3.1868 \(\int \frac{1}{(a+\frac{b}{x^2})^2 x^4} \, dx\)

Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 \sqrt{a} b^{3/2}}+\frac{x}{2 b \left (a x^2+b\right )} \]

[Out]

x/(2*b*(b + a*x^2)) + ArcTan[(Sqrt[a]*x)/Sqrt[b]]/(2*Sqrt[a]*b^(3/2))

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Rubi [A]  time = 0.0114527, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 199, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 \sqrt{a} b^{3/2}}+\frac{x}{2 b \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^2*x^4),x]

[Out]

x/(2*b*(b + a*x^2)) + ArcTan[(Sqrt[a]*x)/Sqrt[b]]/(2*Sqrt[a]*b^(3/2))

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^2 x^4} \, dx &=\int \frac{1}{\left (b+a x^2\right )^2} \, dx\\ &=\frac{x}{2 b \left (b+a x^2\right )}+\frac{\int \frac{1}{b+a x^2} \, dx}{2 b}\\ &=\frac{x}{2 b \left (b+a x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 \sqrt{a} b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0236597, size = 45, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 \sqrt{a} b^{3/2}}+\frac{x}{2 b \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^2*x^4),x]

[Out]

x/(2*b*(b + a*x^2)) + ArcTan[(Sqrt[a]*x)/Sqrt[b]]/(2*Sqrt[a]*b^(3/2))

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Maple [A]  time = 0.003, size = 36, normalized size = 0.8 \begin{align*}{\frac{x}{2\,b \left ( a{x}^{2}+b \right ) }}+{\frac{1}{2\,b}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x^2*b)^2/x^4,x)

[Out]

1/2*x/b/(a*x^2+b)+1/2/b/(a*b)^(1/2)*arctan(a*x/(a*b)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^2/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.45185, size = 261, normalized size = 5.8 \begin{align*} \left [\frac{2 \, a b x -{\left (a x^{2} + b\right )} \sqrt{-a b} \log \left (\frac{a x^{2} - 2 \, \sqrt{-a b} x - b}{a x^{2} + b}\right )}{4 \,{\left (a^{2} b^{2} x^{2} + a b^{3}\right )}}, \frac{a b x +{\left (a x^{2} + b\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{b}\right )}{2 \,{\left (a^{2} b^{2} x^{2} + a b^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^2/x^4,x, algorithm="fricas")

[Out]

[1/4*(2*a*b*x - (a*x^2 + b)*sqrt(-a*b)*log((a*x^2 - 2*sqrt(-a*b)*x - b)/(a*x^2 + b)))/(a^2*b^2*x^2 + a*b^3), 1
/2*(a*b*x + (a*x^2 + b)*sqrt(a*b)*arctan(sqrt(a*b)*x/b))/(a^2*b^2*x^2 + a*b^3)]

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Sympy [B]  time = 0.49636, size = 78, normalized size = 1.73 \begin{align*} \frac{x}{2 a b x^{2} + 2 b^{2}} - \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (- b^{2} \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (b^{2} \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**2/x**4,x)

[Out]

x/(2*a*b*x**2 + 2*b**2) - sqrt(-1/(a*b**3))*log(-b**2*sqrt(-1/(a*b**3)) + x)/4 + sqrt(-1/(a*b**3))*log(b**2*sq
rt(-1/(a*b**3)) + x)/4

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Giac [A]  time = 1.15798, size = 47, normalized size = 1.04 \begin{align*} \frac{\arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b} + \frac{x}{2 \,{\left (a x^{2} + b\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^2/x^4,x, algorithm="giac")

[Out]

1/2*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*b) + 1/2*x/((a*x^2 + b)*b)