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

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

[Out]

(Sqrt[b - a/x^2]*x^2)/Sqrt[a - b*x^2]

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Rubi [A]  time = 0.0260666, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {515, 23, 8} \[ \frac{x^2 \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b - a/x^2]*x^2)/Sqrt[a - b*x^2]

Rule 515

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Dist[(x^(n*FracPa
rt[q])*(c + d/x^n)^FracPart[q])/(d + c*x^n)^FracPart[q], Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q, x], x] /
; FreeQ[{a, b, c, d, m, 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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0081602, size = 28, normalized size = 1. \[ \frac{x^2 \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b - a/x^2]*x^2)/Sqrt[a - b*x^2]

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Maple [A]  time = 0.01, size = 42, normalized size = 1.5 \begin{align*} -{\frac{{x}^{2}}{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(x*(b-a/x^2)^(1/2)/(-b*x^2+a)^(1/2),x)

[Out]

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

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Maxima [C]  time = 1.1588, size = 9, normalized size = 0.32 \begin{align*} -i \, \sqrt{x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*sqrt(x^2)

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Fricas [A]  time = 1.44236, size = 77, normalized size = 2.75 \begin{align*} -\frac{\sqrt{-b x^{2} + a} x^{2} \sqrt{\frac{b x^{2} - a}{x^{2}}}}{b x^{2} - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \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(x*(b-a/x**2)**(1/2)/(-b*x**2+a)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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