Optimal. Leaf size=74 \[ -\frac{25 \sqrt{-x^4+x^2+2} x}{476 \left (5 x^2+7\right )}-\frac{1}{238} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{5}{476} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{167 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{3332} \]
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Rubi [A] time = 0.262785, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{25 \sqrt{-x^4+x^2+2} x}{476 \left (5 x^2+7\right )}-\frac{1}{238} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{5}{476} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{167 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{3332} \]
Antiderivative was successfully verified.
[In] Int[1/((7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4]),x]
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Rubi in Sympy [A] time = 38.5835, size = 73, normalized size = 0.99 \[ - \frac{25 x \sqrt{- x^{4} + x^{2} + 2}}{2380 x^{2} + 3332} - \frac{5 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{476} - \frac{F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{238} + \frac{167 \Pi \left (- \frac{10}{7}; \operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{3332} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(5*x**2+7)**2/(-x**4+x**2+2)**(1/2),x)
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Mathematica [C] time = 0.316699, size = 196, normalized size = 2.65 \[ \frac{350 x^5-350 x^3+119 i \sqrt{2} \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-70 i \sqrt{2} \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-835 i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )-1169 i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )-700 x}{6664 \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4]),x]
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Maple [B] time = 0.026, size = 165, normalized size = 2.2 \[ -{\frac{25\,x}{2380\,{x}^{2}+3332}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{\sqrt{2}}{476}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{5\,\sqrt{2}}{952}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{167\,\sqrt{2}}{3332}\sqrt{1-{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}},-{\frac{10}{7}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(5*x^2+7)^2/(-x^4+x^2+2)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)^2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (25 \, x^{4} + 70 \, x^{2} + 49\right )} \sqrt{-x^{4} + x^{2} + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)^2),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \left (5 x^{2} + 7\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(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. \[ \int \frac{1}{\sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)^2),x, algorithm="giac")
[Out]