3.338 \(\int \frac{7+5 x^2}{\sqrt{2+x^2-x^4}} \, dx\)

Optimal. Leaf size=25 \[ 2 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+5 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

5*EllipticE[ArcSin[x/Sqrt[2]], -2] + 2*EllipticF[ArcSin[x/Sqrt[2]], -2]

_______________________________________________________________________________________

Rubi [A]  time = 0.127942, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ 2 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+5 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(7 + 5*x^2)/Sqrt[2 + x^2 - x^4],x]

[Out]

5*EllipticE[ArcSin[x/Sqrt[2]], -2] + 2*EllipticF[ArcSin[x/Sqrt[2]], -2]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.73, size = 29, normalized size = 1.16 \[ 5 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right ) + 2 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**2+7)/(-x**4+x**2+2)**(1/2),x)

[Out]

5*elliptic_e(asin(sqrt(2)*x/2), -2) + 2*elliptic_f(asin(sqrt(2)*x/2), -2)

_______________________________________________________________________________________

Mathematica [C]  time = 0.0584724, size = 34, normalized size = 1.36 \[ \frac{i \left (10 E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-17 F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(7 + 5*x^2)/Sqrt[2 + x^2 - x^4],x]

[Out]

(I*(10*EllipticE[I*ArcSinh[x], -1/2] - 17*EllipticF[I*ArcSinh[x], -1/2]))/Sqrt[2
]

_______________________________________________________________________________________

Maple [B]  time = 0.007, size = 110, normalized size = 4.4 \[{\frac{7\,\sqrt{2}}{2}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{5\,\sqrt{2}}{2}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^2+7)/(-x^4+x^2+2)^(1/2),x)

[Out]

7/2*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*EllipticF(1/2*2^(1
/2)*x,I*2^(1/2))-5/2*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*(
EllipticF(1/2*2^(1/2)*x,I*2^(1/2))-EllipticE(1/2*2^(1/2)*x,I*2^(1/2)))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{5 \, x^{2} + 7}{\sqrt{-x^{4} + x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 7)/sqrt(-x^4 + x^2 + 2),x, algorithm="maxima")

[Out]

integrate((5*x^2 + 7)/sqrt(-x^4 + x^2 + 2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{5 \, x^{2} + 7}{\sqrt{-x^{4} + x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 7)/sqrt(-x^4 + x^2 + 2),x, algorithm="fricas")

[Out]

integral((5*x^2 + 7)/sqrt(-x^4 + x^2 + 2), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{5 x^{2} + 7}{\sqrt{- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**2+7)/(-x**4+x**2+2)**(1/2),x)

[Out]

Integral((5*x**2 + 7)/sqrt(-(x**2 - 2)*(x**2 + 1)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{5 \, x^{2} + 7}{\sqrt{-x^{4} + x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 7)/sqrt(-x^4 + x^2 + 2),x, algorithm="giac")

[Out]

integrate((5*x^2 + 7)/sqrt(-x^4 + x^2 + 2), x)