Optimal. Leaf size=93 \[ -\frac{17 \sqrt{-x^4+x^2+2} x}{175 \left (5 x^2+7\right )}-\frac{1}{75} \sqrt{-x^4+x^2+2} x+\frac{458}{875} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{97}{525} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{1241 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{6125} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.722154, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{17 \sqrt{-x^4+x^2+2} x}{175 \left (5 x^2+7\right )}-\frac{1}{75} \sqrt{-x^4+x^2+2} x+\frac{458}{875} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{97}{525} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{1241 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{6125} \]
Antiderivative was successfully verified.
[In] Int[(2 + x^2 - x^4)^(3/2)/(7 + 5*x^2)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 114.061, size = 241, normalized size = 2.59 \[ - \frac{x \sqrt{- x^{4} + x^{2} + 2}}{75} - \frac{578 x \sqrt{- x^{4} + x^{2} + 2}}{25 \left (1190 x^{2} + 1666\right )} - \frac{97 \sqrt{2} \sqrt{- x^{4} + x^{2} + 2} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{1050 \sqrt{- \frac{x^{2}}{2} + 1} \sqrt{x^{2} + 1}} - \frac{209 \sqrt{2} \sqrt{- x^{4} + x^{2} + 2} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{8750 \sqrt{- \frac{x^{2}}{2} + 1} \sqrt{x^{2} + 1}} - \frac{1207 \sqrt{2} \sqrt{- x^{4} + x^{2} + 2} \Pi \left (- \frac{10}{7}; \operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{61250 \sqrt{- \frac{x^{2}}{2} + 1} \sqrt{x^{2} + 1}} + \frac{51 \sqrt{- x^{4} + x^{2} + 2} \Pi \left (\frac{2}{7}; \operatorname{atan}{\left (x \right )}\middle | \frac{3}{2}\right )}{250 \sqrt{\frac{- \frac{x^{2}}{2} + 1}{x^{2} + 1}} \left (x^{2} + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**4+x**2+2)**(3/2)/(5*x**2+7)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.292483, size = 201, normalized size = 2.16 \[ \frac{2450 x^7+4550 x^5-11900 x^3+567 i \sqrt{2} \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-6790 i \sqrt{2} \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+18615 i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )+26061 i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )-14000 x}{36750 \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + x^2 - x^4)^(3/2)/(7 + 5*x^2)^2,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.028, size = 180, normalized size = 1.9 \[ -{\frac{17\,x}{875\,{x}^{2}+1225}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{x}{75}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{229\,\sqrt{2}}{875}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{97\,\sqrt{2}}{1050}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{1241\,\sqrt{2}}{6125}\sqrt{1-{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}},-{\frac{10}{7}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^4+x^2+2)^(3/2)/(5*x^2+7)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x^4 + x^2 + 2)^(3/2)/(5*x^2 + 7)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}}{25 \, x^{4} + 70 \, x^{2} + 49}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x^4 + x^2 + 2)^(3/2)/(5*x^2 + 7)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac{3}{2}}}{\left (5 x^{2} + 7\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**4+x**2+2)**(3/2)/(5*x**2+7)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x^4 + x^2 + 2)^(3/2)/(5*x^2 + 7)^2,x, algorithm="giac")
[Out]