3.1593 \(\int (a+\frac{b}{x})^8 x^5 \, dx\)

Optimal. Leaf size=95 \[ 7 a^6 b^2 x^4+\frac{56}{3} a^5 b^3 x^3+35 a^4 b^4 x^2+56 a^3 b^5 x+28 a^2 b^6 \log (x)+\frac{8}{5} a^7 b x^5+\frac{a^8 x^6}{6}-\frac{8 a b^7}{x}-\frac{b^8}{2 x^2} \]

[Out]

-b^8/(2*x^2) - (8*a*b^7)/x + 56*a^3*b^5*x + 35*a^4*b^4*x^2 + (56*a^5*b^3*x^3)/3 + 7*a^6*b^2*x^4 + (8*a^7*b*x^5
)/5 + (a^8*x^6)/6 + 28*a^2*b^6*Log[x]

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Rubi [A]  time = 0.0389603, antiderivative size = 95, 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} \[ 7 a^6 b^2 x^4+\frac{56}{3} a^5 b^3 x^3+35 a^4 b^4 x^2+56 a^3 b^5 x+28 a^2 b^6 \log (x)+\frac{8}{5} a^7 b x^5+\frac{a^8 x^6}{6}-\frac{8 a b^7}{x}-\frac{b^8}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x^5,x]

[Out]

-b^8/(2*x^2) - (8*a*b^7)/x + 56*a^3*b^5*x + 35*a^4*b^4*x^2 + (56*a^5*b^3*x^3)/3 + 7*a^6*b^2*x^4 + (8*a^7*b*x^5
)/5 + (a^8*x^6)/6 + 28*a^2*b^6*Log[x]

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^5 \, dx &=\int \frac{(b+a x)^8}{x^3} \, dx\\ &=\int \left (56 a^3 b^5+\frac{b^8}{x^3}+\frac{8 a b^7}{x^2}+\frac{28 a^2 b^6}{x}+70 a^4 b^4 x+56 a^5 b^3 x^2+28 a^6 b^2 x^3+8 a^7 b x^4+a^8 x^5\right ) \, dx\\ &=-\frac{b^8}{2 x^2}-\frac{8 a b^7}{x}+56 a^3 b^5 x+35 a^4 b^4 x^2+\frac{56}{3} a^5 b^3 x^3+7 a^6 b^2 x^4+\frac{8}{5} a^7 b x^5+\frac{a^8 x^6}{6}+28 a^2 b^6 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0045973, size = 95, normalized size = 1. \[ 7 a^6 b^2 x^4+\frac{56}{3} a^5 b^3 x^3+35 a^4 b^4 x^2+56 a^3 b^5 x+28 a^2 b^6 \log (x)+\frac{8}{5} a^7 b x^5+\frac{a^8 x^6}{6}-\frac{8 a b^7}{x}-\frac{b^8}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x^5,x]

[Out]

-b^8/(2*x^2) - (8*a*b^7)/x + 56*a^3*b^5*x + 35*a^4*b^4*x^2 + (56*a^5*b^3*x^3)/3 + 7*a^6*b^2*x^4 + (8*a^7*b*x^5
)/5 + (a^8*x^6)/6 + 28*a^2*b^6*Log[x]

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Maple [A]  time = 0.003, size = 88, normalized size = 0.9 \begin{align*} -{\frac{{b}^{8}}{2\,{x}^{2}}}-8\,{\frac{{b}^{7}a}{x}}+56\,{a}^{3}{b}^{5}x+35\,{a}^{4}{b}^{4}{x}^{2}+{\frac{56\,{a}^{5}{b}^{3}{x}^{3}}{3}}+7\,{a}^{6}{b}^{2}{x}^{4}+{\frac{8\,{a}^{7}b{x}^{5}}{5}}+{\frac{{a}^{8}{x}^{6}}{6}}+28\,{a}^{2}{b}^{6}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8*x^5,x)

[Out]

-1/2*b^8/x^2-8*a*b^7/x+56*a^3*b^5*x+35*a^4*b^4*x^2+56/3*a^5*b^3*x^3+7*a^6*b^2*x^4+8/5*a^7*b*x^5+1/6*a^8*x^6+28
*a^2*b^6*ln(x)

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Maxima [A]  time = 0.955887, size = 116, normalized size = 1.22 \begin{align*} \frac{1}{6} \, a^{8} x^{6} + \frac{8}{5} \, a^{7} b x^{5} + 7 \, a^{6} b^{2} x^{4} + \frac{56}{3} \, a^{5} b^{3} x^{3} + 35 \, a^{4} b^{4} x^{2} + 56 \, a^{3} b^{5} x + 28 \, a^{2} b^{6} \log \left (x\right ) - \frac{16 \, a b^{7} x + b^{8}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^5,x, algorithm="maxima")

[Out]

1/6*a^8*x^6 + 8/5*a^7*b*x^5 + 7*a^6*b^2*x^4 + 56/3*a^5*b^3*x^3 + 35*a^4*b^4*x^2 + 56*a^3*b^5*x + 28*a^2*b^6*lo
g(x) - 1/2*(16*a*b^7*x + b^8)/x^2

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Fricas [A]  time = 1.43349, size = 215, normalized size = 2.26 \begin{align*} \frac{5 \, a^{8} x^{8} + 48 \, a^{7} b x^{7} + 210 \, a^{6} b^{2} x^{6} + 560 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 1680 \, a^{3} b^{5} x^{3} + 840 \, a^{2} b^{6} x^{2} \log \left (x\right ) - 240 \, a b^{7} x - 15 \, b^{8}}{30 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^5,x, algorithm="fricas")

[Out]

1/30*(5*a^8*x^8 + 48*a^7*b*x^7 + 210*a^6*b^2*x^6 + 560*a^5*b^3*x^5 + 1050*a^4*b^4*x^4 + 1680*a^3*b^5*x^3 + 840
*a^2*b^6*x^2*log(x) - 240*a*b^7*x - 15*b^8)/x^2

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Sympy [A]  time = 0.389573, size = 95, normalized size = 1. \begin{align*} \frac{a^{8} x^{6}}{6} + \frac{8 a^{7} b x^{5}}{5} + 7 a^{6} b^{2} x^{4} + \frac{56 a^{5} b^{3} x^{3}}{3} + 35 a^{4} b^{4} x^{2} + 56 a^{3} b^{5} x + 28 a^{2} b^{6} \log{\left (x \right )} - \frac{16 a b^{7} x + b^{8}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8*x**5,x)

[Out]

a**8*x**6/6 + 8*a**7*b*x**5/5 + 7*a**6*b**2*x**4 + 56*a**5*b**3*x**3/3 + 35*a**4*b**4*x**2 + 56*a**3*b**5*x +
28*a**2*b**6*log(x) - (16*a*b**7*x + b**8)/(2*x**2)

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Giac [A]  time = 1.56671, size = 117, normalized size = 1.23 \begin{align*} \frac{1}{6} \, a^{8} x^{6} + \frac{8}{5} \, a^{7} b x^{5} + 7 \, a^{6} b^{2} x^{4} + \frac{56}{3} \, a^{5} b^{3} x^{3} + 35 \, a^{4} b^{4} x^{2} + 56 \, a^{3} b^{5} x + 28 \, a^{2} b^{6} \log \left ({\left | x \right |}\right ) - \frac{16 \, a b^{7} x + b^{8}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^5,x, algorithm="giac")

[Out]

1/6*a^8*x^6 + 8/5*a^7*b*x^5 + 7*a^6*b^2*x^4 + 56/3*a^5*b^3*x^3 + 35*a^4*b^4*x^2 + 56*a^3*b^5*x + 28*a^2*b^6*lo
g(abs(x)) - 1/2*(16*a*b^7*x + b^8)/x^2