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

Optimal. Leaf size=46 \[ x \sqrt{-x^4+x^2+2} \left (x^2+2\right )+3 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+7 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

x*(2 + x^2)*Sqrt[2 + x^2 - x^4] + 7*EllipticE[ArcSin[x/Sqrt[2]], -2] + 3*Ellipti
cF[ArcSin[x/Sqrt[2]], -2]

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Rubi [A]  time = 0.153491, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ x \sqrt{-x^4+x^2+2} \left (x^2+2\right )+3 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+7 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

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

[Out]

x*(2 + x^2)*Sqrt[2 + x^2 - x^4] + 7*EllipticE[ArcSin[x/Sqrt[2]], -2] + 3*Ellipti
cF[ArcSin[x/Sqrt[2]], -2]

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Rubi in Sympy [A]  time = 24.2568, size = 51, normalized size = 1.11 \[ \frac{x \left (15 x^{2} + 30\right ) \sqrt{- x^{4} + x^{2} + 2}}{15} + 7 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right ) + 3 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right ) \]

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]

x*(15*x**2 + 30)*sqrt(-x**4 + x**2 + 2)/15 + 7*elliptic_e(asin(sqrt(2)*x/2), -2)
+ 3*elliptic_f(asin(sqrt(2)*x/2), -2)

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Mathematica [C]  time = 0.102937, size = 94, normalized size = 2.04 \[ \frac{-x^7-x^5+4 x^3-12 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+7 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+4 x}{\sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*x + 4*x^3 - x^5 - x^7 + (7*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x],
-1/2] - (12*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[x], -1/2])/Sqrt[2 +
x^2 - x^4]

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Maple [B]  time = 0.008, size = 141, normalized size = 3.1 \[ 2\,x\sqrt{-{x}^{4}+{x}^{2}+2}+5\,{\frac{\sqrt{2}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ( 1/2\,\sqrt{2}x,i\sqrt{2} \right ) }{\sqrt{-{x}^{4}+{x}^{2}+2}}}-{\frac{7\,\sqrt{2}}{2}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{x}^{3}\sqrt{-{x}^{4}+{x}^{2}+2} \]

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]

2*x*(-x^4+x^2+2)^(1/2)+5*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))-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))-EllipticE(1/2*2^(1/2)*x,
I*2^(1/2)))+x^3*(-x^4+x^2+2)^(1/2)

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

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. \[{\rm integral}\left (\sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}, x\right ) \]

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. \[ \int \sqrt{- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \left (5 x^{2} + 7\right )\, dx \]

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

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)