3.1213 \(\int \frac{x}{\sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=59 \[ \frac{\sqrt{a} \sqrt [4]{1-\frac{b x^4}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \sqrt [4]{a-b x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0718608, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{\sqrt{a} \sqrt [4]{1-\frac{b x^4}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a - b*x^4)^(1/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.99127, size = 49, normalized size = 0.83 \[ \frac{\sqrt{a} \sqrt [4]{1 - \frac{b x^{4}}{a}} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{\sqrt{b} \sqrt [4]{a - b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0281153, size = 53, normalized size = 0.9 \[ \frac{x^2 \sqrt [4]{\frac{a-b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^4}{a}\right )}{2 \sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \[ \int{x{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-b*x^4+a)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(-b*x^4 + a)^(1/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x/(-b*x^4 + a)^(1/4), x)

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Sympy [A]  time = 2.32549, size = 29, normalized size = 0.49 \[ \frac{x^{2}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{2 \sqrt [4]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(-b*x^4 + a)^(1/4), x)