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

Optimal. Leaf size=93 \[ \frac{27500}{3} \sqrt{-x^4+x^2+2} x+\frac{\left (1419793 x^2+1419985\right ) x}{18 \sqrt{-x^4+x^2+2}}+625 \sqrt{-x^4+x^2+2} x^3+\frac{627857}{6} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{3482293}{18} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

(x*(1419985 + 1419793*x^2))/(18*Sqrt[2 + x^2 - x^4]) + (27500*x*Sqrt[2 + x^2 - x
^4])/3 + 625*x^3*Sqrt[2 + x^2 - x^4] - (3482293*EllipticE[ArcSin[x/Sqrt[2]], -2]
)/18 + (627857*EllipticF[ArcSin[x/Sqrt[2]], -2])/6

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Rubi [A]  time = 0.232379, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{27500}{3} \sqrt{-x^4+x^2+2} x+\frac{\left (1419793 x^2+1419985\right ) x}{18 \sqrt{-x^4+x^2+2}}+625 \sqrt{-x^4+x^2+2} x^3+\frac{627857}{6} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{3482293}{18} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(1419985 + 1419793*x^2))/(18*Sqrt[2 + x^2 - x^4]) + (27500*x*Sqrt[2 + x^2 - x
^4])/3 + 625*x^3*Sqrt[2 + x^2 - x^4] - (3482293*EllipticE[ArcSin[x/Sqrt[2]], -2]
)/18 + (627857*EllipticF[ArcSin[x/Sqrt[2]], -2])/6

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Rubi in Sympy [A]  time = 53.0862, size = 88, normalized size = 0.95 \[ 625 x^{3} \sqrt{- x^{4} + x^{2} + 2} + \frac{x \left (1419793 x^{2} + 1419985\right )}{18 \sqrt{- x^{4} + x^{2} + 2}} + \frac{27500 x \sqrt{- x^{4} + x^{2} + 2}}{3} - \frac{3482293 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{18} + \frac{627857 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

625*x**3*sqrt(-x**4 + x**2 + 2) + x*(1419793*x**2 + 1419985)/(18*sqrt(-x**4 + x*
*2 + 2)) + 27500*x*sqrt(-x**4 + x**2 + 2)/3 - 3482293*elliptic_e(asin(sqrt(2)*x/
2), -2)/18 + 627857*elliptic_f(asin(sqrt(2)*x/2), -2)/6

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Mathematica [C]  time = 0.126462, size = 97, normalized size = 1.04 \[ \frac{-11250 x^7-153750 x^5+1607293 x^3+4281654 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-3482293 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+1749985 x}{18 \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(1749985*x + 1607293*x^3 - 153750*x^5 - 11250*x^7 - (3482293*I)*Sqrt[4 + 2*x^2 -
2*x^4]*EllipticE[I*ArcSinh[x], -1/2] + (4281654*I)*Sqrt[4 + 2*x^2 - 2*x^4]*Elli
pticF[I*ArcSinh[x], -1/2])/(18*Sqrt[2 + x^2 - x^4])

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Maple [B]  time = 0.048, size = 280, normalized size = 3. \[ 33614\,{\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}} \left ({\frac{5\,x}{36}}-1/36\,{x}^{3} \right ) }-{\frac{799361\,\sqrt{2}}{18}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{3482293\,\sqrt{2}}{36}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+120050\,{\frac{1/9\,{x}^{3}-x/18}{\sqrt{-{x}^{4}+{x}^{2}+2}}}+171500\,{\frac{1/18\,{x}^{3}+2/9\,x}{\sqrt{-{x}^{4}+{x}^{2}+2}}}+122500\,{\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}} \left ({\frac{5\,{x}^{3}}{18}}+x/9 \right ) }+43750\,{\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}} \left ({\frac{7\,{x}^{3}}{18}}+5/9\,x \right ) }+{\frac{27500\,x}{3}\sqrt{-{x}^{4}+{x}^{2}+2}}+6250\,{\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}} \left ({\frac{17\,{x}^{3}}{18}}+{\frac{7\,x}{9}} \right ) }+625\,{x}^{3}\sqrt{-{x}^{4}+{x}^{2}+2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

33614*(5/36*x-1/36*x^3)/(-x^4+x^2+2)^(1/2)-799361/18*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))+3482293/36*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)))+120050*(1/9*x^3-1/18*x)/(-x^4+x^2
+2)^(1/2)+171500*(1/18*x^3+2/9*x)/(-x^4+x^2+2)^(1/2)+122500*(5/18*x^3+1/9*x)/(-x
^4+x^2+2)^(1/2)+43750*(7/18*x^3+5/9*x)/(-x^4+x^2+2)^(1/2)+27500/3*x*(-x^4+x^2+2)
^(1/2)+6250*(17/18*x^3+7/9*x)/(-x^4+x^2+2)^(1/2)+625*x^3*(-x^4+x^2+2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{3125 \, x^{10} + 21875 \, x^{8} + 61250 \, x^{6} + 85750 \, x^{4} + 60025 \, x^{2} + 16807}{{\left (x^{4} - x^{2} - 2\right )} \sqrt{-x^{4} + x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(3125*x^10 + 21875*x^8 + 61250*x^6 + 85750*x^4 + 60025*x^2 + 16807)/((
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. \[ \int \frac{\left (5 x^{2} + 7\right )^{5}}{\left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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