Optimal. Leaf size=72 \[ \frac{1}{75} x \sqrt{-x^4+x^2+2} \left (13-3 x^2\right )-\frac{178}{625} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{92}{375} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{1156 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{4375} \]
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Rubi [A] time = 0.440763, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{75} x \sqrt{-x^4+x^2+2} \left (13-3 x^2\right )-\frac{178}{625} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{92}{375} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{1156 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{4375} \]
Antiderivative was successfully verified.
[In] Int[(2 + x^2 - x^4)^(3/2)/(7 + 5*x^2),x]
[Out]
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Rubi in Sympy [A] time = 82.1792, size = 172, normalized size = 2.39 \[ - \frac{x^{3} \sqrt{- x^{4} + x^{2} + 2}}{25} + \frac{13 x \sqrt{- x^{4} + x^{2} + 2}}{75} + \frac{92 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{375} - \frac{22 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{125} - \frac{68 \sqrt{- x^{4} + x^{2} + 2} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{625 \sqrt{- 2 x^{2} + 4} \sqrt{\frac{x^{2}}{2} + \frac{1}{2}}} + \frac{1156 \sqrt{- x^{4} + x^{2} + 2} \Pi \left (- \frac{10}{7}; \operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{4375 \sqrt{- 2 x^{2} + 4} \sqrt{\frac{x^{2}}{2} + \frac{1}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**4+x**2+2)**(3/2)/(5*x**2+7),x)
[Out]
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Mathematica [C] time = 0.177663, size = 130, normalized size = 1.81 \[ \frac{525 x^7-2800 x^5+1225 x^3-2961 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+3220 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-1734 i \sqrt{-2 x^4+2 x^2+4} \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )+4550 x}{13125 \sqrt{-x^4+x^2+2}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + x^2 - x^4)^(3/2)/(7 + 5*x^2),x]
[Out]
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Maple [B] time = 0.021, size = 173, normalized size = 2.4 \[ -{\frac{{x}^{3}}{25}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{13\,x}{75}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{89\,\sqrt{2}}{625}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{46\,\sqrt{2}}{375}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{1156\,\sqrt{2}}{4375}\sqrt{1-{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}},-{\frac{10}{7}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^4+x^2+2)^(3/2)/(5*x^2+7),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}}{5 \, x^{2} + 7}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x^4 + x^2 + 2)^(3/2)/(5*x^2 + 7),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}}{5 \, x^{2} + 7}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x^4 + x^2 + 2)^(3/2)/(5*x^2 + 7),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac{3}{2}}}{5 x^{2} + 7}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**4+x**2+2)**(3/2)/(5*x**2+7),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}}{5 \, x^{2} + 7}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x^4 + x^2 + 2)^(3/2)/(5*x^2 + 7),x, algorithm="giac")
[Out]