3.1228 \(\int \frac{1}{x^2 \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=61 \[ -\frac{\sqrt{b} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt [4]{a-b x^4}} \]

[Out]

-((Sqrt[b]*(1 - a/(b*x^4))^(1/4)*x*EllipticE[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2]
)/(Sqrt[a]*(a - b*x^4)^(1/4)))

_______________________________________________________________________________________

Rubi [A]  time = 0.0908813, antiderivative size = 61, 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 \[ -\frac{\sqrt{b} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a - b*x^4)^(1/4)),x]

[Out]

-((Sqrt[b]*(1 - a/(b*x^4))^(1/4)*x*EllipticE[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2]
)/(Sqrt[a]*(a - b*x^4)^(1/4)))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.1305, size = 53, normalized size = 0.87 \[ - \frac{\sqrt{b} x \sqrt [4]{- \frac{a}{b x^{4}} + 1} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{\sqrt{a} \sqrt [4]{a - b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(b)*x*(-a/(b*x**4) + 1)**(1/4)*elliptic_e(asin(sqrt(a)/(sqrt(b)*x**2))/2, 2
)/(sqrt(a)*(a - b*x**4)**(1/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0494668, size = 71, normalized size = 1.16 \[ \frac{-2 b x^4 \sqrt [4]{1-\frac{b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )-3 a+3 b x^4}{3 a x \sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a - b*x^4)^(1/4)),x]

[Out]

(-3*a + 3*b*x^4 - 2*b*x^4*(1 - (b*x^4)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4,
(b*x^4)/a])/(3*a*x*(a - b*x^4)^(1/4))

_______________________________________________________________________________________

Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(-b*x^4+a)^(1/4),x)

[Out]

int(1/x^2/(-b*x^4+a)^(1/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^2),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^2),x, algorithm="fricas")

[Out]

integral(1/((-b*x^4 + a)^(1/4)*x^2), x)

_______________________________________________________________________________________

Sympy [A]  time = 2.78362, size = 31, normalized size = 0.51 \[ \frac{i e^{\frac{i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{2 \sqrt [4]{b} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

I*exp(I*pi/4)*hyper((1/4, 1/2), (3/2,), a/(b*x**4))/(2*b**(1/4)*x**2)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^2),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^2), x)