3.1853 \(\int \frac{1}{(a+\frac{b}{x^2}) x^4} \, dx\)

Optimal. Leaf size=34 \[ -\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]

[Out]

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

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Rubi [A]  time = 0.0118095, antiderivative size = 34, 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, 325, 205} \[ -\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)*x^4),x]

[Out]

-(1/(b*x)) - (Sqrt[a]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/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 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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

Mathematica [A]  time = 0.0131259, size = 34, normalized size = 1. \[ -\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)*x^4),x]

[Out]

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

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Maple [A]  time = 0.005, size = 30, normalized size = 0.9 \begin{align*} -{\frac{a}{b}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{bx}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.61264, size = 173, normalized size = 5.09 \begin{align*} \left [\frac{x \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) - 2}{2 \, b x}, -\frac{x \sqrt{\frac{a}{b}} \arctan \left (x \sqrt{\frac{a}{b}}\right ) + 1}{b x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.320201, size = 65, normalized size = 1.91 \begin{align*} \frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (x - \frac{b^{2} \sqrt{- \frac{a}{b^{3}}}}{a} \right )}}{2} - \frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (x + \frac{b^{2} \sqrt{- \frac{a}{b^{3}}}}{a} \right )}}{2} - \frac{1}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15348, size = 39, normalized size = 1.15 \begin{align*} -\frac{a \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} b} - \frac{1}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*b) - 1/(b*x)