3.333 \(\int \frac{\left (2+x^2-x^4\right )^{3/2}}{7+5 x^2} \, dx\)

Optimal. Leaf size=72 \[ \frac{1}{75} x \sqrt{-x^4+x^2+2} \left (13-3 x^2\right )-\frac{178}{625} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{92}{375} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{1156 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{4375} \]

[Out]

(x*(13 - 3*x^2)*Sqrt[2 + x^2 - x^4])/75 + (92*EllipticE[ArcSin[x/Sqrt[2]], -2])/
375 - (178*EllipticF[ArcSin[x/Sqrt[2]], -2])/625 + (1156*EllipticPi[-10/7, ArcSi
n[x/Sqrt[2]], -2])/4375

_______________________________________________________________________________________

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]

(x*(13 - 3*x^2)*Sqrt[2 + x^2 - x^4])/75 + (92*EllipticE[ArcSin[x/Sqrt[2]], -2])/
375 - (178*EllipticF[ArcSin[x/Sqrt[2]], -2])/625 + (1156*EllipticPi[-10/7, ArcSi
n[x/Sqrt[2]], -2])/4375

_______________________________________________________________________________________

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]

-x**3*sqrt(-x**4 + x**2 + 2)/25 + 13*x*sqrt(-x**4 + x**2 + 2)/75 + 92*elliptic_e
(asin(sqrt(2)*x/2), -2)/375 - 22*elliptic_f(asin(sqrt(2)*x/2), -2)/125 - 68*sqrt
(-x**4 + x**2 + 2)*elliptic_f(asin(sqrt(2)*x/2), -2)/(625*sqrt(-2*x**2 + 4)*sqrt
(x**2/2 + 1/2)) + 1156*sqrt(-x**4 + x**2 + 2)*elliptic_pi(-10/7, asin(sqrt(2)*x/
2), -2)/(4375*sqrt(-2*x**2 + 4)*sqrt(x**2/2 + 1/2))

_______________________________________________________________________________________

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]

(4550*x + 1225*x^3 - 2800*x^5 + 525*x^7 + (3220*I)*Sqrt[4 + 2*x^2 - 2*x^4]*Ellip
ticE[I*ArcSinh[x], -1/2] - (2961*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[
x], -1/2] - (1734*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2]
)/(13125*Sqrt[2 + x^2 - x^4])

_______________________________________________________________________________________

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]

-1/25*x^3*(-x^4+x^2+2)^(1/2)+13/75*x*(-x^4+x^2+2)^(1/2)-89/625*2^(1/2)*(-2*x^2+4
)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*EllipticF(1/2*2^(1/2)*x,I*2^(1/2))+46/3
75*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*EllipticE(1/2*2^(1/
2)*x,I*2^(1/2))+1156/4375*2^(1/2)*(1-1/2*x^2)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(
1/2)*EllipticPi(1/2*2^(1/2)*x,-10/7,I*2^(1/2))

_______________________________________________________________________________________

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]

integrate((-x^4 + x^2 + 2)^(3/2)/(5*x^2 + 7), x)

_______________________________________________________________________________________

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]

integral((-x^4 + x^2 + 2)^(3/2)/(5*x^2 + 7), x)

_______________________________________________________________________________________

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]

Integral((-(x**2 - 2)*(x**2 + 1))**(3/2)/(5*x**2 + 7), x)

_______________________________________________________________________________________

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]

integrate((-x^4 + x^2 + 2)^(3/2)/(5*x^2 + 7), x)