3.598 \(\int \frac{\sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \, dx\)

Optimal. Leaf size=28 \[ \frac{x \log (x) \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0186658, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {435, 23, 29} \[ \frac{x \log (x) \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b - a/x^2]/Sqrt[a - b*x^2],x]

[Out]

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

Rule 435

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(x^(n*FracPart[q])*(c +
d/x^n)^FracPart[q])/(d + c*x^n)^FracPart[q], Int[((a + b*x^n)^p*(d + c*x^n)^q)/x^(n*q), x], x] /; FreeQ[{a, b,
c, d, n, p, q}, x] && EqQ[mn, -n] &&  !IntegerQ[q] &&  !IntegerQ[p]

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0082065, size = 28, normalized size = 1. \[ \frac{x \log (x) \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b - a/x^2]/Sqrt[a - b*x^2],x]

[Out]

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

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Maple [A]  time = 0.007, size = 42, normalized size = 1.5 \begin{align*} -{\frac{x\ln \left ( x \right ) }{b{x}^{2}-a}\sqrt{{\frac{b{x}^{2}-a}{{x}^{2}}}}\sqrt{-b{x}^{2}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 1.06245, size = 5, normalized size = 0.18 \begin{align*} -i \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*log(x)

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Fricas [B]  time = 1.61596, size = 115, normalized size = 4.11 \begin{align*} -\arctan \left (\frac{\sqrt{-b x^{2} + a}{\left (x^{3} + x\right )} \sqrt{\frac{b x^{2} - a}{x^{2}}}}{b x^{4} -{\left (a + b\right )} x^{2} + a}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(sqrt(-b*x^2 + a)*(x^3 + x)*sqrt((b*x^2 - a)/x^2)/(b*x^4 - (a + b)*x^2 + a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \frac{a}{x^{2}} + b}}{\sqrt{a - b x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-a/x**2 + b)/sqrt(a - b*x**2), x)

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Giac [A]  time = 1.10214, size = 42, normalized size = 1.5 \begin{align*} -\frac{1}{2} \, i \log \left ({\left (b x^{2} - a\right )} i + a i\right ) \mathrm{sgn}\left (b x^{2} - a\right ) \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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