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

Optimal. Leaf size=116 \[ -\frac{116100}{77} \left (-x^4+x^2+2\right )^{3/2} x+\frac{1}{231} \left (717372 x^2+177953\right ) \sqrt{-x^4+x^2+2} x-\frac{625}{11} \left (-x^4+x^2+2\right )^{3/2} x^5-\frac{14500}{33} \left (-x^4+x^2+2\right )^{3/2} x^3-\frac{539419}{77} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{3764813}{231} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

(x*(177953 + 717372*x^2)*Sqrt[2 + x^2 - x^4])/231 - (116100*x*(2 + x^2 - x^4)^(3
/2))/77 - (14500*x^3*(2 + x^2 - x^4)^(3/2))/33 - (625*x^5*(2 + x^2 - x^4)^(3/2))
/11 + (3764813*EllipticE[ArcSin[x/Sqrt[2]], -2])/231 - (539419*EllipticF[ArcSin[
x/Sqrt[2]], -2])/77

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Rubi [A]  time = 0.262908, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{116100}{77} \left (-x^4+x^2+2\right )^{3/2} x+\frac{1}{231} \left (717372 x^2+177953\right ) \sqrt{-x^4+x^2+2} x-\frac{625}{11} \left (-x^4+x^2+2\right )^{3/2} x^5-\frac{14500}{33} \left (-x^4+x^2+2\right )^{3/2} x^3-\frac{539419}{77} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{3764813}{231} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(177953 + 717372*x^2)*Sqrt[2 + x^2 - x^4])/231 - (116100*x*(2 + x^2 - x^4)^(3
/2))/77 - (14500*x^3*(2 + x^2 - x^4)^(3/2))/33 - (625*x^5*(2 + x^2 - x^4)^(3/2))
/11 + (3764813*EllipticE[ArcSin[x/Sqrt[2]], -2])/231 - (539419*EllipticF[ArcSin[
x/Sqrt[2]], -2])/77

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Rubi in Sympy [A]  time = 54.1714, size = 112, normalized size = 0.97 \[ - \frac{625 x^{5} \left (- x^{4} + x^{2} + 2\right )^{\frac{3}{2}}}{11} - \frac{14500 x^{3} \left (- x^{4} + x^{2} + 2\right )^{\frac{3}{2}}}{33} + \frac{x \left (\frac{3586860 x^{2}}{77} + \frac{889765}{77}\right ) \sqrt{- x^{4} + x^{2} + 2}}{15} - \frac{116100 x \left (- x^{4} + x^{2} + 2\right )^{\frac{3}{2}}}{77} + \frac{3764813 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{231} - \frac{539419 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{77} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-625*x**5*(-x**4 + x**2 + 2)**(3/2)/11 - 14500*x**3*(-x**4 + x**2 + 2)**(3/2)/33
+ x*(3586860*x**2/77 + 889765/77)*sqrt(-x**4 + x**2 + 2)/15 - 116100*x*(-x**4 +
x**2 + 2)**(3/2)/77 + 3764813*elliptic_e(asin(sqrt(2)*x/2), -2)/231 - 539419*el
liptic_f(asin(sqrt(2)*x/2), -2)/77

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Mathematica [C]  time = 0.124879, size = 112, normalized size = 0.97 \[ \frac{-13125 x^{13}-75250 x^{11}-105925 x^9+231228 x^7+1125819 x^5-186503 x^3-4838091 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+3764813 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-1037294 x}{231 \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-1037294*x - 186503*x^3 + 1125819*x^5 + 231228*x^7 - 105925*x^9 - 75250*x^11 -
13125*x^13 + (3764813*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] -
(4838091*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[x], -1/2])/(231*Sqrt[2
+ x^2 - x^4])

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Maple [A]  time = 0.035, size = 193, normalized size = 1.7 \[ -{\frac{518647\,x}{231}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{1073278\,\sqrt{2}}{231}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{3764813\,\sqrt{2}}{462}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{166072\,{x}^{3}}{231}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{20050\,{x}^{5}}{21}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{12625\,{x}^{7}}{33}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{625\,{x}^{9}}{11}\sqrt{-{x}^{4}+{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-518647/231*x*(-x^4+x^2+2)^(1/2)+1073278/231*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))-3764813/462*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)))+166072/231*x^3*(-x^4+x^2+2)^(1/2)+20050/
21*x^5*(-x^4+x^2+2)^(1/2)+12625/33*x^7*(-x^4+x^2+2)^(1/2)+625/11*x^9*(-x^4+x^2+2
)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (625 \, x^{8} + 3500 \, x^{6} + 7350 \, x^{4} + 6860 \, x^{2} + 2401\right )} \sqrt{-x^{4} + x^{2} + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((625*x^8 + 3500*x^6 + 7350*x^4 + 6860*x^2 + 2401)*sqrt(-x^4 + x^2 + 2),
x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \left (5 x^{2} + 7\right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-(x**2 - 2)*(x**2 + 1))*(5*x**2 + 7)**4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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