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

Optimal. Leaf size=31 \[ \frac{x^3 \sqrt{b-\frac{a}{x^2}}}{2 \sqrt{a-b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0351915, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {515, 23, 30} \[ \frac{x^3 \sqrt{b-\frac{a}{x^2}}}{2 \sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b - a/x^2]*x^3)/(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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0095304, size = 31, normalized size = 1. \[ \frac{x^3 \sqrt{b-\frac{a}{x^2}}}{2 \sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 31, normalized size = 1. \begin{align*}{\frac{{x}^{3}}{2}\sqrt{-{\frac{-b{x}^{2}+a}{{x}^{2}}}}{\frac{1}{\sqrt{-b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 1.18261, size = 7, normalized size = 0.23 \begin{align*} -\frac{1}{2} i \, x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*x^2

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Fricas [A]  time = 1.46176, size = 82, normalized size = 2.65 \begin{align*} -\frac{\sqrt{-b x^{2} + a} x^{3} \sqrt{\frac{b x^{2} - a}{x^{2}}}}{2 \,{\left (b x^{2} - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-b*x^2 + a)*x^3*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^{2} \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**2*(b-a/x**2)**(1/2)/(-b*x**2+a)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.10978, size = 53, normalized size = 1.71 \begin{align*} -\frac{{\left (b x^{2} - a\right )} i \mathrm{sgn}\left (b x^{2} - a\right ) \mathrm{sgn}\left (x\right )}{2 \, b} + \frac{a i \mathrm{sgn}\left (a\right ) \mathrm{sgn}\left (x\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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