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

Optimal. Leaf size=46 \[ \frac{17}{25} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{1}{5} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{34}{175} \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

-EllipticE[ArcSin[x/Sqrt[2]], -2]/5 + (17*EllipticF[ArcSin[x/Sqrt[2]], -2])/25 -
(34*EllipticPi[-10/7, ArcSin[x/Sqrt[2]], -2])/175

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Rubi [A]  time = 0.264677, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{17}{25} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{1}{5} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{34}{175} \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

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

[Out]

-EllipticE[ArcSin[x/Sqrt[2]], -2]/5 + (17*EllipticF[ArcSin[x/Sqrt[2]], -2])/25 -
(34*EllipticPi[-10/7, ArcSin[x/Sqrt[2]], -2])/175

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Rubi in Sympy [A]  time = 39.1256, size = 51, normalized size = 1.11 \[ - \frac{E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{5} + \frac{17 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{25} - \frac{34 \Pi \left (- \frac{10}{7}; \operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{175} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-elliptic_e(asin(sqrt(2)*x/2), -2)/5 + 17*elliptic_f(asin(sqrt(2)*x/2), -2)/25 -
34*elliptic_pi(-10/7, asin(sqrt(2)*x/2), -2)/175

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Mathematica [C]  time = 0.0853295, size = 51, normalized size = 1.11 \[ -\frac{1}{175} i \sqrt{2} \left (7 F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+35 E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-17 \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-I/175)*Sqrt[2]*(35*EllipticE[I*ArcSinh[x], -1/2] + 7*EllipticF[I*ArcSinh[x], -
1/2] - 17*EllipticPi[5/7, I*ArcSinh[x], -1/2])

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Maple [B]  time = 0.02, size = 141, normalized size = 3.1 \[{\frac{17\,\sqrt{2}}{50}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{\sqrt{2}}{10}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{34\,\sqrt{2}}{175}\sqrt{1-{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}},-{\frac{10}{7}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17/50*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))-1/10*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2
)*EllipticE(1/2*2^(1/2)*x,I*2^(1/2))-34/175*2^(1/2)*(1-1/2*x^2)^(1/2)*(x^2+1)^(1
/2)/(-x^4+x^2+2)^(1/2)*EllipticPi(1/2*2^(1/2)*x,-10/7,I*2^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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