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

Optimal. Leaf size=74 \[ -\frac{25}{7} x \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{21} x \left (354 x^2+275\right ) \sqrt{-x^4+x^2+2}-\frac{79}{7} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{2045}{21} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

(x*(275 + 354*x^2)*Sqrt[2 + x^2 - x^4])/21 - (25*x*(2 + x^2 - x^4)^(3/2))/7 + (2
045*EllipticE[ArcSin[x/Sqrt[2]], -2])/21 - (79*EllipticF[ArcSin[x/Sqrt[2]], -2])
/7

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Rubi [A]  time = 0.189563, antiderivative size = 74, 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 \[ -\frac{25}{7} x \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{21} x \left (354 x^2+275\right ) \sqrt{-x^4+x^2+2}-\frac{79}{7} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{2045}{21} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(275 + 354*x^2)*Sqrt[2 + x^2 - x^4])/21 - (25*x*(2 + x^2 - x^4)^(3/2))/7 + (2
045*EllipticE[ArcSin[x/Sqrt[2]], -2])/21 - (79*EllipticF[ArcSin[x/Sqrt[2]], -2])
/7

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Rubi in Sympy [A]  time = 33.0303, size = 75, normalized size = 1.01 \[ \frac{x \left (\frac{1770 x^{2}}{7} + \frac{1375}{7}\right ) \sqrt{- x^{4} + x^{2} + 2}}{15} - \frac{25 x \left (- x^{4} + x^{2} + 2\right )^{\frac{3}{2}}}{7} + \frac{2045 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{21} - \frac{79 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(1770*x**2/7 + 1375/7)*sqrt(-x**4 + x**2 + 2)/15 - 25*x*(-x**4 + x**2 + 2)**(3
/2)/7 + 2045*elliptic_e(asin(sqrt(2)*x/2), -2)/21 - 79*elliptic_f(asin(sqrt(2)*x
/2), -2)/7

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Mathematica [C]  time = 0.100961, size = 102, normalized size = 1.38 \[ \frac{-75 x^9-204 x^7+304 x^5+683 x^3-2949 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+2045 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+250 x}{21 \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(250*x + 683*x^3 + 304*x^5 - 204*x^7 - 75*x^9 + (2045*I)*Sqrt[4 + 2*x^2 - 2*x^4]
*EllipticE[I*ArcSinh[x], -1/2] - (2949*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*Ar
cSinh[x], -1/2])/(21*Sqrt[2 + x^2 - x^4])

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Maple [B]  time = 0.011, size = 159, normalized size = 2.2 \[{\frac{125\,x}{21}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{904\,\sqrt{2}}{21}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{2045\,\sqrt{2}}{42}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{93\,{x}^{3}}{7}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{25\,{x}^{5}}{7}\sqrt{-{x}^{4}+{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/21*x*(-x^4+x^2+2)^(1/2)+904/21*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))-2045/42*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)))+93/7*x^3*(-x^4+x^2+2)^(1/2)+25/7*x^5*(-x^4+x^2+2)^(1/2
)

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

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)