Integrand size = 11, antiderivative size = 112 \[ \int x^6 (a+b x)^{10} \, dx=\frac {a^6 (a+b x)^{11}}{11 b^7}-\frac {a^5 (a+b x)^{12}}{2 b^7}+\frac {15 a^4 (a+b x)^{13}}{13 b^7}-\frac {10 a^3 (a+b x)^{14}}{7 b^7}+\frac {a^2 (a+b x)^{15}}{b^7}-\frac {3 a (a+b x)^{16}}{8 b^7}+\frac {(a+b x)^{17}}{17 b^7} \]
1/11*a^6*(b*x+a)^11/b^7-1/2*a^5*(b*x+a)^12/b^7+15/13*a^4*(b*x+a)^13/b^7-10 /7*a^3*(b*x+a)^14/b^7+a^2*(b*x+a)^15/b^7-3/8*a*(b*x+a)^16/b^7+1/17*(b*x+a) ^17/b^7
Time = 0.00 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.12 \[ \int x^6 (a+b x)^{10} \, dx=\frac {a^{10} x^7}{7}+\frac {5}{4} a^9 b x^8+5 a^8 b^2 x^9+12 a^7 b^3 x^{10}+\frac {210}{11} a^6 b^4 x^{11}+21 a^5 b^5 x^{12}+\frac {210}{13} a^4 b^6 x^{13}+\frac {60}{7} a^3 b^7 x^{14}+3 a^2 b^8 x^{15}+\frac {5}{8} a b^9 x^{16}+\frac {b^{10} x^{17}}{17} \]
(a^10*x^7)/7 + (5*a^9*b*x^8)/4 + 5*a^8*b^2*x^9 + 12*a^7*b^3*x^10 + (210*a^ 6*b^4*x^11)/11 + 21*a^5*b^5*x^12 + (210*a^4*b^6*x^13)/13 + (60*a^3*b^7*x^1 4)/7 + 3*a^2*b^8*x^15 + (5*a*b^9*x^16)/8 + (b^10*x^17)/17
Time = 0.23 (sec) , antiderivative size = 112, 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.
\(\displaystyle \int x^6 (a+b x)^{10} \, dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (\frac {a^6 (a+b x)^{10}}{b^6}-\frac {6 a^5 (a+b x)^{11}}{b^6}+\frac {15 a^4 (a+b x)^{12}}{b^6}-\frac {20 a^3 (a+b x)^{13}}{b^6}+\frac {15 a^2 (a+b x)^{14}}{b^6}+\frac {(a+b x)^{16}}{b^6}-\frac {6 a (a+b x)^{15}}{b^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^6 (a+b x)^{11}}{11 b^7}-\frac {a^5 (a+b x)^{12}}{2 b^7}+\frac {15 a^4 (a+b x)^{13}}{13 b^7}-\frac {10 a^3 (a+b x)^{14}}{7 b^7}+\frac {a^2 (a+b x)^{15}}{b^7}+\frac {(a+b x)^{17}}{17 b^7}-\frac {3 a (a+b x)^{16}}{8 b^7}\) |
(a^6*(a + b*x)^11)/(11*b^7) - (a^5*(a + b*x)^12)/(2*b^7) + (15*a^4*(a + b* x)^13)/(13*b^7) - (10*a^3*(a + b*x)^14)/(7*b^7) + (a^2*(a + b*x)^15)/b^7 - (3*a*(a + b*x)^16)/(8*b^7) + (a + b*x)^17/(17*b^7)
3.2.28.3.1 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 = 0.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.01
method | result | size |
gosper | \(\frac {1}{7} a^{10} x^{7}+\frac {5}{4} a^{9} b \,x^{8}+5 a^{8} b^{2} x^{9}+12 a^{7} b^{3} x^{10}+\frac {210}{11} a^{6} b^{4} x^{11}+21 a^{5} b^{5} x^{12}+\frac {210}{13} a^{4} b^{6} x^{13}+\frac {60}{7} a^{3} b^{7} x^{14}+3 a^{2} b^{8} x^{15}+\frac {5}{8} a \,b^{9} x^{16}+\frac {1}{17} b^{10} x^{17}\) | \(113\) |
default | \(\frac {1}{7} a^{10} x^{7}+\frac {5}{4} a^{9} b \,x^{8}+5 a^{8} b^{2} x^{9}+12 a^{7} b^{3} x^{10}+\frac {210}{11} a^{6} b^{4} x^{11}+21 a^{5} b^{5} x^{12}+\frac {210}{13} a^{4} b^{6} x^{13}+\frac {60}{7} a^{3} b^{7} x^{14}+3 a^{2} b^{8} x^{15}+\frac {5}{8} a \,b^{9} x^{16}+\frac {1}{17} b^{10} x^{17}\) | \(113\) |
norman | \(\frac {1}{7} a^{10} x^{7}+\frac {5}{4} a^{9} b \,x^{8}+5 a^{8} b^{2} x^{9}+12 a^{7} b^{3} x^{10}+\frac {210}{11} a^{6} b^{4} x^{11}+21 a^{5} b^{5} x^{12}+\frac {210}{13} a^{4} b^{6} x^{13}+\frac {60}{7} a^{3} b^{7} x^{14}+3 a^{2} b^{8} x^{15}+\frac {5}{8} a \,b^{9} x^{16}+\frac {1}{17} b^{10} x^{17}\) | \(113\) |
risch | \(\frac {1}{7} a^{10} x^{7}+\frac {5}{4} a^{9} b \,x^{8}+5 a^{8} b^{2} x^{9}+12 a^{7} b^{3} x^{10}+\frac {210}{11} a^{6} b^{4} x^{11}+21 a^{5} b^{5} x^{12}+\frac {210}{13} a^{4} b^{6} x^{13}+\frac {60}{7} a^{3} b^{7} x^{14}+3 a^{2} b^{8} x^{15}+\frac {5}{8} a \,b^{9} x^{16}+\frac {1}{17} b^{10} x^{17}\) | \(113\) |
parallelrisch | \(\frac {1}{7} a^{10} x^{7}+\frac {5}{4} a^{9} b \,x^{8}+5 a^{8} b^{2} x^{9}+12 a^{7} b^{3} x^{10}+\frac {210}{11} a^{6} b^{4} x^{11}+21 a^{5} b^{5} x^{12}+\frac {210}{13} a^{4} b^{6} x^{13}+\frac {60}{7} a^{3} b^{7} x^{14}+3 a^{2} b^{8} x^{15}+\frac {5}{8} a \,b^{9} x^{16}+\frac {1}{17} b^{10} x^{17}\) | \(113\) |
1/7*a^10*x^7+5/4*a^9*b*x^8+5*a^8*b^2*x^9+12*a^7*b^3*x^10+210/11*a^6*b^4*x^ 11+21*a^5*b^5*x^12+210/13*a^4*b^6*x^13+60/7*a^3*b^7*x^14+3*a^2*b^8*x^15+5/ 8*a*b^9*x^16+1/17*b^10*x^17
Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int x^6 (a+b x)^{10} \, dx=\frac {1}{17} \, b^{10} x^{17} + \frac {5}{8} \, a b^{9} x^{16} + 3 \, a^{2} b^{8} x^{15} + \frac {60}{7} \, a^{3} b^{7} x^{14} + \frac {210}{13} \, a^{4} b^{6} x^{13} + 21 \, a^{5} b^{5} x^{12} + \frac {210}{11} \, a^{6} b^{4} x^{11} + 12 \, a^{7} b^{3} x^{10} + 5 \, a^{8} b^{2} x^{9} + \frac {5}{4} \, a^{9} b x^{8} + \frac {1}{7} \, a^{10} x^{7} \]
1/17*b^10*x^17 + 5/8*a*b^9*x^16 + 3*a^2*b^8*x^15 + 60/7*a^3*b^7*x^14 + 210 /13*a^4*b^6*x^13 + 21*a^5*b^5*x^12 + 210/11*a^6*b^4*x^11 + 12*a^7*b^3*x^10 + 5*a^8*b^2*x^9 + 5/4*a^9*b*x^8 + 1/7*a^10*x^7
Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.14 \[ \int x^6 (a+b x)^{10} \, dx=\frac {a^{10} x^{7}}{7} + \frac {5 a^{9} b x^{8}}{4} + 5 a^{8} b^{2} x^{9} + 12 a^{7} b^{3} x^{10} + \frac {210 a^{6} b^{4} x^{11}}{11} + 21 a^{5} b^{5} x^{12} + \frac {210 a^{4} b^{6} x^{13}}{13} + \frac {60 a^{3} b^{7} x^{14}}{7} + 3 a^{2} b^{8} x^{15} + \frac {5 a b^{9} x^{16}}{8} + \frac {b^{10} x^{17}}{17} \]
a**10*x**7/7 + 5*a**9*b*x**8/4 + 5*a**8*b**2*x**9 + 12*a**7*b**3*x**10 + 2 10*a**6*b**4*x**11/11 + 21*a**5*b**5*x**12 + 210*a**4*b**6*x**13/13 + 60*a **3*b**7*x**14/7 + 3*a**2*b**8*x**15 + 5*a*b**9*x**16/8 + b**10*x**17/17
Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int x^6 (a+b x)^{10} \, dx=\frac {1}{17} \, b^{10} x^{17} + \frac {5}{8} \, a b^{9} x^{16} + 3 \, a^{2} b^{8} x^{15} + \frac {60}{7} \, a^{3} b^{7} x^{14} + \frac {210}{13} \, a^{4} b^{6} x^{13} + 21 \, a^{5} b^{5} x^{12} + \frac {210}{11} \, a^{6} b^{4} x^{11} + 12 \, a^{7} b^{3} x^{10} + 5 \, a^{8} b^{2} x^{9} + \frac {5}{4} \, a^{9} b x^{8} + \frac {1}{7} \, a^{10} x^{7} \]
1/17*b^10*x^17 + 5/8*a*b^9*x^16 + 3*a^2*b^8*x^15 + 60/7*a^3*b^7*x^14 + 210 /13*a^4*b^6*x^13 + 21*a^5*b^5*x^12 + 210/11*a^6*b^4*x^11 + 12*a^7*b^3*x^10 + 5*a^8*b^2*x^9 + 5/4*a^9*b*x^8 + 1/7*a^10*x^7
Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int x^6 (a+b x)^{10} \, dx=\frac {1}{17} \, b^{10} x^{17} + \frac {5}{8} \, a b^{9} x^{16} + 3 \, a^{2} b^{8} x^{15} + \frac {60}{7} \, a^{3} b^{7} x^{14} + \frac {210}{13} \, a^{4} b^{6} x^{13} + 21 \, a^{5} b^{5} x^{12} + \frac {210}{11} \, a^{6} b^{4} x^{11} + 12 \, a^{7} b^{3} x^{10} + 5 \, a^{8} b^{2} x^{9} + \frac {5}{4} \, a^{9} b x^{8} + \frac {1}{7} \, a^{10} x^{7} \]
1/17*b^10*x^17 + 5/8*a*b^9*x^16 + 3*a^2*b^8*x^15 + 60/7*a^3*b^7*x^14 + 210 /13*a^4*b^6*x^13 + 21*a^5*b^5*x^12 + 210/11*a^6*b^4*x^11 + 12*a^7*b^3*x^10 + 5*a^8*b^2*x^9 + 5/4*a^9*b*x^8 + 1/7*a^10*x^7
Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int x^6 (a+b x)^{10} \, dx=\frac {a^{10}\,x^7}{7}+\frac {5\,a^9\,b\,x^8}{4}+5\,a^8\,b^2\,x^9+12\,a^7\,b^3\,x^{10}+\frac {210\,a^6\,b^4\,x^{11}}{11}+21\,a^5\,b^5\,x^{12}+\frac {210\,a^4\,b^6\,x^{13}}{13}+\frac {60\,a^3\,b^7\,x^{14}}{7}+3\,a^2\,b^8\,x^{15}+\frac {5\,a\,b^9\,x^{16}}{8}+\frac {b^{10}\,x^{17}}{17} \]
(a^10*x^7)/7 + (b^10*x^17)/17 + (5*a^9*b*x^8)/4 + (5*a*b^9*x^16)/8 + 5*a^8 *b^2*x^9 + 12*a^7*b^3*x^10 + (210*a^6*b^4*x^11)/11 + 21*a^5*b^5*x^12 + (21 0*a^4*b^6*x^13)/13 + (60*a^3*b^7*x^14)/7 + 3*a^2*b^8*x^15
Time = 0.00 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int x^6 (a+b x)^{10} \, dx=\frac {x^{7} \left (8008 b^{10} x^{10}+85085 a \,b^{9} x^{9}+408408 a^{2} b^{8} x^{8}+1166880 a^{3} b^{7} x^{7}+2199120 a^{4} b^{6} x^{6}+2858856 a^{5} b^{5} x^{5}+2598960 a^{6} b^{4} x^{4}+1633632 a^{7} b^{3} x^{3}+680680 a^{8} b^{2} x^{2}+170170 a^{9} b x +19448 a^{10}\right )}{136136} \]
int(x**6*(a**10 + 10*a**9*b*x + 45*a**8*b**2*x**2 + 120*a**7*b**3*x**3 + 2 10*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)