3.1873 \(\int \frac{1}{(a+\frac{b}{x^2})^2 x^9} \, dx\)

Optimal. Leaf size=66 \[ \frac{a^2}{2 b^3 \left (a x^2+b\right )}-\frac{3 a^2 \log \left (a x^2+b\right )}{2 b^4}+\frac{3 a^2 \log (x)}{b^4}+\frac{a}{b^3 x^2}-\frac{1}{4 b^2 x^4} \]

[Out]

-1/(4*b^2*x^4) + a/(b^3*x^2) + a^2/(2*b^3*(b + a*x^2)) + (3*a^2*Log[x])/b^4 - (3*a^2*Log[b + a*x^2])/(2*b^4)

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Rubi [A]  time = 0.0424086, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 266, 44} \[ \frac{a^2}{2 b^3 \left (a x^2+b\right )}-\frac{3 a^2 \log \left (a x^2+b\right )}{2 b^4}+\frac{3 a^2 \log (x)}{b^4}+\frac{a}{b^3 x^2}-\frac{1}{4 b^2 x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^2*x^9),x]

[Out]

-1/(4*b^2*x^4) + a/(b^3*x^2) + a^2/(2*b^3*(b + a*x^2)) + (3*a^2*Log[x])/b^4 - (3*a^2*Log[b + a*x^2])/(2*b^4)

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

Mathematica [A]  time = 0.0488867, size = 57, normalized size = 0.86 \[ \frac{b \left (\frac{2 a^2}{a x^2+b}+\frac{4 a}{x^2}-\frac{b}{x^4}\right )-6 a^2 \log \left (a x^2+b\right )+12 a^2 \log (x)}{4 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^2*x^9),x]

[Out]

(b*(-(b/x^4) + (4*a)/x^2 + (2*a^2)/(b + a*x^2)) + 12*a^2*Log[x] - 6*a^2*Log[b + a*x^2])/(4*b^4)

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Maple [A]  time = 0.012, size = 61, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{b}^{2}{x}^{4}}}+{\frac{a}{{b}^{3}{x}^{2}}}+{\frac{{a}^{2}}{2\,{b}^{3} \left ( a{x}^{2}+b \right ) }}+3\,{\frac{{a}^{2}\ln \left ( x \right ) }{{b}^{4}}}-{\frac{3\,{a}^{2}\ln \left ( a{x}^{2}+b \right ) }{2\,{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02335, size = 95, normalized size = 1.44 \begin{align*} \frac{6 \, a^{2} x^{4} + 3 \, a b x^{2} - b^{2}}{4 \,{\left (a b^{3} x^{6} + b^{4} x^{4}\right )}} - \frac{3 \, a^{2} \log \left (a x^{2} + b\right )}{2 \, b^{4}} + \frac{3 \, a^{2} \log \left (x^{2}\right )}{2 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(6*a^2*x^4 + 3*a*b*x^2 - b^2)/(a*b^3*x^6 + b^4*x^4) - 3/2*a^2*log(a*x^2 + b)/b^4 + 3/2*a^2*log(x^2)/b^4

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Fricas [A]  time = 1.4122, size = 184, normalized size = 2.79 \begin{align*} \frac{6 \, a^{2} b x^{4} + 3 \, a b^{2} x^{2} - b^{3} - 6 \,{\left (a^{3} x^{6} + a^{2} b x^{4}\right )} \log \left (a x^{2} + b\right ) + 12 \,{\left (a^{3} x^{6} + a^{2} b x^{4}\right )} \log \left (x\right )}{4 \,{\left (a b^{4} x^{6} + b^{5} x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(6*a^2*b*x^4 + 3*a*b^2*x^2 - b^3 - 6*(a^3*x^6 + a^2*b*x^4)*log(a*x^2 + b) + 12*(a^3*x^6 + a^2*b*x^4)*log(x
))/(a*b^4*x^6 + b^5*x^4)

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Sympy [A]  time = 0.874484, size = 68, normalized size = 1.03 \begin{align*} \frac{3 a^{2} \log{\left (x \right )}}{b^{4}} - \frac{3 a^{2} \log{\left (x^{2} + \frac{b}{a} \right )}}{2 b^{4}} + \frac{6 a^{2} x^{4} + 3 a b x^{2} - b^{2}}{4 a b^{3} x^{6} + 4 b^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a**2*log(x)/b**4 - 3*a**2*log(x**2 + b/a)/(2*b**4) + (6*a**2*x**4 + 3*a*b*x**2 - b**2)/(4*a*b**3*x**6 + 4*b*
*4*x**4)

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Giac [A]  time = 1.15918, size = 116, normalized size = 1.76 \begin{align*} \frac{3 \, a^{2} \log \left (x^{2}\right )}{2 \, b^{4}} - \frac{3 \, a^{2} \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, b^{4}} + \frac{3 \, a^{3} x^{2} + 4 \, a^{2} b}{2 \,{\left (a x^{2} + b\right )} b^{4}} - \frac{9 \, a^{2} x^{4} - 4 \, a b x^{2} + b^{2}}{4 \, b^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a^2*log(x^2)/b^4 - 3/2*a^2*log(abs(a*x^2 + b))/b^4 + 1/2*(3*a^3*x^2 + 4*a^2*b)/((a*x^2 + b)*b^4) - 1/4*(9*
a^2*x^4 - 4*a*b*x^2 + b^2)/(b^4*x^4)