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

Optimal. Leaf size=55 \[ \frac{x \left (4897 x^2+4945\right )}{18 \sqrt{-x^4+x^2+2}}+\frac{1763}{6} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{7147}{18} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

(x*(4945 + 4897*x^2))/(18*Sqrt[2 + x^2 - x^4]) - (7147*EllipticE[ArcSin[x/Sqrt[2
]], -2])/18 + (1763*EllipticF[ArcSin[x/Sqrt[2]], -2])/6

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Rubi [A]  time = 0.168468, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{x \left (4897 x^2+4945\right )}{18 \sqrt{-x^4+x^2+2}}+\frac{1763}{6} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{7147}{18} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(4945 + 4897*x^2))/(18*Sqrt[2 + x^2 - x^4]) - (7147*EllipticE[ArcSin[x/Sqrt[2
]], -2])/18 + (1763*EllipticF[ArcSin[x/Sqrt[2]], -2])/6

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Rubi in Sympy [A]  time = 29.5979, size = 54, normalized size = 0.98 \[ \frac{x \left (4897 x^{2} + 4945\right )}{18 \sqrt{- x^{4} + x^{2} + 2}} - \frac{7147 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{18} + \frac{1763 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(4897*x**2 + 4945)/(18*sqrt(-x**4 + x**2 + 2)) - 7147*elliptic_e(asin(sqrt(2)*
x/2), -2)/18 + 1763*elliptic_f(asin(sqrt(2)*x/2), -2)/6

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Mathematica [C]  time = 0.125959, size = 79, normalized size = 1.44 \[ \frac{1}{18} \left (\frac{4945 x}{\sqrt{-x^4+x^2+2}}+\frac{4897 x^3}{\sqrt{-x^4+x^2+2}}+8076 i \sqrt{2} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-7147 i \sqrt{2} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((4945*x)/Sqrt[2 + x^2 - x^4] + (4897*x^3)/Sqrt[2 + x^2 - x^4] - (7147*I)*Sqrt[2
]*EllipticE[I*ArcSinh[x], -1/2] + (8076*I)*Sqrt[2]*EllipticF[I*ArcSinh[x], -1/2]
)/18

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Maple [B]  time = 0.01, size = 202, normalized size = 3.7 \[ 686\,{\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}} \left ({\frac{5\,x}{36}}-1/36\,{x}^{3} \right ) }-{\frac{929\,\sqrt{2}}{18}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{7147\,\sqrt{2}}{36}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+1470\,{\frac{1/9\,{x}^{3}-x/18}{\sqrt{-{x}^{4}+{x}^{2}+2}}}+1050\,{\frac{1/18\,{x}^{3}+2/9\,x}{\sqrt{-{x}^{4}+{x}^{2}+2}}}+250\,{\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}} \left ({\frac{5\,{x}^{3}}{18}}+x/9 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

686*(5/36*x-1/36*x^3)/(-x^4+x^2+2)^(1/2)-929/18*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))+7147/36*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)))+1470*(1/9*x^3-1/18*x)/(-x^4+x^2+2)^(1/2)+
1050*(1/18*x^3+2/9*x)/(-x^4+x^2+2)^(1/2)+250*(5/18*x^3+1/9*x)/(-x^4+x^2+2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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