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

Optimal. Leaf size=55 \[ \frac{x \left (281 x^2+305\right )}{18 \sqrt{-x^4+x^2+2}}+\frac{139}{6} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{281}{18} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

(x*(305 + 281*x^2))/(18*Sqrt[2 + x^2 - x^4]) - (281*EllipticE[ArcSin[x/Sqrt[2]],
-2])/18 + (139*EllipticF[ArcSin[x/Sqrt[2]], -2])/6

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Rubi [A]  time = 0.169027, 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 \[ \frac{x \left (281 x^2+305\right )}{18 \sqrt{-x^4+x^2+2}}+\frac{139}{6} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{281}{18} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(305 + 281*x^2))/(18*Sqrt[2 + x^2 - x^4]) - (281*EllipticE[ArcSin[x/Sqrt[2]],
-2])/18 + (139*EllipticF[ArcSin[x/Sqrt[2]], -2])/6

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Rubi in Sympy [A]  time = 29.5493, size = 54, normalized size = 0.98 \[ \frac{x \left (281 x^{2} + 305\right )}{18 \sqrt{- x^{4} + x^{2} + 2}} - \frac{281 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{18} + \frac{139 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(281*x**2 + 305)/(18*sqrt(-x**4 + x**2 + 2)) - 281*elliptic_e(asin(sqrt(2)*x/2
), -2)/18 + 139*elliptic_f(asin(sqrt(2)*x/2), -2)/6

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Mathematica [C]  time = 0.124175, size = 79, normalized size = 1.44 \[ \frac{1}{18} \left (\frac{305 x}{\sqrt{-x^4+x^2+2}}+\frac{281 x^3}{\sqrt{-x^4+x^2+2}}+213 i \sqrt{2} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-281 i \sqrt{2} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((305*x)/Sqrt[2 + x^2 - x^4] + (281*x^3)/Sqrt[2 + x^2 - x^4] - (281*I)*Sqrt[2]*E
llipticE[I*ArcSinh[x], -1/2] + (213*I)*Sqrt[2]*EllipticF[I*ArcSinh[x], -1/2])/18

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Maple [B]  time = 0.01, size = 179, normalized size = 3.3 \[ 98\,{\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}} \left ({\frac{5\,x}{36}}-1/36\,{x}^{3} \right ) }+{\frac{34\,\sqrt{2}}{9}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{281\,\sqrt{2}}{36}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+140\,{\frac{1/9\,{x}^{3}-x/18}{\sqrt{-{x}^{4}+{x}^{2}+2}}}+50\,{\frac{1/18\,{x}^{3}+2/9\,x}{\sqrt{-{x}^{4}+{x}^{2}+2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

98*(5/36*x-1/36*x^3)/(-x^4+x^2+2)^(1/2)+34/9*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))+281/36*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))-E
llipticE(1/2*2^(1/2)*x,I*2^(1/2)))+140*(1/9*x^3-1/18*x)/(-x^4+x^2+2)^(1/2)+50*(1
/18*x^3+2/9*x)/(-x^4+x^2+2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{25 \, x^{4} + 70 \, x^{2} + 49}{{\left (x^{4} - x^{2} - 2\right )} \sqrt{-x^{4} + x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(25*x^4 + 70*x^2 + 49)/((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. \[ \int \frac{\left (5 x^{2} + 7\right )^{2}}{\left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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