3.1597 \(\int (a+\frac{b}{x})^8 x \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0392794, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {263, 43} \[ -\frac{35 a^4 b^4}{x^2}-\frac{56 a^3 b^5}{3 x^3}-\frac{7 a^2 b^6}{x^4}-\frac{56 a^5 b^3}{x}+28 a^6 b^2 \log (x)+8 a^7 b x+\frac{a^8 x^2}{2}-\frac{8 a b^7}{5 x^5}-\frac{b^8}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x,x]

[Out]

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

Mathematica [A]  time = 0.004581, size = 95, normalized size = 1. \[ -\frac{35 a^4 b^4}{x^2}-\frac{56 a^3 b^5}{3 x^3}-\frac{7 a^2 b^6}{x^4}-\frac{56 a^5 b^3}{x}+28 a^6 b^2 \log (x)+8 a^7 b x+\frac{a^8 x^2}{2}-\frac{8 a b^7}{5 x^5}-\frac{b^8}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x,x]

[Out]

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

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Maple [A]  time = 0.006, size = 88, normalized size = 0.9 \begin{align*} -{\frac{{b}^{8}}{6\,{x}^{6}}}-{\frac{8\,{b}^{7}a}{5\,{x}^{5}}}-7\,{\frac{{a}^{2}{b}^{6}}{{x}^{4}}}-{\frac{56\,{a}^{3}{b}^{5}}{3\,{x}^{3}}}-35\,{\frac{{a}^{4}{b}^{4}}{{x}^{2}}}-56\,{\frac{{a}^{5}{b}^{3}}{x}}+8\,{a}^{7}bx+{\frac{{a}^{8}{x}^{2}}{2}}+28\,{a}^{6}{b}^{2}\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]

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

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Maxima [A]  time = 0.977334, size = 119, normalized size = 1.25 \begin{align*} \frac{1}{2} \, a^{8} x^{2} + 8 \, a^{7} b x + 28 \, a^{6} b^{2} \log \left (x\right ) - \frac{1680 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 560 \, a^{3} b^{5} x^{3} + 210 \, a^{2} b^{6} x^{2} + 48 \, a b^{7} x + 5 \, b^{8}}{30 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^8*x^2 + 8*a^7*b*x + 28*a^6*b^2*log(x) - 1/30*(1680*a^5*b^3*x^5 + 1050*a^4*b^4*x^4 + 560*a^3*b^5*x^3 + 21
0*a^2*b^6*x^2 + 48*a*b^7*x + 5*b^8)/x^6

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Fricas [A]  time = 1.40989, size = 215, normalized size = 2.26 \begin{align*} \frac{15 \, a^{8} x^{8} + 240 \, a^{7} b x^{7} + 840 \, a^{6} b^{2} x^{6} \log \left (x\right ) - 1680 \, a^{5} b^{3} x^{5} - 1050 \, a^{4} b^{4} x^{4} - 560 \, a^{3} b^{5} x^{3} - 210 \, a^{2} b^{6} x^{2} - 48 \, a b^{7} x - 5 \, b^{8}}{30 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.575822, size = 94, normalized size = 0.99 \begin{align*} \frac{a^{8} x^{2}}{2} + 8 a^{7} b x + 28 a^{6} b^{2} \log{\left (x \right )} - \frac{1680 a^{5} b^{3} x^{5} + 1050 a^{4} b^{4} x^{4} + 560 a^{3} b^{5} x^{3} + 210 a^{2} b^{6} x^{2} + 48 a b^{7} x + 5 b^{8}}{30 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**8*x**2/2 + 8*a**7*b*x + 28*a**6*b**2*log(x) - (1680*a**5*b**3*x**5 + 1050*a**4*b**4*x**4 + 560*a**3*b**5*x*
*3 + 210*a**2*b**6*x**2 + 48*a*b**7*x + 5*b**8)/(30*x**6)

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Giac [A]  time = 1.12208, size = 120, normalized size = 1.26 \begin{align*} \frac{1}{2} \, a^{8} x^{2} + 8 \, a^{7} b x + 28 \, a^{6} b^{2} \log \left ({\left | x \right |}\right ) - \frac{1680 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 560 \, a^{3} b^{5} x^{3} + 210 \, a^{2} b^{6} x^{2} + 48 \, a b^{7} x + 5 \, b^{8}}{30 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^8*x^2 + 8*a^7*b*x + 28*a^6*b^2*log(abs(x)) - 1/30*(1680*a^5*b^3*x^5 + 1050*a^4*b^4*x^4 + 560*a^3*b^5*x^3
+ 210*a^2*b^6*x^2 + 48*a*b^7*x + 5*b^8)/x^6