3.1600 \(\int \frac{(a+\frac{b}{x})^8}{x^2} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\left (a+\frac{b}{x}\right )^9}{9 b} \]

[Out]

-(a + b/x)^9/(9*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0034522, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ -\frac{\left (a+\frac{b}{x}\right )^9}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8/x^2,x]

[Out]

-(a + b/x)^9/(9*b)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^8}{x^2} \, dx &=-\frac{\left (a+\frac{b}{x}\right )^9}{9 b}\\ \end{align*}

Mathematica [B]  time = 0.0081863, size = 96, normalized size = 6. \[ -\frac{28 a^6 b^2}{3 x^3}-\frac{14 a^5 b^3}{x^4}-\frac{14 a^4 b^4}{x^5}-\frac{28 a^3 b^5}{3 x^6}-\frac{4 a^2 b^6}{x^7}-\frac{4 a^7 b}{x^2}-\frac{a^8}{x}-\frac{a b^7}{x^8}-\frac{b^8}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8/x^2,x]

[Out]

-b^8/(9*x^9) - (a*b^7)/x^8 - (4*a^2*b^6)/x^7 - (28*a^3*b^5)/(3*x^6) - (14*a^4*b^4)/x^5 - (14*a^5*b^3)/x^4 - (2
8*a^6*b^2)/(3*x^3) - (4*a^7*b)/x^2 - a^8/x

________________________________________________________________________________________

Maple [B]  time = 0.007, size = 91, normalized size = 5.7 \begin{align*} -{\frac{28\,{a}^{6}{b}^{2}}{3\,{x}^{3}}}-14\,{\frac{{a}^{4}{b}^{4}}{{x}^{5}}}-14\,{\frac{{a}^{5}{b}^{3}}{{x}^{4}}}-4\,{\frac{{a}^{7}b}{{x}^{2}}}-{\frac{{b}^{7}a}{{x}^{8}}}-{\frac{28\,{a}^{3}{b}^{5}}{3\,{x}^{6}}}-4\,{\frac{{a}^{2}{b}^{6}}{{x}^{7}}}-{\frac{{a}^{8}}{x}}-{\frac{{b}^{8}}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-28/3*a^6*b^2/x^3-14*a^4*b^4/x^5-14*a^5*b^3/x^4-4*a^7*b/x^2-b^7*a/x^8-28/3*a^3*b^5/x^6-4*a^2*b^6/x^7-a^8/x-1/9
*b^8/x^9

________________________________________________________________________________________

Maxima [A]  time = 1.00066, size = 19, normalized size = 1.19 \begin{align*} -\frac{{\left (a + \frac{b}{x}\right )}^{9}}{9 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(a + b/x)^9/b

________________________________________________________________________________________

Fricas [B]  time = 1.33535, size = 192, normalized size = 12. \begin{align*} -\frac{9 \, a^{8} x^{8} + 36 \, a^{7} b x^{7} + 84 \, a^{6} b^{2} x^{6} + 126 \, a^{5} b^{3} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{3} b^{5} x^{3} + 36 \, a^{2} b^{6} x^{2} + 9 \, a b^{7} x + b^{8}}{9 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(9*a^8*x^8 + 36*a^7*b*x^7 + 84*a^6*b^2*x^6 + 126*a^5*b^3*x^5 + 126*a^4*b^4*x^4 + 84*a^3*b^5*x^3 + 36*a^2*
b^6*x^2 + 9*a*b^7*x + b^8)/x^9

________________________________________________________________________________________

Sympy [B]  time = 0.744885, size = 95, normalized size = 5.94 \begin{align*} - \frac{9 a^{8} x^{8} + 36 a^{7} b x^{7} + 84 a^{6} b^{2} x^{6} + 126 a^{5} b^{3} x^{5} + 126 a^{4} b^{4} x^{4} + 84 a^{3} b^{5} x^{3} + 36 a^{2} b^{6} x^{2} + 9 a b^{7} x + b^{8}}{9 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(9*a**8*x**8 + 36*a**7*b*x**7 + 84*a**6*b**2*x**6 + 126*a**5*b**3*x**5 + 126*a**4*b**4*x**4 + 84*a**3*b**5*x*
*3 + 36*a**2*b**6*x**2 + 9*a*b**7*x + b**8)/(9*x**9)

________________________________________________________________________________________

Giac [A]  time = 1.10413, size = 19, normalized size = 1.19 \begin{align*} -\frac{{\left (a + \frac{b}{x}\right )}^{9}}{9 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(a + b/x)^9/b