3.1869 \(\int \frac{1}{(a+\frac{b}{x^2})^2 x^5} \, dx\)

Optimal. Leaf size=38 \[ -\frac{\log \left (a x^2+b\right )}{2 b^2}+\frac{1}{2 b \left (a x^2+b\right )}+\frac{\log (x)}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.025249, antiderivative size = 38, 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} \[ -\frac{\log \left (a x^2+b\right )}{2 b^2}+\frac{1}{2 b \left (a x^2+b\right )}+\frac{\log (x)}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^2*x^5),x]

[Out]

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

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

Mathematica [A]  time = 0.0121957, size = 33, normalized size = 0.87 \[ \frac{\frac{b}{a x^2+b}-\log \left (a x^2+b\right )+2 \log (x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^2*x^5),x]

[Out]

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

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Maple [A]  time = 0.011, size = 35, normalized size = 0.9 \begin{align*}{\frac{1}{2\,b \left ( a{x}^{2}+b \right ) }}+{\frac{\ln \left ( x \right ) }{{b}^{2}}}-{\frac{\ln \left ( a{x}^{2}+b \right ) }{2\,{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x^2*b)^2/x^5,x)

[Out]

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

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Maxima [A]  time = 0.992228, size = 50, normalized size = 1.32 \begin{align*} \frac{1}{2 \,{\left (a b x^{2} + b^{2}\right )}} - \frac{\log \left (a x^{2} + b\right )}{2 \, b^{2}} + \frac{\log \left (x^{2}\right )}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^2/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45399, size = 108, normalized size = 2.84 \begin{align*} -\frac{{\left (a x^{2} + b\right )} \log \left (a x^{2} + b\right ) - 2 \,{\left (a x^{2} + b\right )} \log \left (x\right ) - b}{2 \,{\left (a b^{2} x^{2} + b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^2/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.569055, size = 34, normalized size = 0.89 \begin{align*} \frac{1}{2 a b x^{2} + 2 b^{2}} + \frac{\log{\left (x \right )}}{b^{2}} - \frac{\log{\left (x^{2} + \frac{b}{a} \right )}}{2 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**2/x**5,x)

[Out]

1/(2*a*b*x**2 + 2*b**2) + log(x)/b**2 - log(x**2 + b/a)/(2*b**2)

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Giac [A]  time = 1.1681, size = 63, normalized size = 1.66 \begin{align*} \frac{\log \left (x^{2}\right )}{2 \, b^{2}} - \frac{\log \left ({\left | a x^{2} + b \right |}\right )}{2 \, b^{2}} + \frac{a x^{2} + 2 \, b}{2 \,{\left (a x^{2} + b\right )} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^2/x^5,x, algorithm="giac")

[Out]

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