\(\int (a+\frac {b}{x})^8 x^{15} \, dx\) [1584]

Optimal result

Integrand size = 13, antiderivative size = 106 \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {b^8 x^8}{8}+\frac {8}{9} a b^7 x^9+\frac {14}{5} a^2 b^6 x^{10}+\frac {56}{11} a^3 b^5 x^{11}+\frac {35}{6} a^4 b^4 x^{12}+\frac {56}{13} a^5 b^3 x^{13}+2 a^6 b^2 x^{14}+\frac {8}{15} a^7 b x^{15}+\frac {a^8 x^{16}}{16} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (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} \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {a^8 x^{16}}{16}+\frac {8}{15} a^7 b x^{15}+2 a^6 b^2 x^{14}+\frac {56}{13} a^5 b^3 x^{13}+\frac {35}{6} a^4 b^4 x^{12}+\frac {56}{11} a^3 b^5 x^{11}+\frac {14}{5} a^2 b^6 x^{10}+\frac {8}{9} a b^7 x^9+\frac {b^8 x^8}{8} \]

[In]

Int[(a + b/x)^8*x^15,x]

[Out]

(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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {b^8 x^8}{8}+\frac {8}{9} a b^7 x^9+\frac {14}{5} a^2 b^6 x^{10}+\frac {56}{11} a^3 b^5 x^{11}+\frac {35}{6} a^4 b^4 x^{12}+\frac {56}{13} a^5 b^3 x^{13}+2 a^6 b^2 x^{14}+\frac {8}{15} a^7 b x^{15}+\frac {a^8 x^{16}}{16} \]

[In]

Integrate[(a + b/x)^8*x^15,x]

[Out]

(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

Maple [A] (verified)

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

method	result	size
gosper	\(\frac {x^{8} \left (6435 a^{8} x^{8}+54912 x^{7} b \,a^{7}+205920 a^{6} b^{2} x^{6}+443520 a^{5} b^{3} x^{5}+600600 a^{4} x^{4} b^{4}+524160 a^{3} b^{5} x^{3}+288288 a^{2} b^{6} x^{2}+91520 a \,b^{7} x +12870 b^{8}\right )}{102960}\)	\(91\)
default	\(\frac {1}{8} b^{8} x^{8}+\frac {8}{9} a \,b^{7} x^{9}+\frac {14}{5} a^{2} b^{6} x^{10}+\frac {56}{11} x^{11} b^{5} a^{3}+\frac {35}{6} a^{4} b^{4} x^{12}+\frac {56}{13} x^{13} b^{3} a^{5}+2 a^{6} b^{2} x^{14}+\frac {8}{15} a^{7} b \,x^{15}+\frac {1}{16} a^{8} x^{16}\)	\(91\)
risch	\(\frac {1}{8} b^{8} x^{8}+\frac {8}{9} a \,b^{7} x^{9}+\frac {14}{5} a^{2} b^{6} x^{10}+\frac {56}{11} x^{11} b^{5} a^{3}+\frac {35}{6} a^{4} b^{4} x^{12}+\frac {56}{13} x^{13} b^{3} a^{5}+2 a^{6} b^{2} x^{14}+\frac {8}{15} a^{7} b \,x^{15}+\frac {1}{16} a^{8} x^{16}\)	\(91\)
parallelrisch	\(\frac {1}{8} b^{8} x^{8}+\frac {8}{9} a \,b^{7} x^{9}+\frac {14}{5} a^{2} b^{6} x^{10}+\frac {56}{11} x^{11} b^{5} a^{3}+\frac {35}{6} a^{4} b^{4} x^{12}+\frac {56}{13} x^{13} b^{3} a^{5}+2 a^{6} b^{2} x^{14}+\frac {8}{15} a^{7} b \,x^{15}+\frac {1}{16} a^{8} x^{16}\)	\(91\)
norman	\(\frac {\frac {1}{16} a^{8} x^{23}+\frac {1}{8} b^{8} x^{15}+\frac {8}{9} a \,b^{7} x^{16}+\frac {14}{5} a^{2} b^{6} x^{17}+\frac {56}{11} a^{3} b^{5} x^{18}+\frac {35}{6} a^{4} b^{4} x^{19}+\frac {56}{13} a^{5} b^{3} x^{20}+2 a^{6} b^{2} x^{21}+\frac {8}{15} a^{7} b \,x^{22}}{x^{7}}\)	\(95\)

[In]

int((a+b/x)^8*x^15,x,method=_RETURNVERBOSE)

[Out]

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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {1}{16} \, a^{8} x^{16} + \frac {8}{15} \, a^{7} b x^{15} + 2 \, a^{6} b^{2} x^{14} + \frac {56}{13} \, a^{5} b^{3} x^{13} + \frac {35}{6} \, a^{4} b^{4} x^{12} + \frac {56}{11} \, a^{3} b^{5} x^{11} + \frac {14}{5} \, a^{2} b^{6} x^{10} + \frac {8}{9} \, a b^{7} x^{9} + \frac {1}{8} \, b^{8} x^{8} \]

[In]

integrate((a+b/x)^8*x^15,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.99 \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {a^{8} x^{16}}{16} + \frac {8 a^{7} b x^{15}}{15} + 2 a^{6} b^{2} x^{14} + \frac {56 a^{5} b^{3} x^{13}}{13} + \frac {35 a^{4} b^{4} x^{12}}{6} + \frac {56 a^{3} b^{5} x^{11}}{11} + \frac {14 a^{2} b^{6} x^{10}}{5} + \frac {8 a b^{7} x^{9}}{9} + \frac {b^{8} x^{8}}{8} \]

[In]

integrate((a+b/x)**8*x**15,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {1}{16} \, a^{8} x^{16} + \frac {8}{15} \, a^{7} b x^{15} + 2 \, a^{6} b^{2} x^{14} + \frac {56}{13} \, a^{5} b^{3} x^{13} + \frac {35}{6} \, a^{4} b^{4} x^{12} + \frac {56}{11} \, a^{3} b^{5} x^{11} + \frac {14}{5} \, a^{2} b^{6} x^{10} + \frac {8}{9} \, a b^{7} x^{9} + \frac {1}{8} \, b^{8} x^{8} \]

[In]

integrate((a+b/x)^8*x^15,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {1}{16} \, a^{8} x^{16} + \frac {8}{15} \, a^{7} b x^{15} + 2 \, a^{6} b^{2} x^{14} + \frac {56}{13} \, a^{5} b^{3} x^{13} + \frac {35}{6} \, a^{4} b^{4} x^{12} + \frac {56}{11} \, a^{3} b^{5} x^{11} + \frac {14}{5} \, a^{2} b^{6} x^{10} + \frac {8}{9} \, a b^{7} x^{9} + \frac {1}{8} \, b^{8} x^{8} \]

[In]

integrate((a+b/x)^8*x^15,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.74 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^8 x^{15} \, dx=\frac {a^8\,x^{16}}{16}+\frac {8\,a^7\,b\,x^{15}}{15}+2\,a^6\,b^2\,x^{14}+\frac {56\,a^5\,b^3\,x^{13}}{13}+\frac {35\,a^4\,b^4\,x^{12}}{6}+\frac {56\,a^3\,b^5\,x^{11}}{11}+\frac {14\,a^2\,b^6\,x^{10}}{5}+\frac {8\,a\,b^7\,x^9}{9}+\frac {b^8\,x^8}{8} \]

[In]

int(x^15*(a + b/x)^8,x)

[Out]

(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