Integrand size = 13, antiderivative size = 64 \[ \int \left (a+\frac {b}{x}\right )^8 x^{11} \, dx=-\frac {b^3 (b+a x)^9}{9 a^4}+\frac {3 b^2 (b+a x)^{10}}{10 a^4}-\frac {3 b (b+a x)^{11}}{11 a^4}+\frac {(b+a x)^{12}}{12 a^4} \] Output:
-1/9*b^3*(a*x+b)^9/a^4+3/10*b^2*(a*x+b)^10/a^4-3/11*b*(a*x+b)^11/a^4+1/12* (a*x+b)^12/a^4
Time = 0.00 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.66 \[ \int \left (a+\frac {b}{x}\right )^8 x^{11} \, dx=\frac {b^8 x^4}{4}+\frac {8}{5} a b^7 x^5+\frac {14}{3} a^2 b^6 x^6+8 a^3 b^5 x^7+\frac {35}{4} a^4 b^4 x^8+\frac {56}{9} a^5 b^3 x^9+\frac {14}{5} a^6 b^2 x^{10}+\frac {8}{11} a^7 b x^{11}+\frac {a^8 x^{12}}{12} \] Input:
Integrate[(a + b/x)^8*x^11,x]
Output:
(b^8*x^4)/4 + (8*a*b^7*x^5)/5 + (14*a^2*b^6*x^6)/3 + 8*a^3*b^5*x^7 + (35*a ^4*b^4*x^8)/4 + (56*a^5*b^3*x^9)/9 + (14*a^6*b^2*x^10)/5 + (8*a^7*b*x^11)/ 11 + (a^8*x^12)/12
Time = 0.31 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {795, 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.
\(\displaystyle \int x^{11} \left (a+\frac {b}{x}\right )^8 \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int x^3 (a x+b)^8dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (-\frac {b^3 (a x+b)^8}{a^3}+\frac {3 b^2 (a x+b)^9}{a^3}+\frac {(a x+b)^{11}}{a^3}-\frac {3 b (a x+b)^{10}}{a^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b^3 (a x+b)^9}{9 a^4}+\frac {3 b^2 (a x+b)^{10}}{10 a^4}+\frac {(a x+b)^{12}}{12 a^4}-\frac {3 b (a x+b)^{11}}{11 a^4}\) |
Input:
Int[(a + b/x)^8*x^11,x]
Output:
-1/9*(b^3*(b + a*x)^9)/a^4 + (3*b^2*(b + a*x)^10)/(10*a^4) - (3*b*(b + a*x )^11)/(11*a^4) + (b + a*x)^12/(12*a^4)
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.42
method | result | size |
gosper | \(\frac {x^{4} \left (165 a^{8} x^{8}+1440 a^{7} b \,x^{7}+5544 a^{6} b^{2} x^{6}+12320 a^{5} b^{3} x^{5}+17325 a^{4} x^{4} b^{4}+15840 a^{3} b^{5} x^{3}+9240 a^{2} b^{6} x^{2}+3168 a \,b^{7} x +495 b^{8}\right )}{1980}\) | \(91\) |
default | \(\frac {1}{12} a^{8} x^{12}+\frac {8}{11} a^{7} b \,x^{11}+\frac {14}{5} a^{6} b^{2} x^{10}+\frac {56}{9} a^{5} b^{3} x^{9}+\frac {35}{4} a^{4} b^{4} x^{8}+8 a^{3} b^{5} x^{7}+\frac {14}{3} a^{2} x^{6} b^{6}+\frac {8}{5} a \,b^{7} x^{5}+\frac {1}{4} b^{8} x^{4}\) | \(91\) |
risch | \(\frac {1}{12} a^{8} x^{12}+\frac {8}{11} a^{7} b \,x^{11}+\frac {14}{5} a^{6} b^{2} x^{10}+\frac {56}{9} a^{5} b^{3} x^{9}+\frac {35}{4} a^{4} b^{4} x^{8}+8 a^{3} b^{5} x^{7}+\frac {14}{3} a^{2} x^{6} b^{6}+\frac {8}{5} a \,b^{7} x^{5}+\frac {1}{4} b^{8} x^{4}\) | \(91\) |
parallelrisch | \(\frac {1}{12} a^{8} x^{12}+\frac {8}{11} a^{7} b \,x^{11}+\frac {14}{5} a^{6} b^{2} x^{10}+\frac {56}{9} a^{5} b^{3} x^{9}+\frac {35}{4} a^{4} b^{4} x^{8}+8 a^{3} b^{5} x^{7}+\frac {14}{3} a^{2} x^{6} b^{6}+\frac {8}{5} a \,b^{7} x^{5}+\frac {1}{4} b^{8} x^{4}\) | \(91\) |
norman | \(\frac {\frac {1}{12} a^{8} x^{19}+\frac {1}{4} b^{8} x^{11}+\frac {8}{5} a \,b^{7} x^{12}+\frac {14}{3} a^{2} b^{6} x^{13}+8 a^{3} b^{5} x^{14}+\frac {35}{4} a^{4} b^{4} x^{15}+\frac {56}{9} a^{5} b^{3} x^{16}+\frac {14}{5} a^{6} b^{2} x^{17}+\frac {8}{11} a^{7} b \,x^{18}}{x^{7}}\) | \(95\) |
orering | \(\frac {x^{12} \left (165 a^{8} x^{8}+1440 a^{7} b \,x^{7}+5544 a^{6} b^{2} x^{6}+12320 a^{5} b^{3} x^{5}+17325 a^{4} x^{4} b^{4}+15840 a^{3} b^{5} x^{3}+9240 a^{2} b^{6} x^{2}+3168 a \,b^{7} x +495 b^{8}\right ) \left (a +\frac {b}{x}\right )^{8}}{1980 \left (a x +b \right )^{8}}\) | \(107\) |
Input:
int((a+b/x)^8*x^11,x,method=_RETURNVERBOSE)
Output:
1/1980*x^4*(165*a^8*x^8+1440*a^7*b*x^7+5544*a^6*b^2*x^6+12320*a^5*b^3*x^5+ 17325*a^4*b^4*x^4+15840*a^3*b^5*x^3+9240*a^2*b^6*x^2+3168*a*b^7*x+495*b^8)
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.41 \[ \int \left (a+\frac {b}{x}\right )^8 x^{11} \, dx=\frac {1}{12} \, a^{8} x^{12} + \frac {8}{11} \, a^{7} b x^{11} + \frac {14}{5} \, a^{6} b^{2} x^{10} + \frac {56}{9} \, a^{5} b^{3} x^{9} + \frac {35}{4} \, a^{4} b^{4} x^{8} + 8 \, a^{3} b^{5} x^{7} + \frac {14}{3} \, a^{2} b^{6} x^{6} + \frac {8}{5} \, a b^{7} x^{5} + \frac {1}{4} \, b^{8} x^{4} \] Input:
integrate((a+b/x)^8*x^11,x, algorithm="fricas")
Output:
1/12*a^8*x^12 + 8/11*a^7*b*x^11 + 14/5*a^6*b^2*x^10 + 56/9*a^5*b^3*x^9 + 3 5/4*a^4*b^4*x^8 + 8*a^3*b^5*x^7 + 14/3*a^2*b^6*x^6 + 8/5*a*b^7*x^5 + 1/4*b ^8*x^4
Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.64 \[ \int \left (a+\frac {b}{x}\right )^8 x^{11} \, dx=\frac {a^{8} x^{12}}{12} + \frac {8 a^{7} b x^{11}}{11} + \frac {14 a^{6} b^{2} x^{10}}{5} + \frac {56 a^{5} b^{3} x^{9}}{9} + \frac {35 a^{4} b^{4} x^{8}}{4} + 8 a^{3} b^{5} x^{7} + \frac {14 a^{2} b^{6} x^{6}}{3} + \frac {8 a b^{7} x^{5}}{5} + \frac {b^{8} x^{4}}{4} \] Input:
integrate((a+b/x)**8*x**11,x)
Output:
a**8*x**12/12 + 8*a**7*b*x**11/11 + 14*a**6*b**2*x**10/5 + 56*a**5*b**3*x* *9/9 + 35*a**4*b**4*x**8/4 + 8*a**3*b**5*x**7 + 14*a**2*b**6*x**6/3 + 8*a* b**7*x**5/5 + b**8*x**4/4
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.41 \[ \int \left (a+\frac {b}{x}\right )^8 x^{11} \, dx=\frac {1}{12} \, a^{8} x^{12} + \frac {8}{11} \, a^{7} b x^{11} + \frac {14}{5} \, a^{6} b^{2} x^{10} + \frac {56}{9} \, a^{5} b^{3} x^{9} + \frac {35}{4} \, a^{4} b^{4} x^{8} + 8 \, a^{3} b^{5} x^{7} + \frac {14}{3} \, a^{2} b^{6} x^{6} + \frac {8}{5} \, a b^{7} x^{5} + \frac {1}{4} \, b^{8} x^{4} \] Input:
integrate((a+b/x)^8*x^11,x, algorithm="maxima")
Output:
1/12*a^8*x^12 + 8/11*a^7*b*x^11 + 14/5*a^6*b^2*x^10 + 56/9*a^5*b^3*x^9 + 3 5/4*a^4*b^4*x^8 + 8*a^3*b^5*x^7 + 14/3*a^2*b^6*x^6 + 8/5*a*b^7*x^5 + 1/4*b ^8*x^4
Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.41 \[ \int \left (a+\frac {b}{x}\right )^8 x^{11} \, dx=\frac {1}{12} \, a^{8} x^{12} + \frac {8}{11} \, a^{7} b x^{11} + \frac {14}{5} \, a^{6} b^{2} x^{10} + \frac {56}{9} \, a^{5} b^{3} x^{9} + \frac {35}{4} \, a^{4} b^{4} x^{8} + 8 \, a^{3} b^{5} x^{7} + \frac {14}{3} \, a^{2} b^{6} x^{6} + \frac {8}{5} \, a b^{7} x^{5} + \frac {1}{4} \, b^{8} x^{4} \] Input:
integrate((a+b/x)^8*x^11,x, algorithm="giac")
Output:
1/12*a^8*x^12 + 8/11*a^7*b*x^11 + 14/5*a^6*b^2*x^10 + 56/9*a^5*b^3*x^9 + 3 5/4*a^4*b^4*x^8 + 8*a^3*b^5*x^7 + 14/3*a^2*b^6*x^6 + 8/5*a*b^7*x^5 + 1/4*b ^8*x^4
Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.41 \[ \int \left (a+\frac {b}{x}\right )^8 x^{11} \, dx=\frac {a^8\,x^{12}}{12}+\frac {8\,a^7\,b\,x^{11}}{11}+\frac {14\,a^6\,b^2\,x^{10}}{5}+\frac {56\,a^5\,b^3\,x^9}{9}+\frac {35\,a^4\,b^4\,x^8}{4}+8\,a^3\,b^5\,x^7+\frac {14\,a^2\,b^6\,x^6}{3}+\frac {8\,a\,b^7\,x^5}{5}+\frac {b^8\,x^4}{4} \] Input:
int(x^11*(a + b/x)^8,x)
Output:
(a^8*x^12)/12 + (b^8*x^4)/4 + (8*a*b^7*x^5)/5 + (8*a^7*b*x^11)/11 + (14*a^ 2*b^6*x^6)/3 + 8*a^3*b^5*x^7 + (35*a^4*b^4*x^8)/4 + (56*a^5*b^3*x^9)/9 + ( 14*a^6*b^2*x^10)/5
Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.41 \[ \int \left (a+\frac {b}{x}\right )^8 x^{11} \, dx=\frac {x^{4} \left (165 a^{8} x^{8}+1440 a^{7} b \,x^{7}+5544 a^{6} b^{2} x^{6}+12320 a^{5} b^{3} x^{5}+17325 a^{4} b^{4} x^{4}+15840 a^{3} b^{5} x^{3}+9240 a^{2} b^{6} x^{2}+3168 a \,b^{7} x +495 b^{8}\right )}{1980} \] Input:
int((a+b/x)^8*x^11,x)
Output:
(x**4*(165*a**8*x**8 + 1440*a**7*b*x**7 + 5544*a**6*b**2*x**6 + 12320*a**5 *b**3*x**5 + 17325*a**4*b**4*x**4 + 15840*a**3*b**5*x**3 + 9240*a**2*b**6* x**2 + 3168*a*b**7*x + 495*b**8))/1980