3.167 \(\int \frac{1}{x^9 (a+b x^2)^2} \, dx\)

Optimal. Leaf size=93 \[ \frac{b^4}{2 a^5 \left (a+b x^2\right )}+\frac{2 b^3}{a^5 x^2}-\frac{3 b^2}{4 a^4 x^4}-\frac{5 b^4 \log \left (a+b x^2\right )}{2 a^6}+\frac{5 b^4 \log (x)}{a^6}+\frac{b}{3 a^3 x^6}-\frac{1}{8 a^2 x^8} \]

[Out]

-1/(8*a^2*x^8) + b/(3*a^3*x^6) - (3*b^2)/(4*a^4*x^4) + (2*b^3)/(a^5*x^2) + b^4/(2*a^5*(a + b*x^2)) + (5*b^4*Lo
g[x])/a^6 - (5*b^4*Log[a + b*x^2])/(2*a^6)

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Rubi [A]  time = 0.0652697, 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 = {266, 44} \[ \frac{b^4}{2 a^5 \left (a+b x^2\right )}+\frac{2 b^3}{a^5 x^2}-\frac{3 b^2}{4 a^4 x^4}-\frac{5 b^4 \log \left (a+b x^2\right )}{2 a^6}+\frac{5 b^4 \log (x)}{a^6}+\frac{b}{3 a^3 x^6}-\frac{1}{8 a^2 x^8} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^9*(a + b*x^2)^2),x]

[Out]

-1/(8*a^2*x^8) + b/(3*a^3*x^6) - (3*b^2)/(4*a^4*x^4) + (2*b^3)/(a^5*x^2) + b^4/(2*a^5*(a + b*x^2)) + (5*b^4*Lo
g[x])/a^6 - (5*b^4*Log[a + b*x^2])/(2*a^6)

Rule 266

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

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^9 \left (a+b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^5 (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^5}-\frac{2 b}{a^3 x^4}+\frac{3 b^2}{a^4 x^3}-\frac{4 b^3}{a^5 x^2}+\frac{5 b^4}{a^6 x}-\frac{b^5}{a^5 (a+b x)^2}-\frac{5 b^5}{a^6 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{8 a^2 x^8}+\frac{b}{3 a^3 x^6}-\frac{3 b^2}{4 a^4 x^4}+\frac{2 b^3}{a^5 x^2}+\frac{b^4}{2 a^5 \left (a+b x^2\right )}+\frac{5 b^4 \log (x)}{a^6}-\frac{5 b^4 \log \left (a+b x^2\right )}{2 a^6}\\ \end{align*}

Mathematica [A]  time = 0.0711963, size = 79, normalized size = 0.85 \[ \frac{a \left (\frac{8 a^2 b}{x^6}-\frac{3 a^3}{x^8}-\frac{18 a b^2}{x^4}+12 b^3 \left (\frac{b}{a+b x^2}+\frac{4}{x^2}\right )\right )-60 b^4 \log \left (a+b x^2\right )+120 b^4 \log (x)}{24 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^9*(a + b*x^2)^2),x]

[Out]

(a*((-3*a^3)/x^8 + (8*a^2*b)/x^6 - (18*a*b^2)/x^4 + 12*b^3*(4/x^2 + b/(a + b*x^2))) + 120*b^4*Log[x] - 60*b^4*
Log[a + b*x^2])/(24*a^6)

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Maple [A]  time = 0.013, size = 84, normalized size = 0.9 \begin{align*} -{\frac{1}{8\,{a}^{2}{x}^{8}}}+{\frac{b}{3\,{a}^{3}{x}^{6}}}-{\frac{3\,{b}^{2}}{4\,{a}^{4}{x}^{4}}}+2\,{\frac{{b}^{3}}{{a}^{5}{x}^{2}}}+{\frac{{b}^{4}}{2\,{a}^{5} \left ( b{x}^{2}+a \right ) }}+5\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{6}}}-{\frac{5\,{b}^{4}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^9/(b*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 2.10876, size = 124, normalized size = 1.33 \begin{align*} \frac{60 \, b^{4} x^{8} + 30 \, a b^{3} x^{6} - 10 \, a^{2} b^{2} x^{4} + 5 \, a^{3} b x^{2} - 3 \, a^{4}}{24 \,{\left (a^{5} b x^{10} + a^{6} x^{8}\right )}} - \frac{5 \, b^{4} \log \left (b x^{2} + a\right )}{2 \, a^{6}} + \frac{5 \, b^{4} \log \left (x^{2}\right )}{2 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

1/24*(60*b^4*x^8 + 30*a*b^3*x^6 - 10*a^2*b^2*x^4 + 5*a^3*b*x^2 - 3*a^4)/(a^5*b*x^10 + a^6*x^8) - 5/2*b^4*log(b
*x^2 + a)/a^6 + 5/2*b^4*log(x^2)/a^6

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Fricas [A]  time = 1.14426, size = 242, normalized size = 2.6 \begin{align*} \frac{60 \, a b^{4} x^{8} + 30 \, a^{2} b^{3} x^{6} - 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} - 3 \, a^{5} - 60 \,{\left (b^{5} x^{10} + a b^{4} x^{8}\right )} \log \left (b x^{2} + a\right ) + 120 \,{\left (b^{5} x^{10} + a b^{4} x^{8}\right )} \log \left (x\right )}{24 \,{\left (a^{6} b x^{10} + a^{7} x^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/24*(60*a*b^4*x^8 + 30*a^2*b^3*x^6 - 10*a^3*b^2*x^4 + 5*a^4*b*x^2 - 3*a^5 - 60*(b^5*x^10 + a*b^4*x^8)*log(b*x
^2 + a) + 120*(b^5*x^10 + a*b^4*x^8)*log(x))/(a^6*b*x^10 + a^7*x^8)

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Sympy [A]  time = 1.29343, size = 94, normalized size = 1.01 \begin{align*} \frac{- 3 a^{4} + 5 a^{3} b x^{2} - 10 a^{2} b^{2} x^{4} + 30 a b^{3} x^{6} + 60 b^{4} x^{8}}{24 a^{6} x^{8} + 24 a^{5} b x^{10}} + \frac{5 b^{4} \log{\left (x \right )}}{a^{6}} - \frac{5 b^{4} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**9/(b*x**2+a)**2,x)

[Out]

(-3*a**4 + 5*a**3*b*x**2 - 10*a**2*b**2*x**4 + 30*a*b**3*x**6 + 60*b**4*x**8)/(24*a**6*x**8 + 24*a**5*b*x**10)
+ 5*b**4*log(x)/a**6 - 5*b**4*log(a/b + x**2)/(2*a**6)

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Giac [A]  time = 2.67453, size = 149, normalized size = 1.6 \begin{align*} \frac{5 \, b^{4} \log \left (x^{2}\right )}{2 \, a^{6}} - \frac{5 \, b^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{6}} + \frac{5 \, b^{5} x^{2} + 6 \, a b^{4}}{2 \,{\left (b x^{2} + a\right )} a^{6}} - \frac{125 \, b^{4} x^{8} - 48 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} - 8 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, a^{6} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^2+a)^2,x, algorithm="giac")

[Out]

5/2*b^4*log(x^2)/a^6 - 5/2*b^4*log(abs(b*x^2 + a))/a^6 + 1/2*(5*b^5*x^2 + 6*a*b^4)/((b*x^2 + a)*a^6) - 1/24*(1
25*b^4*x^8 - 48*a*b^3*x^6 + 18*a^2*b^2*x^4 - 8*a^3*b*x^2 + 3*a^4)/(a^6*x^8)