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

Optimal. Leaf size=112 \[ \frac{a \sqrt{1-x^2} \sqrt{\frac{a \left (x^2+1\right )}{a+b x^2}} \Pi \left (\frac{b}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b} x}{\sqrt{b x^2+a}}\right )|-\frac{a-b}{a+b}\right )}{\sqrt{x^2+1} \sqrt{a+b} \sqrt{\frac{a \left (1-x^2\right )}{a+b x^2}}} \]

[Out]

(a*Sqrt[1 - x^2]*Sqrt[(a*(1 + x^2))/(a + b*x^2)]*EllipticPi[b/(a + b), ArcSin[(Sqrt[a + b]*x)/Sqrt[a + b*x^2]]
, -((a - b)/(a + b))])/(Sqrt[a + b]*Sqrt[1 + x^2]*Sqrt[(a*(1 - x^2))/(a + b*x^2)])

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Rubi [F]  time = 0.0094724, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt{a+b x^2}}{\sqrt{1-x^4}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[a + b*x^2]/Sqrt[1 - x^4],x]

[Out]

Defer[Int][Sqrt[a + b*x^2]/Sqrt[1 - x^4], x]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b x^2}}{\sqrt{1-x^4}} \, dx &=\int \frac{\sqrt{a+b x^2}}{\sqrt{1-x^4}} \, dx\\ \end{align*}

Mathematica [F]  time = 0.0598551, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b x^2}}{\sqrt{1-x^4}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[a + b*x^2]/Sqrt[1 - x^4],x]

[Out]

Integrate[Sqrt[a + b*x^2]/Sqrt[1 - x^4], x]

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Maple [F]  time = 0.144, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{-{x}^{4}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x^{2} + a}}{\sqrt{-x^{4} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{4} + 1} \sqrt{b x^{2} + a}}{x^{4} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-x^4 + 1)*sqrt(b*x^2 + a)/(x^4 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x^{2} + a}}{\sqrt{-x^{4} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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