3.1599 \(\int \frac{(a+\frac{b}{x})^8}{x} \, dx\)

Optimal. Leaf size=100 \[ -\frac{14 a^6 b^2}{x^2}-\frac{56 a^5 b^3}{3 x^3}-\frac{35 a^4 b^4}{2 x^4}-\frac{56 a^3 b^5}{5 x^5}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a^7 b}{x}+a^8 \log (x)-\frac{8 a b^7}{7 x^7}-\frac{b^8}{8 x^8} \]

[Out]

-b^8/(8*x^8) - (8*a*b^7)/(7*x^7) - (14*a^2*b^6)/(3*x^6) - (56*a^3*b^5)/(5*x^5) - (35*a^4*b^4)/(2*x^4) - (56*a^
5*b^3)/(3*x^3) - (14*a^6*b^2)/x^2 - (8*a^7*b)/x + a^8*Log[x]

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Rubi [A]  time = 0.0420873, antiderivative size = 100, 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} \[ -\frac{14 a^6 b^2}{x^2}-\frac{56 a^5 b^3}{3 x^3}-\frac{35 a^4 b^4}{2 x^4}-\frac{56 a^3 b^5}{5 x^5}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a^7 b}{x}+a^8 \log (x)-\frac{8 a b^7}{7 x^7}-\frac{b^8}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8/x,x]

[Out]

-b^8/(8*x^8) - (8*a*b^7)/(7*x^7) - (14*a^2*b^6)/(3*x^6) - (56*a^3*b^5)/(5*x^5) - (35*a^4*b^4)/(2*x^4) - (56*a^
5*b^3)/(3*x^3) - (14*a^6*b^2)/x^2 - (8*a^7*b)/x + a^8*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 \frac{\left (a+\frac{b}{x}\right )^8}{x} \, dx &=\int \frac{(b+a x)^8}{x^9} \, dx\\ &=\int \left (\frac{b^8}{x^9}+\frac{8 a b^7}{x^8}+\frac{28 a^2 b^6}{x^7}+\frac{56 a^3 b^5}{x^6}+\frac{70 a^4 b^4}{x^5}+\frac{56 a^5 b^3}{x^4}+\frac{28 a^6 b^2}{x^3}+\frac{8 a^7 b}{x^2}+\frac{a^8}{x}\right ) \, dx\\ &=-\frac{b^8}{8 x^8}-\frac{8 a b^7}{7 x^7}-\frac{14 a^2 b^6}{3 x^6}-\frac{56 a^3 b^5}{5 x^5}-\frac{35 a^4 b^4}{2 x^4}-\frac{56 a^5 b^3}{3 x^3}-\frac{14 a^6 b^2}{x^2}-\frac{8 a^7 b}{x}+a^8 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0041736, size = 100, normalized size = 1. \[ -\frac{14 a^6 b^2}{x^2}-\frac{56 a^5 b^3}{3 x^3}-\frac{35 a^4 b^4}{2 x^4}-\frac{56 a^3 b^5}{5 x^5}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a^7 b}{x}+a^8 \log (x)-\frac{8 a b^7}{7 x^7}-\frac{b^8}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8/x,x]

[Out]

-b^8/(8*x^8) - (8*a*b^7)/(7*x^7) - (14*a^2*b^6)/(3*x^6) - (56*a^3*b^5)/(5*x^5) - (35*a^4*b^4)/(2*x^4) - (56*a^
5*b^3)/(3*x^3) - (14*a^6*b^2)/x^2 - (8*a^7*b)/x + a^8*Log[x]

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Maple [A]  time = 0.007, size = 89, normalized size = 0.9 \begin{align*} -{\frac{{b}^{8}}{8\,{x}^{8}}}-{\frac{8\,{b}^{7}a}{7\,{x}^{7}}}-{\frac{14\,{a}^{2}{b}^{6}}{3\,{x}^{6}}}-{\frac{56\,{a}^{3}{b}^{5}}{5\,{x}^{5}}}-{\frac{35\,{a}^{4}{b}^{4}}{2\,{x}^{4}}}-{\frac{56\,{a}^{5}{b}^{3}}{3\,{x}^{3}}}-14\,{\frac{{a}^{6}{b}^{2}}{{x}^{2}}}-8\,{\frac{{a}^{7}b}{x}}+{a}^{8}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*b^8/x^8-8/7*a*b^7/x^7-14/3*a^2*b^6/x^6-56/5*a^3*b^5/x^5-35/2*a^4*b^4/x^4-56/3*a^5*b^3/x^3-14*a^6*b^2/x^2-
8*a^7*b/x+a^8*ln(x)

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Maxima [A]  time = 0.990892, size = 120, normalized size = 1.2 \begin{align*} a^{8} \log \left (x\right ) - \frac{6720 \, a^{7} b x^{7} + 11760 \, a^{6} b^{2} x^{6} + 15680 \, a^{5} b^{3} x^{5} + 14700 \, a^{4} b^{4} x^{4} + 9408 \, a^{3} b^{5} x^{3} + 3920 \, a^{2} b^{6} x^{2} + 960 \, a b^{7} x + 105 \, b^{8}}{840 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^8*log(x) - 1/840*(6720*a^7*b*x^7 + 11760*a^6*b^2*x^6 + 15680*a^5*b^3*x^5 + 14700*a^4*b^4*x^4 + 9408*a^3*b^5*
x^3 + 3920*a^2*b^6*x^2 + 960*a*b^7*x + 105*b^8)/x^8

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Fricas [A]  time = 1.42111, size = 231, normalized size = 2.31 \begin{align*} \frac{840 \, a^{8} x^{8} \log \left (x\right ) - 6720 \, a^{7} b x^{7} - 11760 \, a^{6} b^{2} x^{6} - 15680 \, a^{5} b^{3} x^{5} - 14700 \, a^{4} b^{4} x^{4} - 9408 \, a^{3} b^{5} x^{3} - 3920 \, a^{2} b^{6} x^{2} - 960 \, a b^{7} x - 105 \, b^{8}}{840 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(840*a^8*x^8*log(x) - 6720*a^7*b*x^7 - 11760*a^6*b^2*x^6 - 15680*a^5*b^3*x^5 - 14700*a^4*b^4*x^4 - 9408*
a^3*b^5*x^3 - 3920*a^2*b^6*x^2 - 960*a*b^7*x - 105*b^8)/x^8

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Sympy [A]  time = 0.71769, size = 94, normalized size = 0.94 \begin{align*} a^{8} \log{\left (x \right )} - \frac{6720 a^{7} b x^{7} + 11760 a^{6} b^{2} x^{6} + 15680 a^{5} b^{3} x^{5} + 14700 a^{4} b^{4} x^{4} + 9408 a^{3} b^{5} x^{3} + 3920 a^{2} b^{6} x^{2} + 960 a b^{7} x + 105 b^{8}}{840 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**8*log(x) - (6720*a**7*b*x**7 + 11760*a**6*b**2*x**6 + 15680*a**5*b**3*x**5 + 14700*a**4*b**4*x**4 + 9408*a*
*3*b**5*x**3 + 3920*a**2*b**6*x**2 + 960*a*b**7*x + 105*b**8)/(840*x**8)

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Giac [A]  time = 1.09003, size = 122, normalized size = 1.22 \begin{align*} a^{8} \log \left ({\left | x \right |}\right ) - \frac{6720 \, a^{7} b x^{7} + 11760 \, a^{6} b^{2} x^{6} + 15680 \, a^{5} b^{3} x^{5} + 14700 \, a^{4} b^{4} x^{4} + 9408 \, a^{3} b^{5} x^{3} + 3920 \, a^{2} b^{6} x^{2} + 960 \, a b^{7} x + 105 \, b^{8}}{840 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^8*log(abs(x)) - 1/840*(6720*a^7*b*x^7 + 11760*a^6*b^2*x^6 + 15680*a^5*b^3*x^5 + 14700*a^4*b^4*x^4 + 9408*a^3
*b^5*x^3 + 3920*a^2*b^6*x^2 + 960*a*b^7*x + 105*b^8)/x^8