Optimal. Leaf size=65 \[ -\frac{625}{3} \sqrt{-x^4+x^2+2} x-25 \sqrt{-x^4+x^2+2} x^3-542 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{3905}{3} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
[Out]
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Rubi [A] time = 0.198078, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{625}{3} \sqrt{-x^4+x^2+2} x-25 \sqrt{-x^4+x^2+2} x^3-542 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{3905}{3} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
[In] Int[(7 + 5*x^2)^3/Sqrt[2 + x^2 - x^4],x]
[Out]
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Rubi in Sympy [A] time = 39.0364, size = 65, normalized size = 1. \[ - 25 x^{3} \sqrt{- x^{4} + x^{2} + 2} - \frac{625 x \sqrt{- x^{4} + x^{2} + 2}}{3} + \frac{3905 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{3} - 542 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5*x**2+7)**3/(-x**4+x**2+2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.123721, size = 97, normalized size = 1.49 \[ \frac{150 x^7+1100 x^5-1550 x^3-10089 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+7810 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-2500 x}{6 \sqrt{-x^4+x^2+2}} \]
Antiderivative was successfully verified.
[In] Integrate[(7 + 5*x^2)^3/Sqrt[2 + x^2 - x^4],x]
[Out]
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Maple [B] time = 0.024, size = 142, normalized size = 2.2 \[{\frac{2279\,\sqrt{2}}{6}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{3905\,\sqrt{2}}{6}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{625\,x}{3}\sqrt{-{x}^{4}+{x}^{2}+2}}-25\,{x}^{3}\sqrt{-{x}^{4}+{x}^{2}+2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5*x^2+7)^3/(-x^4+x^2+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x^{2} + 7\right )}^{3}}{\sqrt{-x^{4} + x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^2 + 7)^3/sqrt(-x^4 + x^2 + 2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343}{\sqrt{-x^{4} + x^{2} + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^2 + 7)^3/sqrt(-x^4 + x^2 + 2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (5 x^{2} + 7\right )^{3}}{\sqrt{- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x**2+7)**3/(-x**4+x**2+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x^{2} + 7\right )}^{3}}{\sqrt{-x^{4} + x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^2 + 7)^3/sqrt(-x^4 + x^2 + 2),x, algorithm="giac")
[Out]