Optimal. Leaf size=63 \[ \frac{\sqrt{x^2-3} \sqrt{2 x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{7} x}{\sqrt{x^2-3}}\right )|\frac{1}{7}\right )}{\sqrt{7} \sqrt{2 x^4-5 x^2-3}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0285873, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\sqrt{x^2-3} \sqrt{2 x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{7} x}{\sqrt{x^2-3}}\right )|\frac{1}{7}\right )}{\sqrt{7} \sqrt{2 x^4-5 x^2-3}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[-3 - 5*x^2 + 2*x^4],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 3.70724, size = 70, normalized size = 1.11 \[ \frac{\sqrt{3} \sqrt{\frac{2 x^{2}}{7} - \frac{6}{7}} \sqrt{12 x^{2} + 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{\sqrt{\frac{2 x^{2}}{7} - \frac{6}{7}}} \right )}\middle | \frac{1}{7}\right )}{6 \sqrt{2 x^{4} - 5 x^{2} - 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2*x**4-5*x**2-3)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0416615, size = 65, normalized size = 1.03 \[ -\frac{i \sqrt{1-\frac{x^2}{3}} \sqrt{2 x^2+1} F\left (i \sinh ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{6}\right )}{\sqrt{2} \sqrt{2 x^4-5 x^2-3}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[-3 - 5*x^2 + 2*x^4],x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.022, size = 53, normalized size = 0.8 \[{-{\frac{i}{6}}\sqrt{2}{\it EllipticF} \left ( i\sqrt{2}x,{\frac{i}{6}}\sqrt{6} \right ) \sqrt{2\,{x}^{2}+1}\sqrt{-3\,{x}^{2}+9}{\frac{1}{\sqrt{2\,{x}^{4}-5\,{x}^{2}-3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2*x^4-5*x^2-3)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} - 5 \, x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(2*x^4 - 5*x^2 - 3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{2 \, x^{4} - 5 \, x^{2} - 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(2*x^4 - 5*x^2 - 3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 x^{4} - 5 x^{2} - 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2*x**4-5*x**2-3)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} - 5 \, x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(2*x^4 - 5*x^2 - 3),x, algorithm="giac")
[Out]