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]

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

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Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.03, size = 45, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 \sqrt {a} b^{3/2}}+\frac {x}{2 b \left (a x^2+b\right )} \]

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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fricas [A]  time = 1.51, size = 120, normalized size = 2.67 \[ \left [\frac {2 \, a b x - {\left (a x^{2} + b\right )} \sqrt {-a b} \log \left (\frac {a x^{2} - 2 \, \sqrt {-a b} x - b}{a x^{2} + b}\right )}{4 \, {\left (a^{2} b^{2} x^{2} + a b^{3}\right )}}, \frac {a b x + {\left (a x^{2} + b\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{b}\right )}{2 \, {\left (a^{2} b^{2} x^{2} + a b^{3}\right )}}\right ] \]

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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giac [A]  time = 0.16, size = 35, normalized size = 0.78 \[ \frac {\arctan \left (\frac {a x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b} + \frac {x}{2 \, {\left (a x^{2} + b\right )} b} \]

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)

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maple [A]  time = 0.00, size = 36, normalized size = 0.80 \[ \frac {x}{2 \left (a \,x^{2}+b \right ) b}+\frac {\arctan \left (\frac {a x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.99, size = 35, normalized size = 0.78 \[ \frac {x}{2 \, {\left (a b x^{2} + b^{2}\right )}} + \frac {\arctan \left (\frac {a x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 33, normalized size = 0.73 \[ \frac {x}{2\,b\,\left (a\,x^2+b\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{2\,\sqrt {a}\,b^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b/x^2)^2),x)

[Out]

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

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sympy [B]  time = 0.23, size = 78, normalized size = 1.73 \[ \frac {x}{2 a b x^{2} + 2 b^{2}} - \frac {\sqrt {- \frac {1}{a b^{3}}} \log {\left (- b^{2} \sqrt {- \frac {1}{a b^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a b^{3}}} \log {\left (b^{2} \sqrt {- \frac {1}{a b^{3}}} + x \right )}}{4} \]

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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