Integrand size = 13, antiderivative size = 98 \[ \int \left (a+\frac {b}{x}\right )^8 x^{13} \, dx=-\frac {b^5 (b+a x)^9}{9 a^6}+\frac {b^4 (b+a x)^{10}}{2 a^6}-\frac {10 b^3 (b+a x)^{11}}{11 a^6}+\frac {5 b^2 (b+a x)^{12}}{6 a^6}-\frac {5 b (b+a x)^{13}}{13 a^6}+\frac {(b+a x)^{14}}{14 a^6} \] Output:
-1/9*b^5*(a*x+b)^9/a^6+1/2*b^4*(a*x+b)^10/a^6-10/11*b^3*(a*x+b)^11/a^6+5/6 *b^2*(a*x+b)^12/a^6-5/13*b*(a*x+b)^13/a^6+1/14*(a*x+b)^14/a^6
Time = 0.00 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.08 \[ \int \left (a+\frac {b}{x}\right )^8 x^{13} \, dx=\frac {b^8 x^6}{6}+\frac {8}{7} a b^7 x^7+\frac {7}{2} a^2 b^6 x^8+\frac {56}{9} a^3 b^5 x^9+7 a^4 b^4 x^{10}+\frac {56}{11} a^5 b^3 x^{11}+\frac {7}{3} a^6 b^2 x^{12}+\frac {8}{13} a^7 b x^{13}+\frac {a^8 x^{14}}{14} \] Input:
Integrate[(a + b/x)^8*x^13,x]
Output:
(b^8*x^6)/6 + (8*a*b^7*x^7)/7 + (7*a^2*b^6*x^8)/2 + (56*a^3*b^5*x^9)/9 + 7 *a^4*b^4*x^10 + (56*a^5*b^3*x^11)/11 + (7*a^6*b^2*x^12)/3 + (8*a^7*b*x^13) /13 + (a^8*x^14)/14
Time = 0.37 (sec) , antiderivative size = 98, 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.
\(\displaystyle \int x^{13} \left (a+\frac {b}{x}\right )^8 \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int x^5 (a x+b)^8dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (-\frac {b^5 (a x+b)^8}{a^5}+\frac {5 b^4 (a x+b)^9}{a^5}-\frac {10 b^3 (a x+b)^{10}}{a^5}+\frac {10 b^2 (a x+b)^{11}}{a^5}+\frac {(a x+b)^{13}}{a^5}-\frac {5 b (a x+b)^{12}}{a^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b^5 (a x+b)^9}{9 a^6}+\frac {b^4 (a x+b)^{10}}{2 a^6}-\frac {10 b^3 (a x+b)^{11}}{11 a^6}+\frac {5 b^2 (a x+b)^{12}}{6 a^6}+\frac {(a x+b)^{14}}{14 a^6}-\frac {5 b (a x+b)^{13}}{13 a^6}\) |
Input:
Int[(a + b/x)^8*x^13,x]
Output:
-1/9*(b^5*(b + a*x)^9)/a^6 + (b^4*(b + a*x)^10)/(2*a^6) - (10*b^3*(b + a*x )^11)/(11*a^6) + (5*b^2*(b + a*x)^12)/(6*a^6) - (5*b*(b + a*x)^13)/(13*a^6 ) + (b + a*x)^14/(14*a^6)
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 = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.93
method | result | size |
gosper | \(\frac {x^{6} \left (1287 a^{8} x^{8}+11088 a^{7} b \,x^{7}+42042 a^{6} b^{2} x^{6}+91728 a^{5} b^{3} x^{5}+126126 a^{4} x^{4} b^{4}+112112 a^{3} b^{5} x^{3}+63063 a^{2} b^{6} x^{2}+20592 a \,b^{7} x +3003 b^{8}\right )}{18018}\) | \(91\) |
default | \(\frac {1}{6} b^{8} x^{6}+\frac {8}{7} a \,x^{7} b^{7}+\frac {7}{2} a^{2} b^{6} x^{8}+\frac {56}{9} a^{3} b^{5} x^{9}+7 a^{4} b^{4} x^{10}+\frac {56}{11} a^{5} b^{3} x^{11}+\frac {7}{3} a^{6} b^{2} x^{12}+\frac {8}{13} a^{7} b \,x^{13}+\frac {1}{14} a^{8} x^{14}\) | \(91\) |
risch | \(\frac {1}{6} b^{8} x^{6}+\frac {8}{7} a \,x^{7} b^{7}+\frac {7}{2} a^{2} b^{6} x^{8}+\frac {56}{9} a^{3} b^{5} x^{9}+7 a^{4} b^{4} x^{10}+\frac {56}{11} a^{5} b^{3} x^{11}+\frac {7}{3} a^{6} b^{2} x^{12}+\frac {8}{13} a^{7} b \,x^{13}+\frac {1}{14} a^{8} x^{14}\) | \(91\) |
parallelrisch | \(\frac {1}{6} b^{8} x^{6}+\frac {8}{7} a \,x^{7} b^{7}+\frac {7}{2} a^{2} b^{6} x^{8}+\frac {56}{9} a^{3} b^{5} x^{9}+7 a^{4} b^{4} x^{10}+\frac {56}{11} a^{5} b^{3} x^{11}+\frac {7}{3} a^{6} b^{2} x^{12}+\frac {8}{13} a^{7} b \,x^{13}+\frac {1}{14} a^{8} x^{14}\) | \(91\) |
norman | \(\frac {\frac {1}{14} a^{8} x^{21}+\frac {1}{6} b^{8} x^{13}+\frac {8}{7} a \,b^{7} x^{14}+\frac {7}{2} a^{2} b^{6} x^{15}+\frac {56}{9} a^{3} b^{5} x^{16}+7 a^{4} b^{4} x^{17}+\frac {56}{11} a^{5} b^{3} x^{18}+\frac {7}{3} a^{6} b^{2} x^{19}+\frac {8}{13} a^{7} b \,x^{20}}{x^{7}}\) | \(95\) |
orering | \(\frac {x^{14} \left (1287 a^{8} x^{8}+11088 a^{7} b \,x^{7}+42042 a^{6} b^{2} x^{6}+91728 a^{5} b^{3} x^{5}+126126 a^{4} x^{4} b^{4}+112112 a^{3} b^{5} x^{3}+63063 a^{2} b^{6} x^{2}+20592 a \,b^{7} x +3003 b^{8}\right ) \left (a +\frac {b}{x}\right )^{8}}{18018 \left (a x +b \right )^{8}}\) | \(107\) |
Input:
int((a+b/x)^8*x^13,x,method=_RETURNVERBOSE)
Output:
1/18018*x^6*(1287*a^8*x^8+11088*a^7*b*x^7+42042*a^6*b^2*x^6+91728*a^5*b^3* x^5+126126*a^4*b^4*x^4+112112*a^3*b^5*x^3+63063*a^2*b^6*x^2+20592*a*b^7*x+ 3003*b^8)
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.92 \[ \int \left (a+\frac {b}{x}\right )^8 x^{13} \, dx=\frac {1}{14} \, a^{8} x^{14} + \frac {8}{13} \, a^{7} b x^{13} + \frac {7}{3} \, a^{6} b^{2} x^{12} + \frac {56}{11} \, a^{5} b^{3} x^{11} + 7 \, a^{4} b^{4} x^{10} + \frac {56}{9} \, a^{3} b^{5} x^{9} + \frac {7}{2} \, a^{2} b^{6} x^{8} + \frac {8}{7} \, a b^{7} x^{7} + \frac {1}{6} \, b^{8} x^{6} \] Input:
integrate((a+b/x)^8*x^13,x, algorithm="fricas")
Output:
1/14*a^8*x^14 + 8/13*a^7*b*x^13 + 7/3*a^6*b^2*x^12 + 56/11*a^5*b^3*x^11 + 7*a^4*b^4*x^10 + 56/9*a^3*b^5*x^9 + 7/2*a^2*b^6*x^8 + 8/7*a*b^7*x^7 + 1/6* b^8*x^6
Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.07 \[ \int \left (a+\frac {b}{x}\right )^8 x^{13} \, dx=\frac {a^{8} x^{14}}{14} + \frac {8 a^{7} b x^{13}}{13} + \frac {7 a^{6} b^{2} x^{12}}{3} + \frac {56 a^{5} b^{3} x^{11}}{11} + 7 a^{4} b^{4} x^{10} + \frac {56 a^{3} b^{5} x^{9}}{9} + \frac {7 a^{2} b^{6} x^{8}}{2} + \frac {8 a b^{7} x^{7}}{7} + \frac {b^{8} x^{6}}{6} \] Input:
integrate((a+b/x)**8*x**13,x)
Output:
a**8*x**14/14 + 8*a**7*b*x**13/13 + 7*a**6*b**2*x**12/3 + 56*a**5*b**3*x** 11/11 + 7*a**4*b**4*x**10 + 56*a**3*b**5*x**9/9 + 7*a**2*b**6*x**8/2 + 8*a *b**7*x**7/7 + b**8*x**6/6
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.92 \[ \int \left (a+\frac {b}{x}\right )^8 x^{13} \, dx=\frac {1}{14} \, a^{8} x^{14} + \frac {8}{13} \, a^{7} b x^{13} + \frac {7}{3} \, a^{6} b^{2} x^{12} + \frac {56}{11} \, a^{5} b^{3} x^{11} + 7 \, a^{4} b^{4} x^{10} + \frac {56}{9} \, a^{3} b^{5} x^{9} + \frac {7}{2} \, a^{2} b^{6} x^{8} + \frac {8}{7} \, a b^{7} x^{7} + \frac {1}{6} \, b^{8} x^{6} \] Input:
integrate((a+b/x)^8*x^13,x, algorithm="maxima")
Output:
1/14*a^8*x^14 + 8/13*a^7*b*x^13 + 7/3*a^6*b^2*x^12 + 56/11*a^5*b^3*x^11 + 7*a^4*b^4*x^10 + 56/9*a^3*b^5*x^9 + 7/2*a^2*b^6*x^8 + 8/7*a*b^7*x^7 + 1/6* b^8*x^6
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.92 \[ \int \left (a+\frac {b}{x}\right )^8 x^{13} \, dx=\frac {1}{14} \, a^{8} x^{14} + \frac {8}{13} \, a^{7} b x^{13} + \frac {7}{3} \, a^{6} b^{2} x^{12} + \frac {56}{11} \, a^{5} b^{3} x^{11} + 7 \, a^{4} b^{4} x^{10} + \frac {56}{9} \, a^{3} b^{5} x^{9} + \frac {7}{2} \, a^{2} b^{6} x^{8} + \frac {8}{7} \, a b^{7} x^{7} + \frac {1}{6} \, b^{8} x^{6} \] Input:
integrate((a+b/x)^8*x^13,x, algorithm="giac")
Output:
1/14*a^8*x^14 + 8/13*a^7*b*x^13 + 7/3*a^6*b^2*x^12 + 56/11*a^5*b^3*x^11 + 7*a^4*b^4*x^10 + 56/9*a^3*b^5*x^9 + 7/2*a^2*b^6*x^8 + 8/7*a*b^7*x^7 + 1/6* b^8*x^6
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.92 \[ \int \left (a+\frac {b}{x}\right )^8 x^{13} \, dx=\frac {a^8\,x^{14}}{14}+\frac {8\,a^7\,b\,x^{13}}{13}+\frac {7\,a^6\,b^2\,x^{12}}{3}+\frac {56\,a^5\,b^3\,x^{11}}{11}+7\,a^4\,b^4\,x^{10}+\frac {56\,a^3\,b^5\,x^9}{9}+\frac {7\,a^2\,b^6\,x^8}{2}+\frac {8\,a\,b^7\,x^7}{7}+\frac {b^8\,x^6}{6} \] Input:
int(x^13*(a + b/x)^8,x)
Output:
(a^8*x^14)/14 + (b^8*x^6)/6 + (8*a*b^7*x^7)/7 + (8*a^7*b*x^13)/13 + (7*a^2 *b^6*x^8)/2 + (56*a^3*b^5*x^9)/9 + 7*a^4*b^4*x^10 + (56*a^5*b^3*x^11)/11 + (7*a^6*b^2*x^12)/3
Time = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.92 \[ \int \left (a+\frac {b}{x}\right )^8 x^{13} \, dx=\frac {x^{6} \left (1287 a^{8} x^{8}+11088 a^{7} b \,x^{7}+42042 a^{6} b^{2} x^{6}+91728 a^{5} b^{3} x^{5}+126126 a^{4} b^{4} x^{4}+112112 a^{3} b^{5} x^{3}+63063 a^{2} b^{6} x^{2}+20592 a \,b^{7} x +3003 b^{8}\right )}{18018} \] Input:
int((a+b/x)^8*x^13,x)
Output:
(x**6*(1287*a**8*x**8 + 11088*a**7*b*x**7 + 42042*a**6*b**2*x**6 + 91728*a **5*b**3*x**5 + 126126*a**4*b**4*x**4 + 112112*a**3*b**5*x**3 + 63063*a**2 *b**6*x**2 + 20592*a*b**7*x + 3003*b**8))/18018