3.332 \(\int \left (2+x^2-x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=74 \[ \frac{1}{7} x \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{35} x \left (3 x^2+19\right ) \sqrt{-x^4+x^2+2}+\frac{48}{35} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{34}{35} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

(x*(19 + 3*x^2)*Sqrt[2 + x^2 - x^4])/35 + (x*(2 + x^2 - x^4)^(3/2))/7 + (34*Elli
pticE[ArcSin[x/Sqrt[2]], -2])/35 + (48*EllipticF[ArcSin[x/Sqrt[2]], -2])/35

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Rubi [A]  time = 0.161289, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ \frac{1}{7} x \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{35} x \left (3 x^2+19\right ) \sqrt{-x^4+x^2+2}+\frac{48}{35} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{34}{35} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 + x^2 - x^4)^(3/2),x]

[Out]

(x*(19 + 3*x^2)*Sqrt[2 + x^2 - x^4])/35 + (x*(2 + x^2 - x^4)^(3/2))/7 + (34*Elli
pticE[ArcSin[x/Sqrt[2]], -2])/35 + (48*EllipticF[ArcSin[x/Sqrt[2]], -2])/35

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Rubi in Sympy [A]  time = 25.7945, size = 70, normalized size = 0.95 \[ \frac{x \left (3 x^{2} + 19\right ) \sqrt{- x^{4} + x^{2} + 2}}{35} + \frac{x \left (- x^{4} + x^{2} + 2\right )^{\frac{3}{2}}}{7} + \frac{34 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{35} + \frac{48 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{35} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(3*x**2 + 19)*sqrt(-x**4 + x**2 + 2)/35 + x*(-x**4 + x**2 + 2)**(3/2)/7 + 34*e
lliptic_e(asin(sqrt(2)*x/2), -2)/35 + 48*elliptic_f(asin(sqrt(2)*x/2), -2)/35

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Mathematica [C]  time = 0.0888932, size = 102, normalized size = 1.38 \[ \frac{5 x^9-13 x^7-31 x^5+45 x^3-75 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+34 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+58 x}{35 \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + x^2 - x^4)^(3/2),x]

[Out]

(58*x + 45*x^3 - 31*x^5 - 13*x^7 + 5*x^9 + (34*I)*Sqrt[4 + 2*x^2 - 2*x^4]*Ellipt
icE[I*ArcSinh[x], -1/2] - (75*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[x],
-1/2])/(35*Sqrt[2 + x^2 - x^4])

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Maple [B]  time = 0.005, size = 159, normalized size = 2.2 \[ -{\frac{{x}^{5}}{7}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{8\,{x}^{3}}{35}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{29\,x}{35}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{41\,\sqrt{2}}{35}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{17\,\sqrt{2}}{35}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+x^2+2)^(3/2),x)

[Out]

-1/7*x^5*(-x^4+x^2+2)^(1/2)+8/35*x^3*(-x^4+x^2+2)^(1/2)+29/35*x*(-x^4+x^2+2)^(1/
2)+41/35*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))-17/35*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)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-x^4 + x^2 + 2)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-x^4 + x^2 + 2)^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (- x^{4} + x^{2} + 2\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((-x**4 + x**2 + 2)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-x^4 + x^2 + 2)^(3/2), x)