3.341 \(\int \frac{1}{\left (7+5 x^2\right )^2 \sqrt{2+x^2-x^4}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{25 \sqrt{-x^4+x^2+2} x}{476 \left (5 x^2+7\right )}-\frac{1}{238} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{5}{476} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{167 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{3332} \]

[Out]

(-25*x*Sqrt[2 + x^2 - x^4])/(476*(7 + 5*x^2)) - (5*EllipticE[ArcSin[x/Sqrt[2]],
-2])/476 - EllipticF[ArcSin[x/Sqrt[2]], -2]/238 + (167*EllipticPi[-10/7, ArcSin[
x/Sqrt[2]], -2])/3332

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Rubi [A]  time = 0.262785, 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 \[ -\frac{25 \sqrt{-x^4+x^2+2} x}{476 \left (5 x^2+7\right )}-\frac{1}{238} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{5}{476} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{167 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{3332} \]

Antiderivative was successfully verified.

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

[Out]

(-25*x*Sqrt[2 + x^2 - x^4])/(476*(7 + 5*x^2)) - (5*EllipticE[ArcSin[x/Sqrt[2]],
-2])/476 - EllipticF[ArcSin[x/Sqrt[2]], -2]/238 + (167*EllipticPi[-10/7, ArcSin[
x/Sqrt[2]], -2])/3332

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Rubi in Sympy [A]  time = 38.5835, size = 73, normalized size = 0.99 \[ - \frac{25 x \sqrt{- x^{4} + x^{2} + 2}}{2380 x^{2} + 3332} - \frac{5 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{476} - \frac{F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{238} + \frac{167 \Pi \left (- \frac{10}{7}; \operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{3332} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25*x*sqrt(-x**4 + x**2 + 2)/(2380*x**2 + 3332) - 5*elliptic_e(asin(sqrt(2)*x/2)
, -2)/476 - elliptic_f(asin(sqrt(2)*x/2), -2)/238 + 167*elliptic_pi(-10/7, asin(
sqrt(2)*x/2), -2)/3332

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Mathematica [C]  time = 0.316699, size = 196, normalized size = 2.65 \[ \frac{350 x^5-350 x^3+119 i \sqrt{2} \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-70 i \sqrt{2} \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-835 i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )-1169 i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )-700 x}{6664 \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-700*x - 350*x^3 + 350*x^5 - (70*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4]*Ell
ipticE[I*ArcSinh[x], -1/2] + (119*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4]*Ell
ipticF[I*ArcSinh[x], -1/2] - (1169*I)*Sqrt[2]*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7
, I*ArcSinh[x], -1/2] - (835*I)*Sqrt[2]*x^2*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7,
I*ArcSinh[x], -1/2])/(6664*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4])

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Maple [B]  time = 0.026, size = 165, normalized size = 2.2 \[ -{\frac{25\,x}{2380\,{x}^{2}+3332}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{\sqrt{2}}{476}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{5\,\sqrt{2}}{952}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{167\,\sqrt{2}}{3332}\sqrt{1-{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}},-{\frac{10}{7}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/476*x*(-x^4+x^2+2)^(1/2)/(5*x^2+7)-1/476*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))-5/952*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))+167
/3332*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. \[ \int \frac{1}{\sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)^2),x, algorithm="maxima")

[Out]

integrate(1/(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. \[{\rm integral}\left (\frac{1}{{\left (25 \, x^{4} + 70 \, x^{2} + 49\right )} \sqrt{-x^{4} + x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)^2),x, algorithm="fricas")

[Out]

integral(1/((25*x^4 + 70*x^2 + 49)*sqrt(-x^4 + x^2 + 2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(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. \[ \int \frac{1}{\sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)^2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)^2), x)