3.1595 \(\int (a+\frac{b}{x})^8 x^3 \, dx\)

Optimal. Leaf size=95 \[ 14 a^6 b^2 x^2-\frac{14 a^2 b^6}{x^2}+56 a^5 b^3 x-\frac{56 a^3 b^5}{x}+70 a^4 b^4 \log (x)+\frac{8}{3} a^7 b x^3+\frac{a^8 x^4}{4}-\frac{8 a b^7}{3 x^3}-\frac{b^8}{4 x^4} \]

[Out]

-b^8/(4*x^4) - (8*a*b^7)/(3*x^3) - (14*a^2*b^6)/x^2 - (56*a^3*b^5)/x + 56*a^5*b^3*x + 14*a^6*b^2*x^2 + (8*a^7*
b*x^3)/3 + (a^8*x^4)/4 + 70*a^4*b^4*Log[x]

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Rubi [A]  time = 0.0392652, 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} \[ 14 a^6 b^2 x^2-\frac{14 a^2 b^6}{x^2}+56 a^5 b^3 x-\frac{56 a^3 b^5}{x}+70 a^4 b^4 \log (x)+\frac{8}{3} a^7 b x^3+\frac{a^8 x^4}{4}-\frac{8 a b^7}{3 x^3}-\frac{b^8}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x^3,x]

[Out]

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

Mathematica [A]  time = 0.0047304, size = 95, normalized size = 1. \[ 14 a^6 b^2 x^2-\frac{14 a^2 b^6}{x^2}+56 a^5 b^3 x-\frac{56 a^3 b^5}{x}+70 a^4 b^4 \log (x)+\frac{8}{3} a^7 b x^3+\frac{a^8 x^4}{4}-\frac{8 a b^7}{3 x^3}-\frac{b^8}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x^3,x]

[Out]

-b^8/(4*x^4) - (8*a*b^7)/(3*x^3) - (14*a^2*b^6)/x^2 - (56*a^3*b^5)/x + 56*a^5*b^3*x + 14*a^6*b^2*x^2 + (8*a^7*
b*x^3)/3 + (a^8*x^4)/4 + 70*a^4*b^4*Log[x]

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Maple [A]  time = 0.007, size = 88, normalized size = 0.9 \begin{align*} -{\frac{{b}^{8}}{4\,{x}^{4}}}-{\frac{8\,{b}^{7}a}{3\,{x}^{3}}}-14\,{\frac{{a}^{2}{b}^{6}}{{x}^{2}}}-56\,{\frac{{a}^{3}{b}^{5}}{x}}+56\,{a}^{5}{b}^{3}x+14\,{a}^{6}{b}^{2}{x}^{2}+{\frac{8\,{a}^{7}b{x}^{3}}{3}}+{\frac{{a}^{8}{x}^{4}}{4}}+70\,{a}^{4}{b}^{4}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8*x^3,x)

[Out]

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

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Maxima [A]  time = 1.00472, size = 119, normalized size = 1.25 \begin{align*} \frac{1}{4} \, a^{8} x^{4} + \frac{8}{3} \, a^{7} b x^{3} + 14 \, a^{6} b^{2} x^{2} + 56 \, a^{5} b^{3} x + 70 \, a^{4} b^{4} \log \left (x\right ) - \frac{672 \, a^{3} b^{5} x^{3} + 168 \, a^{2} b^{6} x^{2} + 32 \, a b^{7} x + 3 \, b^{8}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^3,x, algorithm="maxima")

[Out]

1/4*a^8*x^4 + 8/3*a^7*b*x^3 + 14*a^6*b^2*x^2 + 56*a^5*b^3*x + 70*a^4*b^4*log(x) - 1/12*(672*a^3*b^5*x^3 + 168*
a^2*b^6*x^2 + 32*a*b^7*x + 3*b^8)/x^4

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Fricas [A]  time = 1.43669, size = 209, normalized size = 2.2 \begin{align*} \frac{3 \, a^{8} x^{8} + 32 \, a^{7} b x^{7} + 168 \, a^{6} b^{2} x^{6} + 672 \, a^{5} b^{3} x^{5} + 840 \, a^{4} b^{4} x^{4} \log \left (x\right ) - 672 \, a^{3} b^{5} x^{3} - 168 \, a^{2} b^{6} x^{2} - 32 \, a b^{7} x - 3 \, b^{8}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^3,x, algorithm="fricas")

[Out]

1/12*(3*a^8*x^8 + 32*a^7*b*x^7 + 168*a^6*b^2*x^6 + 672*a^5*b^3*x^5 + 840*a^4*b^4*x^4*log(x) - 672*a^3*b^5*x^3
- 168*a^2*b^6*x^2 - 32*a*b^7*x - 3*b^8)/x^4

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Sympy [A]  time = 0.479598, size = 95, normalized size = 1. \begin{align*} \frac{a^{8} x^{4}}{4} + \frac{8 a^{7} b x^{3}}{3} + 14 a^{6} b^{2} x^{2} + 56 a^{5} b^{3} x + 70 a^{4} b^{4} \log{\left (x \right )} - \frac{672 a^{3} b^{5} x^{3} + 168 a^{2} b^{6} x^{2} + 32 a b^{7} x + 3 b^{8}}{12 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8*x**3,x)

[Out]

a**8*x**4/4 + 8*a**7*b*x**3/3 + 14*a**6*b**2*x**2 + 56*a**5*b**3*x + 70*a**4*b**4*log(x) - (672*a**3*b**5*x**3
+ 168*a**2*b**6*x**2 + 32*a*b**7*x + 3*b**8)/(12*x**4)

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Giac [A]  time = 1.11281, size = 120, normalized size = 1.26 \begin{align*} \frac{1}{4} \, a^{8} x^{4} + \frac{8}{3} \, a^{7} b x^{3} + 14 \, a^{6} b^{2} x^{2} + 56 \, a^{5} b^{3} x + 70 \, a^{4} b^{4} \log \left ({\left | x \right |}\right ) - \frac{672 \, a^{3} b^{5} x^{3} + 168 \, a^{2} b^{6} x^{2} + 32 \, a b^{7} x + 3 \, b^{8}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^3,x, algorithm="giac")

[Out]

1/4*a^8*x^4 + 8/3*a^7*b*x^3 + 14*a^6*b^2*x^2 + 56*a^5*b^3*x + 70*a^4*b^4*log(abs(x)) - 1/12*(672*a^3*b^5*x^3 +
168*a^2*b^6*x^2 + 32*a*b^7*x + 3*b^8)/x^4