3.1583 \(\int (a+\frac{b}{x})^8 x^{16} \, dx\)

Optimal. Leaf size=106 \[ \frac{28}{15} a^6 b^2 x^{15}+4 a^5 b^3 x^{14}+\frac{70}{13} a^4 b^4 x^{13}+\frac{14}{3} a^3 b^5 x^{12}+\frac{28}{11} a^2 b^6 x^{11}+\frac{1}{2} a^7 b x^{16}+\frac{a^8 x^{17}}{17}+\frac{4}{5} a b^7 x^{10}+\frac{b^8 x^9}{9} \]

[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

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Rubi [A]  time = 0.0569334, 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} \[ \frac{28}{15} a^6 b^2 x^{15}+4 a^5 b^3 x^{14}+\frac{70}{13} a^4 b^4 x^{13}+\frac{14}{3} a^3 b^5 x^{12}+\frac{28}{11} a^2 b^6 x^{11}+\frac{1}{2} a^7 b x^{16}+\frac{a^8 x^{17}}{17}+\frac{4}{5} a b^7 x^{10}+\frac{b^8 x^9}{9} \]

Antiderivative was successfully verified.

[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 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^{16} \, dx &=\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]  time = 0.0027548, size = 106, normalized size = 1. \[ \frac{28}{15} a^6 b^2 x^{15}+4 a^5 b^3 x^{14}+\frac{70}{13} a^4 b^4 x^{13}+\frac{14}{3} a^3 b^5 x^{12}+\frac{28}{11} a^2 b^6 x^{11}+\frac{1}{2} a^7 b x^{16}+\frac{a^8 x^{17}}{17}+\frac{4}{5} a b^7 x^{10}+\frac{b^8 x^9}{9} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.002, size = 91, normalized size = 0.9 \begin{align*}{\frac{{b}^{8}{x}^{9}}{9}}+{\frac{4\,a{b}^{7}{x}^{10}}{5}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{11}}{11}}+{\frac{14\,{a}^{3}{b}^{5}{x}^{12}}{3}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{13}}{13}}+4\,{a}^{5}{b}^{3}{x}^{14}+{\frac{28\,{a}^{6}{b}^{2}{x}^{15}}{15}}+{\frac{{a}^{7}b{x}^{16}}{2}}+{\frac{{a}^{8}{x}^{17}}{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [A]  time = 0.959201, size = 122, normalized size = 1.15 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.41615, size = 220, normalized size = 2.08 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 0.081245, size = 104, normalized size = 0.98 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.1983, size = 122, normalized size = 1.15 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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