Integrand size = 13, antiderivative size = 106 \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {b^8 x^8}{8}+\frac {8}{9} a b^7 x^9+\frac {14}{5} a^2 b^6 x^{10}+\frac {56}{11} a^3 b^5 x^{11}+\frac {35}{6} a^4 b^4 x^{12}+\frac {56}{13} a^5 b^3 x^{13}+2 a^6 b^2 x^{14}+\frac {8}{15} a^7 b x^{15}+\frac {a^8 x^{16}}{16} \] Output:
1/8*b^8*x^8+8/9*a*b^7*x^9+14/5*a^2*b^6*x^10+56/11*a^3*b^5*x^11+35/6*a^4*b^ 4*x^12+56/13*a^5*b^3*x^13+2*a^6*b^2*x^14+8/15*a^7*b*x^15+1/16*a^8*x^16
Time = 0.00 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {b^8 x^8}{8}+\frac {8}{9} a b^7 x^9+\frac {14}{5} a^2 b^6 x^{10}+\frac {56}{11} a^3 b^5 x^{11}+\frac {35}{6} a^4 b^4 x^{12}+\frac {56}{13} a^5 b^3 x^{13}+2 a^6 b^2 x^{14}+\frac {8}{15} a^7 b x^{15}+\frac {a^8 x^{16}}{16} \] Input:
Integrate[(a + b/x)^8*x^15,x]
Output:
(b^8*x^8)/8 + (8*a*b^7*x^9)/9 + (14*a^2*b^6*x^10)/5 + (56*a^3*b^5*x^11)/11 + (35*a^4*b^4*x^12)/6 + (56*a^5*b^3*x^13)/13 + 2*a^6*b^2*x^14 + (8*a^7*b* x^15)/15 + (a^8*x^16)/16
Time = 0.37 (sec) , antiderivative size = 106, 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.
\(\displaystyle \int x^{15} \left (a+\frac {b}{x}\right )^8 \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int x^7 (a x+b)^8dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (a^8 x^{15}+8 a^7 b x^{14}+28 a^6 b^2 x^{13}+56 a^5 b^3 x^{12}+70 a^4 b^4 x^{11}+56 a^3 b^5 x^{10}+28 a^2 b^6 x^9+8 a b^7 x^8+b^8 x^7\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^8 x^{16}}{16}+\frac {8}{15} a^7 b x^{15}+2 a^6 b^2 x^{14}+\frac {56}{13} a^5 b^3 x^{13}+\frac {35}{6} a^4 b^4 x^{12}+\frac {56}{11} a^3 b^5 x^{11}+\frac {14}{5} a^2 b^6 x^{10}+\frac {8}{9} a b^7 x^9+\frac {b^8 x^8}{8}\) |
Input:
Int[(a + b/x)^8*x^15,x]
Output:
(b^8*x^8)/8 + (8*a*b^7*x^9)/9 + (14*a^2*b^6*x^10)/5 + (56*a^3*b^5*x^11)/11 + (35*a^4*b^4*x^12)/6 + (56*a^5*b^3*x^13)/13 + 2*a^6*b^2*x^14 + (8*a^7*b* x^15)/15 + (a^8*x^16)/16
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 = 0.86
method | result | size |
gosper | \(\frac {x^{8} \left (6435 a^{8} x^{8}+54912 a^{7} b \,x^{7}+205920 a^{6} b^{2} x^{6}+443520 a^{5} b^{3} x^{5}+600600 a^{4} x^{4} b^{4}+524160 a^{3} b^{5} x^{3}+288288 a^{2} b^{6} x^{2}+91520 a \,b^{7} x +12870 b^{8}\right )}{102960}\) | \(91\) |
default | \(\frac {1}{8} b^{8} x^{8}+\frac {8}{9} a \,b^{7} x^{9}+\frac {14}{5} a^{2} b^{6} x^{10}+\frac {56}{11} a^{3} b^{5} x^{11}+\frac {35}{6} a^{4} b^{4} x^{12}+\frac {56}{13} a^{5} b^{3} x^{13}+2 a^{6} b^{2} x^{14}+\frac {8}{15} a^{7} b \,x^{15}+\frac {1}{16} a^{8} x^{16}\) | \(91\) |
risch | \(\frac {1}{8} b^{8} x^{8}+\frac {8}{9} a \,b^{7} x^{9}+\frac {14}{5} a^{2} b^{6} x^{10}+\frac {56}{11} a^{3} b^{5} x^{11}+\frac {35}{6} a^{4} b^{4} x^{12}+\frac {56}{13} a^{5} b^{3} x^{13}+2 a^{6} b^{2} x^{14}+\frac {8}{15} a^{7} b \,x^{15}+\frac {1}{16} a^{8} x^{16}\) | \(91\) |
parallelrisch | \(\frac {1}{8} b^{8} x^{8}+\frac {8}{9} a \,b^{7} x^{9}+\frac {14}{5} a^{2} b^{6} x^{10}+\frac {56}{11} a^{3} b^{5} x^{11}+\frac {35}{6} a^{4} b^{4} x^{12}+\frac {56}{13} a^{5} b^{3} x^{13}+2 a^{6} b^{2} x^{14}+\frac {8}{15} a^{7} b \,x^{15}+\frac {1}{16} a^{8} x^{16}\) | \(91\) |
norman | \(\frac {\frac {1}{16} a^{8} x^{23}+\frac {1}{8} b^{8} x^{15}+\frac {8}{9} a \,b^{7} x^{16}+\frac {14}{5} a^{2} b^{6} x^{17}+\frac {56}{11} a^{3} b^{5} x^{18}+\frac {35}{6} a^{4} b^{4} x^{19}+\frac {56}{13} a^{5} b^{3} x^{20}+2 a^{6} b^{2} x^{21}+\frac {8}{15} a^{7} b \,x^{22}}{x^{7}}\) | \(95\) |
orering | \(\frac {x^{16} \left (6435 a^{8} x^{8}+54912 a^{7} b \,x^{7}+205920 a^{6} b^{2} x^{6}+443520 a^{5} b^{3} x^{5}+600600 a^{4} x^{4} b^{4}+524160 a^{3} b^{5} x^{3}+288288 a^{2} b^{6} x^{2}+91520 a \,b^{7} x +12870 b^{8}\right ) \left (a +\frac {b}{x}\right )^{8}}{102960 \left (a x +b \right )^{8}}\) | \(107\) |
Input:
int((a+b/x)^8*x^15,x,method=_RETURNVERBOSE)
Output:
1/102960*x^8*(6435*a^8*x^8+54912*a^7*b*x^7+205920*a^6*b^2*x^6+443520*a^5*b ^3*x^5+600600*a^4*b^4*x^4+524160*a^3*b^5*x^3+288288*a^2*b^6*x^2+91520*a*b^ 7*x+12870*b^8)
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {1}{16} \, a^{8} x^{16} + \frac {8}{15} \, a^{7} b x^{15} + 2 \, a^{6} b^{2} x^{14} + \frac {56}{13} \, a^{5} b^{3} x^{13} + \frac {35}{6} \, a^{4} b^{4} x^{12} + \frac {56}{11} \, a^{3} b^{5} x^{11} + \frac {14}{5} \, a^{2} b^{6} x^{10} + \frac {8}{9} \, a b^{7} x^{9} + \frac {1}{8} \, b^{8} x^{8} \] Input:
integrate((a+b/x)^8*x^15,x, algorithm="fricas")
Output:
1/16*a^8*x^16 + 8/15*a^7*b*x^15 + 2*a^6*b^2*x^14 + 56/13*a^5*b^3*x^13 + 35 /6*a^4*b^4*x^12 + 56/11*a^3*b^5*x^11 + 14/5*a^2*b^6*x^10 + 8/9*a*b^7*x^9 + 1/8*b^8*x^8
Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.99 \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {a^{8} x^{16}}{16} + \frac {8 a^{7} b x^{15}}{15} + 2 a^{6} b^{2} x^{14} + \frac {56 a^{5} b^{3} x^{13}}{13} + \frac {35 a^{4} b^{4} x^{12}}{6} + \frac {56 a^{3} b^{5} x^{11}}{11} + \frac {14 a^{2} b^{6} x^{10}}{5} + \frac {8 a b^{7} x^{9}}{9} + \frac {b^{8} x^{8}}{8} \] Input:
integrate((a+b/x)**8*x**15,x)
Output:
a**8*x**16/16 + 8*a**7*b*x**15/15 + 2*a**6*b**2*x**14 + 56*a**5*b**3*x**13 /13 + 35*a**4*b**4*x**12/6 + 56*a**3*b**5*x**11/11 + 14*a**2*b**6*x**10/5 + 8*a*b**7*x**9/9 + b**8*x**8/8
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {1}{16} \, a^{8} x^{16} + \frac {8}{15} \, a^{7} b x^{15} + 2 \, a^{6} b^{2} x^{14} + \frac {56}{13} \, a^{5} b^{3} x^{13} + \frac {35}{6} \, a^{4} b^{4} x^{12} + \frac {56}{11} \, a^{3} b^{5} x^{11} + \frac {14}{5} \, a^{2} b^{6} x^{10} + \frac {8}{9} \, a b^{7} x^{9} + \frac {1}{8} \, b^{8} x^{8} \] Input:
integrate((a+b/x)^8*x^15,x, algorithm="maxima")
Output:
1/16*a^8*x^16 + 8/15*a^7*b*x^15 + 2*a^6*b^2*x^14 + 56/13*a^5*b^3*x^13 + 35 /6*a^4*b^4*x^12 + 56/11*a^3*b^5*x^11 + 14/5*a^2*b^6*x^10 + 8/9*a*b^7*x^9 + 1/8*b^8*x^8
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {1}{16} \, a^{8} x^{16} + \frac {8}{15} \, a^{7} b x^{15} + 2 \, a^{6} b^{2} x^{14} + \frac {56}{13} \, a^{5} b^{3} x^{13} + \frac {35}{6} \, a^{4} b^{4} x^{12} + \frac {56}{11} \, a^{3} b^{5} x^{11} + \frac {14}{5} \, a^{2} b^{6} x^{10} + \frac {8}{9} \, a b^{7} x^{9} + \frac {1}{8} \, b^{8} x^{8} \] Input:
integrate((a+b/x)^8*x^15,x, algorithm="giac")
Output:
1/16*a^8*x^16 + 8/15*a^7*b*x^15 + 2*a^6*b^2*x^14 + 56/13*a^5*b^3*x^13 + 35 /6*a^4*b^4*x^12 + 56/11*a^3*b^5*x^11 + 14/5*a^2*b^6*x^10 + 8/9*a*b^7*x^9 + 1/8*b^8*x^8
Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {a^8\,x^{16}}{16}+\frac {8\,a^7\,b\,x^{15}}{15}+2\,a^6\,b^2\,x^{14}+\frac {56\,a^5\,b^3\,x^{13}}{13}+\frac {35\,a^4\,b^4\,x^{12}}{6}+\frac {56\,a^3\,b^5\,x^{11}}{11}+\frac {14\,a^2\,b^6\,x^{10}}{5}+\frac {8\,a\,b^7\,x^9}{9}+\frac {b^8\,x^8}{8} \] Input:
int(x^15*(a + b/x)^8,x)
Output:
(a^8*x^16)/16 + (b^8*x^8)/8 + (8*a*b^7*x^9)/9 + (8*a^7*b*x^15)/15 + (14*a^ 2*b^6*x^10)/5 + (56*a^3*b^5*x^11)/11 + (35*a^4*b^4*x^12)/6 + (56*a^5*b^3*x ^13)/13 + 2*a^6*b^2*x^14
Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {x^{8} \left (6435 a^{8} x^{8}+54912 a^{7} b \,x^{7}+205920 a^{6} b^{2} x^{6}+443520 a^{5} b^{3} x^{5}+600600 a^{4} b^{4} x^{4}+524160 a^{3} b^{5} x^{3}+288288 a^{2} b^{6} x^{2}+91520 a \,b^{7} x +12870 b^{8}\right )}{102960} \] Input:
int((a+b/x)^8*x^15,x)
Output:
(x**8*(6435*a**8*x**8 + 54912*a**7*b*x**7 + 205920*a**6*b**2*x**6 + 443520 *a**5*b**3*x**5 + 600600*a**4*b**4*x**4 + 524160*a**3*b**5*x**3 + 288288*a **2*b**6*x**2 + 91520*a*b**7*x + 12870*b**8))/102960