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

Optimal. Leaf size=16 \[ \frac{E\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b}} \]

[Out]

EllipticE[ArcSin[Sqrt[b]*x], -1]/Sqrt[b]

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Rubi [A]  time = 0.0592676, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{E\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + b*x^2)/Sqrt[1 - b^2*x^4],x]

[Out]

EllipticE[ArcSin[Sqrt[b]*x], -1]/Sqrt[b]

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Rubi in Sympy [A]  time = 12.0469, size = 15, normalized size = 0.94 \[ \frac{E\left (\operatorname{asin}{\left (\sqrt{b} x \right )}\middle | -1\right )}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+1)/(-b**2*x**4+1)**(1/2),x)

[Out]

elliptic_e(asin(sqrt(b)*x), -1)/sqrt(b)

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Mathematica [C]  time = 0.0470417, size = 27, normalized size = 1.69 \[ -\frac{i E\left (\left .i \sinh ^{-1}\left (\sqrt{-b} x\right )\right |-1\right )}{\sqrt{-b}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(1 + b*x^2)/Sqrt[1 - b^2*x^4],x]

[Out]

((-I)*EllipticE[I*ArcSinh[Sqrt[-b]*x], -1])/Sqrt[-b]

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Maple [B]  time = 0.017, size = 100, normalized size = 6.3 \[{1\sqrt{-b{x}^{2}+1}\sqrt{b{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{b},i \right ){\frac{1}{\sqrt{b}}}{\frac{1}{\sqrt{-{b}^{2}{x}^{4}+1}}}}-{1\sqrt{-b{x}^{2}+1}\sqrt{b{x}^{2}+1} \left ({\it EllipticF} \left ( x\sqrt{b},i \right ) -{\it EllipticE} \left ( x\sqrt{b},i \right ) \right ){\frac{1}{\sqrt{b}}}{\frac{1}{\sqrt{-{b}^{2}{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+1)/(-b^2*x^4+1)^(1/2),x)

[Out]

1/b^(1/2)*(-b*x^2+1)^(1/2)*(b*x^2+1)^(1/2)/(-b^2*x^4+1)^(1/2)*EllipticF(x*b^(1/2
),I)-1/b^(1/2)*(-b*x^2+1)^(1/2)*(b*x^2+1)^(1/2)/(-b^2*x^4+1)^(1/2)*(EllipticF(x*
b^(1/2),I)-EllipticE(x*b^(1/2),I))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + 1)/sqrt(-b^2*x^4 + 1),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + 1)/sqrt(-b^2*x^4 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.62017, size = 70, normalized size = 4.38 \[ \frac{b x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{b^{2} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{b^{2} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+1)/(-b**2*x**4+1)**(1/2),x)

[Out]

b*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), b**2*x**4*exp_polar(2*I*pi))/(4*gamm
a(7/4)) + x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), b**2*x**4*exp_polar(2*I*pi))/(4
*gamma(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{b x^{2} + 1}{\sqrt{-b^{2} x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + 1)/sqrt(-b^2*x^4 + 1),x, algorithm="giac")

[Out]

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