3.1596 \(\int (a+\frac{b}{x})^8 x^2 \, dx\)

Optimal. Leaf size=93 \[ -\frac{28 a^3 b^5}{x^2}-\frac{28 a^2 b^6}{3 x^3}+28 a^6 b^2 x-\frac{70 a^4 b^4}{x}+56 a^5 b^3 \log (x)+4 a^7 b x^2+\frac{a^8 x^3}{3}-\frac{2 a b^7}{x^4}-\frac{b^8}{5 x^5} \]

[Out]

-b^8/(5*x^5) - (2*a*b^7)/x^4 - (28*a^2*b^6)/(3*x^3) - (28*a^3*b^5)/x^2 - (70*a^4*b^4)/x + 28*a^6*b^2*x + 4*a^7
*b*x^2 + (a^8*x^3)/3 + 56*a^5*b^3*Log[x]

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Rubi [A]  time = 0.0431441, antiderivative size = 93, 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} \[ -\frac{28 a^3 b^5}{x^2}-\frac{28 a^2 b^6}{3 x^3}+28 a^6 b^2 x-\frac{70 a^4 b^4}{x}+56 a^5 b^3 \log (x)+4 a^7 b x^2+\frac{a^8 x^3}{3}-\frac{2 a b^7}{x^4}-\frac{b^8}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x^2,x]

[Out]

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

Mathematica [A]  time = 0.0090289, size = 93, normalized size = 1. \[ -\frac{28 a^3 b^5}{x^2}-\frac{28 a^2 b^6}{3 x^3}+28 a^6 b^2 x-\frac{70 a^4 b^4}{x}+56 a^5 b^3 \log (x)+4 a^7 b x^2+\frac{a^8 x^3}{3}-\frac{2 a b^7}{x^4}-\frac{b^8}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x^2,x]

[Out]

-b^8/(5*x^5) - (2*a*b^7)/x^4 - (28*a^2*b^6)/(3*x^3) - (28*a^3*b^5)/x^2 - (70*a^4*b^4)/x + 28*a^6*b^2*x + 4*a^7
*b*x^2 + (a^8*x^3)/3 + 56*a^5*b^3*Log[x]

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Maple [A]  time = 0.008, size = 88, normalized size = 1. \begin{align*} -{\frac{{b}^{8}}{5\,{x}^{5}}}-2\,{\frac{{b}^{7}a}{{x}^{4}}}-{\frac{28\,{a}^{2}{b}^{6}}{3\,{x}^{3}}}-28\,{\frac{{a}^{3}{b}^{5}}{{x}^{2}}}-70\,{\frac{{a}^{4}{b}^{4}}{x}}+28\,{a}^{6}{b}^{2}x+4\,{a}^{7}b{x}^{2}+{\frac{{a}^{8}{x}^{3}}{3}}+56\,{a}^{5}{b}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8*x^2,x)

[Out]

-1/5*b^8/x^5-2*a*b^7/x^4-28/3*a^2*b^6/x^3-28*a^3*b^5/x^2-70*a^4*b^4/x+28*a^6*b^2*x+4*a^7*b*x^2+1/3*a^8*x^3+56*
a^5*b^3*ln(x)

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Maxima [A]  time = 0.998388, size = 119, normalized size = 1.28 \begin{align*} \frac{1}{3} \, a^{8} x^{3} + 4 \, a^{7} b x^{2} + 28 \, a^{6} b^{2} x + 56 \, a^{5} b^{3} \log \left (x\right ) - \frac{1050 \, a^{4} b^{4} x^{4} + 420 \, a^{3} b^{5} x^{3} + 140 \, a^{2} b^{6} x^{2} + 30 \, a b^{7} x + 3 \, b^{8}}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^2,x, algorithm="maxima")

[Out]

1/3*a^8*x^3 + 4*a^7*b*x^2 + 28*a^6*b^2*x + 56*a^5*b^3*log(x) - 1/15*(1050*a^4*b^4*x^4 + 420*a^3*b^5*x^3 + 140*
a^2*b^6*x^2 + 30*a*b^7*x + 3*b^8)/x^5

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Fricas [A]  time = 1.44917, size = 211, normalized size = 2.27 \begin{align*} \frac{5 \, a^{8} x^{8} + 60 \, a^{7} b x^{7} + 420 \, a^{6} b^{2} x^{6} + 840 \, a^{5} b^{3} x^{5} \log \left (x\right ) - 1050 \, a^{4} b^{4} x^{4} - 420 \, a^{3} b^{5} x^{3} - 140 \, a^{2} b^{6} x^{2} - 30 \, a b^{7} x - 3 \, b^{8}}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^2,x, algorithm="fricas")

[Out]

1/15*(5*a^8*x^8 + 60*a^7*b*x^7 + 420*a^6*b^2*x^6 + 840*a^5*b^3*x^5*log(x) - 1050*a^4*b^4*x^4 - 420*a^3*b^5*x^3
- 140*a^2*b^6*x^2 - 30*a*b^7*x - 3*b^8)/x^5

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Sympy [A]  time = 0.541665, size = 94, normalized size = 1.01 \begin{align*} \frac{a^{8} x^{3}}{3} + 4 a^{7} b x^{2} + 28 a^{6} b^{2} x + 56 a^{5} b^{3} \log{\left (x \right )} - \frac{1050 a^{4} b^{4} x^{4} + 420 a^{3} b^{5} x^{3} + 140 a^{2} b^{6} x^{2} + 30 a b^{7} x + 3 b^{8}}{15 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8*x**2,x)

[Out]

a**8*x**3/3 + 4*a**7*b*x**2 + 28*a**6*b**2*x + 56*a**5*b**3*log(x) - (1050*a**4*b**4*x**4 + 420*a**3*b**5*x**3
+ 140*a**2*b**6*x**2 + 30*a*b**7*x + 3*b**8)/(15*x**5)

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Giac [A]  time = 1.10933, size = 120, normalized size = 1.29 \begin{align*} \frac{1}{3} \, a^{8} x^{3} + 4 \, a^{7} b x^{2} + 28 \, a^{6} b^{2} x + 56 \, a^{5} b^{3} \log \left ({\left | x \right |}\right ) - \frac{1050 \, a^{4} b^{4} x^{4} + 420 \, a^{3} b^{5} x^{3} + 140 \, a^{2} b^{6} x^{2} + 30 \, a b^{7} x + 3 \, b^{8}}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^2,x, algorithm="giac")

[Out]

1/3*a^8*x^3 + 4*a^7*b*x^2 + 28*a^6*b^2*x + 56*a^5*b^3*log(abs(x)) - 1/15*(1050*a^4*b^4*x^4 + 420*a^3*b^5*x^3 +
140*a^2*b^6*x^2 + 30*a*b^7*x + 3*b^8)/x^5