Optimal. Leaf size=74 \[ -\frac{25}{7} x \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{21} x \left (354 x^2+275\right ) \sqrt{-x^4+x^2+2}-\frac{79}{7} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{2045}{21} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
[Out]
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Rubi [A] time = 0.189563, antiderivative size = 74, 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 \[ -\frac{25}{7} x \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{21} x \left (354 x^2+275\right ) \sqrt{-x^4+x^2+2}-\frac{79}{7} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{2045}{21} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
[In] Int[(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4],x]
[Out]
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Rubi in Sympy [A] time = 33.0303, size = 75, normalized size = 1.01 \[ \frac{x \left (\frac{1770 x^{2}}{7} + \frac{1375}{7}\right ) \sqrt{- x^{4} + x^{2} + 2}}{15} - \frac{25 x \left (- x^{4} + x^{2} + 2\right )^{\frac{3}{2}}}{7} + \frac{2045 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{21} - \frac{79 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5*x**2+7)**2*(-x**4+x**2+2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.100961, size = 102, normalized size = 1.38 \[ \frac{-75 x^9-204 x^7+304 x^5+683 x^3-2949 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+2045 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+250 x}{21 \sqrt{-x^4+x^2+2}} \]
Antiderivative was successfully verified.
[In] Integrate[(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4],x]
[Out]
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Maple [B] time = 0.011, size = 159, normalized size = 2.2 \[{\frac{125\,x}{21}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{904\,\sqrt{2}}{21}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{2045\,\sqrt{2}}{42}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{93\,{x}^{3}}{7}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{25\,{x}^{5}}{7}\sqrt{-{x}^{4}+{x}^{2}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5*x^2+7)^2*(-x^4+x^2+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (25 \, x^{4} + 70 \, x^{2} + 49\right )} \sqrt{-x^{4} + x^{2} + 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \left (5 x^{2} + 7\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x**2+7)**2*(-x**4+x**2+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)^2,x, algorithm="giac")
[Out]