3.336 \(\int \frac{\left (7+5 x^2\right )^3}{\sqrt{2+x^2-x^4}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{625}{3} \sqrt{-x^4+x^2+2} x-25 \sqrt{-x^4+x^2+2} x^3-542 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{3905}{3} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

(-625*x*Sqrt[2 + x^2 - x^4])/3 - 25*x^3*Sqrt[2 + x^2 - x^4] + (3905*EllipticE[Ar
cSin[x/Sqrt[2]], -2])/3 - 542*EllipticF[ArcSin[x/Sqrt[2]], -2]

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Rubi [A]  time = 0.198078, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{625}{3} \sqrt{-x^4+x^2+2} x-25 \sqrt{-x^4+x^2+2} x^3-542 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{3905}{3} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(7 + 5*x^2)^3/Sqrt[2 + x^2 - x^4],x]

[Out]

(-625*x*Sqrt[2 + x^2 - x^4])/3 - 25*x^3*Sqrt[2 + x^2 - x^4] + (3905*EllipticE[Ar
cSin[x/Sqrt[2]], -2])/3 - 542*EllipticF[ArcSin[x/Sqrt[2]], -2]

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Rubi in Sympy [A]  time = 39.0364, size = 65, normalized size = 1. \[ - 25 x^{3} \sqrt{- x^{4} + x^{2} + 2} - \frac{625 x \sqrt{- x^{4} + x^{2} + 2}}{3} + \frac{3905 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{3} - 542 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**2+7)**3/(-x**4+x**2+2)**(1/2),x)

[Out]

-25*x**3*sqrt(-x**4 + x**2 + 2) - 625*x*sqrt(-x**4 + x**2 + 2)/3 + 3905*elliptic
_e(asin(sqrt(2)*x/2), -2)/3 - 542*elliptic_f(asin(sqrt(2)*x/2), -2)

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Mathematica [C]  time = 0.123721, size = 97, normalized size = 1.49 \[ \frac{150 x^7+1100 x^5-1550 x^3-10089 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+7810 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-2500 x}{6 \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(7 + 5*x^2)^3/Sqrt[2 + x^2 - x^4],x]

[Out]

(-2500*x - 1550*x^3 + 1100*x^5 + 150*x^7 + (7810*I)*Sqrt[4 + 2*x^2 - 2*x^4]*Elli
pticE[I*ArcSinh[x], -1/2] - (10089*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSin
h[x], -1/2])/(6*Sqrt[2 + x^2 - x^4])

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Maple [B]  time = 0.024, size = 142, normalized size = 2.2 \[{\frac{2279\,\sqrt{2}}{6}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{3905\,\sqrt{2}}{6}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{625\,x}{3}\sqrt{-{x}^{4}+{x}^{2}+2}}-25\,{x}^{3}\sqrt{-{x}^{4}+{x}^{2}+2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^2+7)^3/(-x^4+x^2+2)^(1/2),x)

[Out]

2279/6*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))-3905/6*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)))-625
/3*x*(-x^4+x^2+2)^(1/2)-25*x^3*(-x^4+x^2+2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x^{2} + 7\right )}^{3}}{\sqrt{-x^{4} + x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 7)^3/sqrt(-x^4 + x^2 + 2),x, algorithm="maxima")

[Out]

integrate((5*x^2 + 7)^3/sqrt(-x^4 + x^2 + 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343}{\sqrt{-x^{4} + x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 7)^3/sqrt(-x^4 + x^2 + 2),x, algorithm="fricas")

[Out]

integral((125*x^6 + 525*x^4 + 735*x^2 + 343)/sqrt(-x^4 + x^2 + 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (5 x^{2} + 7\right )^{3}}{\sqrt{- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**2+7)**3/(-x**4+x**2+2)**(1/2),x)

[Out]

Integral((5*x**2 + 7)**3/sqrt(-(x**2 - 2)*(x**2 + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x^{2} + 7\right )}^{3}}{\sqrt{-x^{4} + x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 7)^3/sqrt(-x^4 + x^2 + 2),x, algorithm="giac")

[Out]

integrate((5*x^2 + 7)^3/sqrt(-x^4 + x^2 + 2), x)