^{13}+\frac {45}{14} a^8 b^2 x^{14}+8 a^7 b^3 x^{15}+\frac {105}{8} a^6 b^4 x^{16}+\frac {252}{17} a^5 b^5 x^{17}+\frac {35}{3} a^4 b^6 x^{18}+\frac {120}{19} a^3 b^7 x^{19}+\frac {9}{4} a^2 b^8 x^{20}+\frac {10}{21} a b^9 x^{21}+\frac {b^{10} x^{22}}{22} \]
1/12*a^10*x^12+10/13*a^9*b*x^13+45/14*a^8*b^2*x^14+8*a^7*b^3*x^15+105/8*a^ 6*b^4*x^16+252/17*a^5*b^5*x^17+35/3*a^4*b^6*x^18+120/19*a^3*b^7*x^19+9/4*a ^2*b^8*x^20+10/21*a*b^9*x^21+1/22*b^10*x^22
Time = 0.00 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00 \[ \int x^{11} (a+b x)^{10} \, dx=\frac {a^{10} x^{12}}{12}+\frac {10}{13} a^9 b x^{13}+\frac {45}{14} a^8 b^2 x^{14}+8 a^7 b^3 x^{15}+\frac {105}{8} a^6 b^4 x^{16}+\frac {252}{17} a^5 b^5 x^{17}+\frac {35}{3} a^4 b^6 x^{18}+\frac {120}{19} a^3 b^7 x^{19}+\frac {9}{4} a^2 b^8 x^{20}+\frac {10}{21} a b^9 x^{21}+\frac {b^{10} x^{22}}{22} \]
(a^10*x^12)/12 + (10*a^9*b*x^13)/13 + (45*a^8*b^2*x^14)/14 + 8*a^7*b^3*x^1 5 + (105*a^6*b^4*x^16)/8 + (252*a^5*b^5*x^17)/17 + (35*a^4*b^6*x^18)/3 + ( 120*a^3*b^7*x^19)/19 + (9*a^2*b^8*x^20)/4 + (10*a*b^9*x^21)/21 + (b^10*x^2 2)/22
Time = 0.25 (sec) , antiderivative size = 132, 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 = {49, 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.
\(\displaystyle \int x^{11} (a+b x)^{10} \, dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (a^{10} x^{11}+10 a^9 b x^{12}+45 a^8 b^2 x^{13}+120 a^7 b^3 x^{14}+210 a^6 b^4 x^{15}+252 a^5 b^5 x^{16}+210 a^4 b^6 x^{17}+120 a^3 b^7 x^{18}+45 a^2 b^8 x^{19}+10 a b^9 x^{20}+b^{10} x^{21}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^{10} x^{12}}{12}+\frac {10}{13} a^9 b x^{13}+\frac {45}{14} a^8 b^2 x^{14}+8 a^7 b^3 x^{15}+\frac {105}{8} a^6 b^4 x^{16}+\frac {252}{17} a^5 b^5 x^{17}+\frac {35}{3} a^4 b^6 x^{18}+\frac {120}{19} a^3 b^7 x^{19}+\frac {9}{4} a^2 b^8 x^{20}+\frac {10}{21} a b^9 x^{21}+\frac {b^{10} x^{22}}{22}\) |
(a^10*x^12)/12 + (10*a^9*b*x^13)/13 + (45*a^8*b^2*x^14)/14 + 8*a^7*b^3*x^1 5 + (105*a^6*b^4*x^16)/8 + (252*a^5*b^5*x^17)/17 + (35*a^4*b^6*x^18)/3 + ( 120*a^3*b^7*x^19)/19 + (9*a^2*b^8*x^20)/4 + (10*a*b^9*x^21)/21 + (b^10*x^2 2)/22
3.2.23.3.1 Defintions of rubi rules used
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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Time = 0.17 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86
method | result | size |
gosper | \(\frac {1}{12} a^{10} x^{12}+\frac {10}{13} a^{9} b \,x^{13}+\frac {45}{14} a^{8} b^{2} x^{14}+8 a^{7} b^{3} x^{15}+\frac {105}{8} a^{6} b^{4} x^{16}+\frac {252}{17} a^{5} b^{5} x^{17}+\frac {35}{3} a^{4} b^{6} x^{18}+\frac {120}{19} a^{3} b^{7} x^{19}+\frac {9}{4} a^{2} b^{8} x^{20}+\frac {10}{21} a \,b^{9} x^{21}+\frac {1}{22} b^{10} x^{22}\) | \(113\) |
default | \(\frac {1}{12} a^{10} x^{12}+\frac {10}{13} a^{9} b \,x^{13}+\frac {45}{14} a^{8} b^{2} x^{14}+8 a^{7} b^{3} x^{15}+\frac {105}{8} a^{6} b^{4} x^{16}+\frac {252}{17} a^{5} b^{5} x^{17}+\frac {35}{3} a^{4} b^{6} x^{18}+\frac {120}{19} a^{3} b^{7} x^{19}+\frac {9}{4} a^{2} b^{8} x^{20}+\frac {10}{21} a \,b^{9} x^{21}+\frac {1}{22} b^{10} x^{22}\) | \(113\) |
norman | \(\frac {1}{12} a^{10} x^{12}+\frac {10}{13} a^{9} b \,x^{13}+\frac {45}{14} a^{8} b^{2} x^{14}+8 a^{7} b^{3} x^{15}+\frac {105}{8} a^{6} b^{4} x^{16}+\frac {252}{17} a^{5} b^{5} x^{17}+\frac {35}{3} a^{4} b^{6} x^{18}+\frac {120}{19} a^{3} b^{7} x^{19}+\frac {9}{4} a^{2} b^{8} x^{20}+\frac {10}{21} a \,b^{9} x^{21}+\frac {1}{22} b^{10} x^{22}\) | \(113\) |
risch | \(\frac {1}{12} a^{10} x^{12}+\frac {10}{13} a^{9} b \,x^{13}+\frac {45}{14} a^{8} b^{2} x^{14}+8 a^{7} b^{3} x^{15}+\frac {105}{8} a^{6} b^{4} x^{16}+\frac {252}{17} a^{5} b^{5} x^{17}+\frac {35}{3} a^{4} b^{6} x^{18}+\frac {120}{19} a^{3} b^{7} x^{19}+\frac {9}{4} a^{2} b^{8} x^{20}+\frac {10}{21} a \,b^{9} x^{21}+\frac {1}{22} b^{10} x^{22}\) | \(113\) |
parallelrisch | \(\frac {1}{12} a^{10} x^{12}+\frac {10}{13} a^{9} b \,x^{13}+\frac {45}{14} a^{8} b^{2} x^{14}+8 a^{7} b^{3} x^{15}+\frac {105}{8} a^{6} b^{4} x^{16}+\frac {252}{17} a^{5} b^{5} x^{17}+\frac {35}{3} a^{4} b^{6} x^{18}+\frac {120}{19} a^{3} b^{7} x^{19}+\frac {9}{4} a^{2} b^{8} x^{20}+\frac {10}{21} a \,b^{9} x^{21}+\frac {1}{22} b^{10} x^{22}\) | \(113\) |
1/12*a^10*x^12+10/13*a^9*b*x^13+45/14*a^8*b^2*x^14+8*a^7*b^3*x^15+105/8*a^ 6*b^4*x^16+252/17*a^5*b^5*x^17+35/3*a^4*b^6*x^18+120/19*a^3*b^7*x^19+9/4*a ^2*b^8*x^20+10/21*a*b^9*x^21+1/22*b^10*x^22
Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^{11} (a+b x)^{10} \, dx=\frac {1}{22} \, b^{10} x^{22} + \frac {10}{21} \, a b^{9} x^{21} + \frac {9}{4} \, a^{2} b^{8} x^{20} + \frac {120}{19} \, a^{3} b^{7} x^{19} + \frac {35}{3} \, a^{4} b^{6} x^{18} + \frac {252}{17} \, a^{5} b^{5} x^{17} + \frac {105}{8} \, a^{6} b^{4} x^{16} + 8 \, a^{7} b^{3} x^{15} + \frac {45}{14} \, a^{8} b^{2} x^{14} + \frac {10}{13} \, a^{9} b x^{13} + \frac {1}{12} \, a^{10} x^{12} \]
1/22*b^10*x^22 + 10/21*a*b^9*x^21 + 9/4*a^2*b^8*x^20 + 120/19*a^3*b^7*x^19 + 35/3*a^4*b^6*x^18 + 252/17*a^5*b^5*x^17 + 105/8*a^6*b^4*x^16 + 8*a^7*b^ 3*x^15 + 45/14*a^8*b^2*x^14 + 10/13*a^9*b*x^13 + 1/12*a^10*x^12
Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.01 \[ \int x^{11} (a+b x)^{10} \, dx=\frac {a^{10} x^{12}}{12} + \frac {10 a^{9} b x^{13}}{13} + \frac {45 a^{8} b^{2} x^{14}}{14} + 8 a^{7} b^{3} x^{15} + \frac {105 a^{6} b^{4} x^{16}}{8} + \frac {252 a^{5} b^{5} x^{17}}{17} + \frac {35 a^{4} b^{6} x^{18}}{3} + \frac {120 a^{3} b^{7} x^{19}}{19} + \frac {9 a^{2} b^{8} x^{20}}{4} + \frac {10 a b^{9} x^{21}}{21} + \frac {b^{10} x^{22}}{22} \]
a**10*x**12/12 + 10*a**9*b*x**13/13 + 45*a**8*b**2*x**14/14 + 8*a**7*b**3* x**15 + 105*a**6*b**4*x**16/8 + 252*a**5*b**5*x**17/17 + 35*a**4*b**6*x**1 8/3 + 120*a**3*b**7*x**19/19 + 9*a**2*b**8*x**20/4 + 10*a*b**9*x**21/21 + b**10*x**22/22
Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^{11} (a+b x)^{10} \, dx=\frac {1}{22} \, b^{10} x^{22} + \frac {10}{21} \, a b^{9} x^{21} + \frac {9}{4} \, a^{2} b^{8} x^{20} + \frac {120}{19} \, a^{3} b^{7} x^{19} + \frac {35}{3} \, a^{4} b^{6} x^{18} + \frac {252}{17} \, a^{5} b^{5} x^{17} + \frac {105}{8} \, a^{6} b^{4} x^{16} + 8 \, a^{7} b^{3} x^{15} + \frac {45}{14} \, a^{8} b^{2} x^{14} + \frac {10}{13} \, a^{9} b x^{13} + \frac {1}{12} \, a^{10} x^{12} \]
1/22*b^10*x^22 + 10/21*a*b^9*x^21 + 9/4*a^2*b^8*x^20 + 120/19*a^3*b^7*x^19 + 35/3*a^4*b^6*x^18 + 252/17*a^5*b^5*x^17 + 105/8*a^6*b^4*x^16 + 8*a^7*b^ 3*x^15 + 45/14*a^8*b^2*x^14 + 10/13*a^9*b*x^13 + 1/12*a^10*x^12
Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^{11} (a+b x)^{10} \, dx=\frac {1}{22} \, b^{10} x^{22} + \frac {10}{21} \, a b^{9} x^{21} + \frac {9}{4} \, a^{2} b^{8} x^{20} + \frac {120}{19} \, a^{3} b^{7} x^{19} + \frac {35}{3} \, a^{4} b^{6} x^{18} + \frac {252}{17} \, a^{5} b^{5} x^{17} + \frac {105}{8} \, a^{6} b^{4} x^{16} + 8 \, a^{7} b^{3} x^{15} + \frac {45}{14} \, a^{8} b^{2} x^{14} + \frac {10}{13} \, a^{9} b x^{13} + \frac {1}{12} \, a^{10} x^{12} \]
1/22*b^10*x^22 + 10/21*a*b^9*x^21 + 9/4*a^2*b^8*x^20 + 120/19*a^3*b^7*x^19 + 35/3*a^4*b^6*x^18 + 252/17*a^5*b^5*x^17 + 105/8*a^6*b^4*x^16 + 8*a^7*b^ 3*x^15 + 45/14*a^8*b^2*x^14 + 10/13*a^9*b*x^13 + 1/12*a^10*x^12
Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^{11} (a+b x)^{10} \, dx=\frac {a^{10}\,x^{12}}{12}+\frac {10\,a^9\,b\,x^{13}}{13}+\frac {45\,a^8\,b^2\,x^{14}}{14}+8\,a^7\,b^3\,x^{15}+\frac {105\,a^6\,b^4\,x^{16}}{8}+\frac {252\,a^5\,b^5\,x^{17}}{17}+\frac {35\,a^4\,b^6\,x^{18}}{3}+\frac {120\,a^3\,b^7\,x^{19}}{19}+\frac {9\,a^2\,b^8\,x^{20}}{4}+\frac {10\,a\,b^9\,x^{21}}{21}+\frac {b^{10}\,x^{22}}{22} \]
(a^10*x^12)/12 + (b^10*x^22)/22 + (10*a^9*b*x^13)/13 + (10*a*b^9*x^21)/21 + (45*a^8*b^2*x^14)/14 + 8*a^7*b^3*x^15 + (105*a^6*b^4*x^16)/8 + (252*a^5* b^5*x^17)/17 + (35*a^4*b^6*x^18)/3 + (120*a^3*b^7*x^19)/19 + (9*a^2*b^8*x^ 20)/4
Time = 0.00 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^{11} (a+b x)^{10} \, dx=\frac {x^{12} \left (352716 b^{10} x^{10}+3695120 a \,b^{9} x^{9}+17459442 a^{2} b^{8} x^{8}+49008960 a^{3} b^{7} x^{7}+90530440 a^{4} b^{6} x^{6}+115026912 a^{5} b^{5} x^{5}+101846745 a^{6} b^{4} x^{4}+62078016 a^{7} b^{3} x^{3}+24942060 a^{8} b^{2} x^{2}+5969040 a^{9} b x +646646 a^{10}\right )}{7759752} \]
int(x**11*(a**10 + 10*a**9*b*x + 45*a**8*b**2*x**2 + 120*a**7*b**3*x**3 + 210*a**6*b**4*x**4 + 252*a**5*b**5*x**5 + 210*a**4*b**6*x**6 + 120*a**3*b* *7*x**7 + 45*a**2*b**8*x**8 + 10*a*b**9*x**9 + b**10*x**10),x)