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

Optimal. Leaf size=74 \[ \frac{\sqrt{-x^4+x^2+2} x}{14 \left (5 x^2+7\right )}-\frac{6}{175} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{1}{70} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{99 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{2450} \]

[Out]

(x*Sqrt[2 + x^2 - x^4])/(14*(7 + 5*x^2)) + EllipticE[ArcSin[x/Sqrt[2]], -2]/70 -
(6*EllipticF[ArcSin[x/Sqrt[2]], -2])/175 + (99*EllipticPi[-10/7, ArcSin[x/Sqrt[
2]], -2])/2450

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Rubi [A]  time = 0.268251, antiderivative size = 74, 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 \[ \frac{\sqrt{-x^4+x^2+2} x}{14 \left (5 x^2+7\right )}-\frac{6}{175} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{1}{70} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{99 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{2450} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[2 + x^2 - x^4])/(14*(7 + 5*x^2)) + EllipticE[ArcSin[x/Sqrt[2]], -2]/70 -
(6*EllipticF[ArcSin[x/Sqrt[2]], -2])/175 + (99*EllipticPi[-10/7, ArcSin[x/Sqrt[
2]], -2])/2450

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Rubi in Sympy [A]  time = 37.3469, size = 178, normalized size = 2.41 \[ \frac{x \sqrt{- x^{4} + x^{2} + 2}}{70 x^{2} + 98} + \frac{\sqrt{2} \sqrt{- x^{4} + x^{2} + 2} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{140 \sqrt{- \frac{x^{2}}{2} + 1} \sqrt{x^{2} + 1}} - \frac{3 \sqrt{2} \sqrt{- x^{4} + x^{2} + 2} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{175 \sqrt{- \frac{x^{2}}{2} + 1} \sqrt{x^{2} + 1}} + \frac{99 \sqrt{2} \sqrt{- x^{4} + x^{2} + 2} \Pi \left (- \frac{10}{7}; \operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{4900 \sqrt{- \frac{x^{2}}{2} + 1} \sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(-x**4 + x**2 + 2)/(70*x**2 + 98) + sqrt(2)*sqrt(-x**4 + x**2 + 2)*ellipti
c_e(asin(sqrt(2)*x/2), -2)/(140*sqrt(-x**2/2 + 1)*sqrt(x**2 + 1)) - 3*sqrt(2)*sq
rt(-x**4 + x**2 + 2)*elliptic_f(asin(sqrt(2)*x/2), -2)/(175*sqrt(-x**2/2 + 1)*sq
rt(x**2 + 1)) + 99*sqrt(2)*sqrt(-x**4 + x**2 + 2)*elliptic_pi(-10/7, asin(sqrt(2
)*x/2), -2)/(4900*sqrt(-x**2/2 + 1)*sqrt(x**2 + 1))

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Mathematica [C]  time = 0.267206, size = 196, normalized size = 2.65 \[ \frac{-350 x^5+350 x^3-21 i \sqrt{2} \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+70 i \sqrt{2} \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-495 i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )-693 i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )+700 x}{4900 \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(700*x + 350*x^3 - 350*x^5 + (70*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4]*Elli
pticE[I*ArcSinh[x], -1/2] - (21*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4]*Ellip
ticF[I*ArcSinh[x], -1/2] - (693*I)*Sqrt[2]*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I
*ArcSinh[x], -1/2] - (495*I)*Sqrt[2]*x^2*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*A
rcSinh[x], -1/2])/(4900*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4])

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Maple [B]  time = 0.027, size = 165, normalized size = 2.2 \[{\frac{x}{70\,{x}^{2}+98}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{3\,\sqrt{2}}{175}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{\sqrt{2}}{140}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{99\,\sqrt{2}}{2450}\sqrt{1-{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}},-{\frac{10}{7}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14*x*(-x^4+x^2+2)^(1/2)/(5*x^2+7)-3/175*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))+1/140*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))+99/245
0*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. \[ \int \frac{\sqrt{-x^{4} + x^{2} + 2}}{{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-x^{4} + x^{2} + 2}}{25 \, x^{4} + 70 \, x^{2} + 49}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(-x^4 + x^2 + 2)/(25*x^4 + 70*x^2 + 49), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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