3.982 \(\int \frac{x^2}{\sqrt{3-b x^4}} \, dx\)

Optimal. Leaf size=54 \[ \frac{\sqrt [4]{3} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}}-\frac{\sqrt [4]{3} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}} \]

[Out]

(3^(1/4)*EllipticE[ArcSin[(b^(1/4)*x)/3^(1/4)], -1])/b^(3/4) - (3^(1/4)*Elliptic
F[ArcSin[(b^(1/4)*x)/3^(1/4)], -1])/b^(3/4)

_______________________________________________________________________________________

Rubi [A]  time = 0.145094, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt [4]{3} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}}-\frac{\sqrt [4]{3} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[3 - b*x^4],x]

[Out]

(3^(1/4)*EllipticE[ArcSin[(b^(1/4)*x)/3^(1/4)], -1])/b^(3/4) - (3^(1/4)*Elliptic
F[ArcSin[(b^(1/4)*x)/3^(1/4)], -1])/b^(3/4)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.8078, size = 56, normalized size = 1.04 \[ \frac{\sqrt [4]{3} E\left (\operatorname{asin}{\left (\frac{3^{\frac{3}{4}} \sqrt [4]{b} x}{3} \right )}\middle | -1\right )}{b^{\frac{3}{4}}} - \frac{\sqrt [4]{3} F\left (\operatorname{asin}{\left (\frac{3^{\frac{3}{4}} \sqrt [4]{b} x}{3} \right )}\middle | -1\right )}{b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(-b*x**4+3)**(1/2),x)

[Out]

3**(1/4)*elliptic_e(asin(3**(3/4)*b**(1/4)*x/3), -1)/b**(3/4) - 3**(1/4)*ellipti
c_f(asin(3**(3/4)*b**(1/4)*x/3), -1)/b**(3/4)

_______________________________________________________________________________________

Mathematica [C]  time = 0.0705489, size = 76, normalized size = 1.41 \[ \frac{i \sqrt [4]{3} \sqrt{-\sqrt{b}} \left (E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b}} x}{\sqrt [4]{3}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b}} x}{\sqrt [4]{3}}\right )\right |-1\right )\right )}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[3 - b*x^4],x]

[Out]

(I*3^(1/4)*Sqrt[-Sqrt[b]]*(EllipticE[I*ArcSinh[(Sqrt[-Sqrt[b]]*x)/3^(1/4)], -1]
- EllipticF[I*ArcSinh[(Sqrt[-Sqrt[b]]*x)/3^(1/4)], -1]))/b

_______________________________________________________________________________________

Maple [B]  time = 0.02, size = 94, normalized size = 1.7 \[ -{\frac{1}{3}\sqrt{9-3\,\sqrt{3}\sqrt{b}{x}^{2}}\sqrt{9+3\,\sqrt{3}\sqrt{b}{x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{3}}{3}\sqrt{\sqrt{3}\sqrt{b}}},i \right ) -{\it EllipticE} \left ({\frac{x\sqrt{3}}{3}\sqrt{\sqrt{3}\sqrt{b}}},i \right ) \right ){\frac{1}{\sqrt{\sqrt{3}\sqrt{b}}}}{\frac{1}{\sqrt{-b{x}^{4}+3}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-b*x^4+3)^(1/2),x)

[Out]

-1/3/(3^(1/2)*b^(1/2))^(1/2)*(9-3*3^(1/2)*b^(1/2)*x^2)^(1/2)*(9+3*3^(1/2)*b^(1/2
)*x^2)^(1/2)/(-b*x^4+3)^(1/2)/b^(1/2)*(EllipticF(1/3*x*3^(1/2)*(3^(1/2)*b^(1/2))
^(1/2),I)-EllipticE(1/3*x*3^(1/2)*(3^(1/2)*b^(1/2))^(1/2),I))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-b x^{4} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-b*x^4 + 3),x, algorithm="maxima")

[Out]

integrate(x^2/sqrt(-b*x^4 + 3), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{-b x^{4} + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-b*x^4 + 3),x, algorithm="fricas")

[Out]

integral(x^2/sqrt(-b*x^4 + 3), x)

_______________________________________________________________________________________

Sympy [A]  time = 2.1119, size = 39, normalized size = 0.72 \[ \frac{\sqrt{3} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{3}} \right )}}{12 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(-b*x**4+3)**(1/2),x)

[Out]

sqrt(3)*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), b*x**4*exp_polar(2*I*pi)/3)/(1
2*gamma(7/4))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-b x^{4} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-b*x^4 + 3),x, algorithm="giac")

[Out]

integrate(x^2/sqrt(-b*x^4 + 3), x)