\(\int (a+\frac {b}{x})^8 x^{16} \, dx\) [1583]

Optimal result

Integrand size = 13, antiderivative size = 106 \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {b^8 x^9}{9}+\frac {4}{5} a b^7 x^{10}+\frac {28}{11} a^2 b^6 x^{11}+\frac {14}{3} a^3 b^5 x^{12}+\frac {70}{13} a^4 b^4 x^{13}+4 a^5 b^3 x^{14}+\frac {28}{15} a^6 b^2 x^{15}+\frac {1}{2} a^7 b x^{16}+\frac {a^8 x^{17}}{17} \]

[Out]

1/9*b^8*x^9+4/5*a*b^7*x^10+28/11*a^2*b^6*x^11+14/3*a^3*b^5*x^12+70/13*a^4*b^4*x^13+4*a^5*b^3*x^14+28/15*a^6*b^
2*x^15+1/2*a^7*b*x^16+1/17*a^8*x^17

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 45} \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {a^8 x^{17}}{17}+\frac {1}{2} a^7 b x^{16}+\frac {28}{15} a^6 b^2 x^{15}+4 a^5 b^3 x^{14}+\frac {70}{13} a^4 b^4 x^{13}+\frac {14}{3} a^3 b^5 x^{12}+\frac {28}{11} a^2 b^6 x^{11}+\frac {4}{5} a b^7 x^{10}+\frac {b^8 x^9}{9} \]

[In]

Int[(a + b/x)^8*x^16,x]

[Out]

(b^8*x^9)/9 + (4*a*b^7*x^10)/5 + (28*a^2*b^6*x^11)/11 + (14*a^3*b^5*x^12)/3 + (70*a^4*b^4*x^13)/13 + 4*a^5*b^3
*x^14 + (28*a^6*b^2*x^15)/15 + (a^7*b*x^16)/2 + (a^8*x^17)/17

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 269

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]

Rubi steps \begin{align*} \text {integral}& = \int x^8 (b+a x)^8 \, dx \\ & = \int \left (b^8 x^8+8 a b^7 x^9+28 a^2 b^6 x^{10}+56 a^3 b^5 x^{11}+70 a^4 b^4 x^{12}+56 a^5 b^3 x^{13}+28 a^6 b^2 x^{14}+8 a^7 b x^{15}+a^8 x^{16}\right ) \, dx \\ & = \frac {b^8 x^9}{9}+\frac {4}{5} a b^7 x^{10}+\frac {28}{11} a^2 b^6 x^{11}+\frac {14}{3} a^3 b^5 x^{12}+\frac {70}{13} a^4 b^4 x^{13}+4 a^5 b^3 x^{14}+\frac {28}{15} a^6 b^2 x^{15}+\frac {1}{2} a^7 b x^{16}+\frac {a^8 x^{17}}{17} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {b^8 x^9}{9}+\frac {4}{5} a b^7 x^{10}+\frac {28}{11} a^2 b^6 x^{11}+\frac {14}{3} a^3 b^5 x^{12}+\frac {70}{13} a^4 b^4 x^{13}+4 a^5 b^3 x^{14}+\frac {28}{15} a^6 b^2 x^{15}+\frac {1}{2} a^7 b x^{16}+\frac {a^8 x^{17}}{17} \]

[In]

Integrate[(a + b/x)^8*x^16,x]

[Out]

(b^8*x^9)/9 + (4*a*b^7*x^10)/5 + (28*a^2*b^6*x^11)/11 + (14*a^3*b^5*x^12)/3 + (70*a^4*b^4*x^13)/13 + 4*a^5*b^3
*x^14 + (28*a^6*b^2*x^15)/15 + (a^7*b*x^16)/2 + (a^8*x^17)/17

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.86

method	result	size
gosper	\(\frac {x^{9} \left (12870 a^{8} x^{8}+109395 x^{7} b \,a^{7}+408408 a^{6} b^{2} x^{6}+875160 a^{5} b^{3} x^{5}+1178100 a^{4} x^{4} b^{4}+1021020 a^{3} b^{5} x^{3}+556920 a^{2} b^{6} x^{2}+175032 a \,b^{7} x +24310 b^{8}\right )}{218790}\)	\(91\)
default	\(\frac {1}{9} b^{8} x^{9}+\frac {4}{5} a \,b^{7} x^{10}+\frac {28}{11} a^{2} b^{6} x^{11}+\frac {14}{3} a^{3} b^{5} x^{12}+\frac {70}{13} x^{13} b^{4} a^{4}+4 a^{5} b^{3} x^{14}+\frac {28}{15} a^{6} b^{2} x^{15}+\frac {1}{2} a^{7} b \,x^{16}+\frac {1}{17} a^{8} x^{17}\)	\(91\)
risch	\(\frac {1}{9} b^{8} x^{9}+\frac {4}{5} a \,b^{7} x^{10}+\frac {28}{11} a^{2} b^{6} x^{11}+\frac {14}{3} a^{3} b^{5} x^{12}+\frac {70}{13} x^{13} b^{4} a^{4}+4 a^{5} b^{3} x^{14}+\frac {28}{15} a^{6} b^{2} x^{15}+\frac {1}{2} a^{7} b \,x^{16}+\frac {1}{17} a^{8} x^{17}\)	\(91\)
parallelrisch	\(\frac {1}{9} b^{8} x^{9}+\frac {4}{5} a \,b^{7} x^{10}+\frac {28}{11} a^{2} b^{6} x^{11}+\frac {14}{3} a^{3} b^{5} x^{12}+\frac {70}{13} x^{13} b^{4} a^{4}+4 a^{5} b^{3} x^{14}+\frac {28}{15} a^{6} b^{2} x^{15}+\frac {1}{2} a^{7} b \,x^{16}+\frac {1}{17} a^{8} x^{17}\)	\(91\)
norman	\(\frac {\frac {1}{17} a^{8} x^{24}+\frac {1}{9} b^{8} x^{16}+\frac {4}{5} a \,b^{7} x^{17}+\frac {28}{11} a^{2} b^{6} x^{18}+\frac {14}{3} a^{3} b^{5} x^{19}+\frac {70}{13} a^{4} b^{4} x^{20}+4 a^{5} b^{3} x^{21}+\frac {28}{15} a^{6} b^{2} x^{22}+\frac {1}{2} a^{7} b \,x^{23}}{x^{7}}\)	\(95\)

[In]

int((a+b/x)^8*x^16,x,method=_RETURNVERBOSE)

[Out]

1/218790*x^9*(12870*a^8*x^8+109395*a^7*b*x^7+408408*a^6*b^2*x^6+875160*a^5*b^3*x^5+1178100*a^4*b^4*x^4+1021020
*a^3*b^5*x^3+556920*a^2*b^6*x^2+175032*a*b^7*x+24310*b^8)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {1}{17} \, a^{8} x^{17} + \frac {1}{2} \, a^{7} b x^{16} + \frac {28}{15} \, a^{6} b^{2} x^{15} + 4 \, a^{5} b^{3} x^{14} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {14}{3} \, a^{3} b^{5} x^{12} + \frac {28}{11} \, a^{2} b^{6} x^{11} + \frac {4}{5} \, a b^{7} x^{10} + \frac {1}{9} \, b^{8} x^{9} \]

[In]

integrate((a+b/x)^8*x^16,x, algorithm="fricas")

[Out]

1/17*a^8*x^17 + 1/2*a^7*b*x^16 + 28/15*a^6*b^2*x^15 + 4*a^5*b^3*x^14 + 70/13*a^4*b^4*x^13 + 14/3*a^3*b^5*x^12
+ 28/11*a^2*b^6*x^11 + 4/5*a*b^7*x^10 + 1/9*b^8*x^9

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.98 \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {a^{8} x^{17}}{17} + \frac {a^{7} b x^{16}}{2} + \frac {28 a^{6} b^{2} x^{15}}{15} + 4 a^{5} b^{3} x^{14} + \frac {70 a^{4} b^{4} x^{13}}{13} + \frac {14 a^{3} b^{5} x^{12}}{3} + \frac {28 a^{2} b^{6} x^{11}}{11} + \frac {4 a b^{7} x^{10}}{5} + \frac {b^{8} x^{9}}{9} \]

[In]

integrate((a+b/x)**8*x**16,x)

[Out]

a**8*x**17/17 + a**7*b*x**16/2 + 28*a**6*b**2*x**15/15 + 4*a**5*b**3*x**14 + 70*a**4*b**4*x**13/13 + 14*a**3*b
**5*x**12/3 + 28*a**2*b**6*x**11/11 + 4*a*b**7*x**10/5 + b**8*x**9/9

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {1}{17} \, a^{8} x^{17} + \frac {1}{2} \, a^{7} b x^{16} + \frac {28}{15} \, a^{6} b^{2} x^{15} + 4 \, a^{5} b^{3} x^{14} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {14}{3} \, a^{3} b^{5} x^{12} + \frac {28}{11} \, a^{2} b^{6} x^{11} + \frac {4}{5} \, a b^{7} x^{10} + \frac {1}{9} \, b^{8} x^{9} \]

[In]

integrate((a+b/x)^8*x^16,x, algorithm="maxima")

[Out]

1/17*a^8*x^17 + 1/2*a^7*b*x^16 + 28/15*a^6*b^2*x^15 + 4*a^5*b^3*x^14 + 70/13*a^4*b^4*x^13 + 14/3*a^3*b^5*x^12
+ 28/11*a^2*b^6*x^11 + 4/5*a*b^7*x^10 + 1/9*b^8*x^9

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {1}{17} \, a^{8} x^{17} + \frac {1}{2} \, a^{7} b x^{16} + \frac {28}{15} \, a^{6} b^{2} x^{15} + 4 \, a^{5} b^{3} x^{14} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {14}{3} \, a^{3} b^{5} x^{12} + \frac {28}{11} \, a^{2} b^{6} x^{11} + \frac {4}{5} \, a b^{7} x^{10} + \frac {1}{9} \, b^{8} x^{9} \]

[In]

integrate((a+b/x)^8*x^16,x, algorithm="giac")

[Out]

1/17*a^8*x^17 + 1/2*a^7*b*x^16 + 28/15*a^6*b^2*x^15 + 4*a^5*b^3*x^14 + 70/13*a^4*b^4*x^13 + 14/3*a^3*b^5*x^12
+ 28/11*a^2*b^6*x^11 + 4/5*a*b^7*x^10 + 1/9*b^8*x^9

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx=\frac {a^8\,x^{17}}{17}+\frac {a^7\,b\,x^{16}}{2}+\frac {28\,a^6\,b^2\,x^{15}}{15}+4\,a^5\,b^3\,x^{14}+\frac {70\,a^4\,b^4\,x^{13}}{13}+\frac {14\,a^3\,b^5\,x^{12}}{3}+\frac {28\,a^2\,b^6\,x^{11}}{11}+\frac {4\,a\,b^7\,x^{10}}{5}+\frac {b^8\,x^9}{9} \]

[In]

int(x^16*(a + b/x)^8,x)

[Out]

(a^8*x^17)/17 + (b^8*x^9)/9 + (4*a*b^7*x^10)/5 + (a^7*b*x^16)/2 + (28*a^2*b^6*x^11)/11 + (14*a^3*b^5*x^12)/3 +
(70*a^4*b^4*x^13)/13 + 4*a^5*b^3*x^14 + (28*a^6*b^2*x^15)/15