3.1584 \(\int (a+\frac{b}{x})^8 x^{15} \, dx\)

Optimal. Leaf size=106 \[ 2 a^6 b^2 x^{14}+\frac{56}{13} a^5 b^3 x^{13}+\frac{35}{6} a^4 b^4 x^{12}+\frac{56}{11} a^3 b^5 x^{11}+\frac{14}{5} a^2 b^6 x^{10}+\frac{8}{15} a^7 b x^{15}+\frac{a^8 x^{16}}{16}+\frac{8}{9} a b^7 x^9+\frac{b^8 x^8}{8} \]

[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

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Rubi [A]  time = 0.0479001, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 43} \[ 2 a^6 b^2 x^{14}+\frac{56}{13} a^5 b^3 x^{13}+\frac{35}{6} a^4 b^4 x^{12}+\frac{56}{11} a^3 b^5 x^{11}+\frac{14}{5} a^2 b^6 x^{10}+\frac{8}{15} a^7 b x^{15}+\frac{a^8 x^{16}}{16}+\frac{8}{9} a b^7 x^9+\frac{b^8 x^8}{8} \]

Antiderivative was successfully verified.

[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 263

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]

Rule 43

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])

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x}\right )^8 x^{15} \, dx &=\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]  time = 0.0025703, size = 106, normalized size = 1. \[ 2 a^6 b^2 x^{14}+\frac{56}{13} a^5 b^3 x^{13}+\frac{35}{6} a^4 b^4 x^{12}+\frac{56}{11} a^3 b^5 x^{11}+\frac{14}{5} a^2 b^6 x^{10}+\frac{8}{15} a^7 b x^{15}+\frac{a^8 x^{16}}{16}+\frac{8}{9} a b^7 x^9+\frac{b^8 x^8}{8} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0., size = 91, normalized size = 0.9 \begin{align*}{\frac{{b}^{8}{x}^{8}}{8}}+{\frac{8\,a{b}^{7}{x}^{9}}{9}}+{\frac{14\,{a}^{2}{b}^{6}{x}^{10}}{5}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{11}}{11}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{12}}{6}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{13}}{13}}+2\,{a}^{6}{b}^{2}{x}^{14}+{\frac{8\,{a}^{7}b{x}^{15}}{15}}+{\frac{{a}^{8}{x}^{16}}{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [A]  time = 0.95764, size = 122, normalized size = 1.15 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.39593, size = 219, normalized size = 2.07 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 0.080222, size = 105, normalized size = 0.99 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.14751, size = 122, normalized size = 1.15 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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