3.1227 \(\int \frac{x^2}{\sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=86 \[ \frac{\sqrt{a} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{b} \sqrt [4]{a-b x^4}}-\frac{\left (a-b x^4\right )^{3/4}}{2 b x} \]

[Out]

-(a - b*x^4)^(3/4)/(2*b*x) + (Sqrt[a]*(1 - a/(b*x^4))^(1/4)*x*EllipticE[ArcCsc[(
Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(2*Sqrt[b]*(a - b*x^4)^(1/4))

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Rubi [A]  time = 0.122977, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ \frac{\sqrt{a} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{b} \sqrt [4]{a-b x^4}}-\frac{\left (a-b x^4\right )^{3/4}}{2 b x} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a - b*x^4)^(1/4),x]

[Out]

-(a - b*x^4)^(3/4)/(2*b*x) + (Sqrt[a]*(1 - a/(b*x^4))^(1/4)*x*EllipticE[ArcCsc[(
Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(2*Sqrt[b]*(a - b*x^4)^(1/4))

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Rubi in Sympy [A]  time = 16.1321, size = 68, normalized size = 0.79 \[ \frac{\sqrt{a} x \sqrt [4]{- \frac{a}{b x^{4}} + 1} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{2 \sqrt{b} \sqrt [4]{a - b x^{4}}} - \frac{\left (a - b x^{4}\right )^{\frac{3}{4}}}{2 b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*x*(-a/(b*x**4) + 1)**(1/4)*elliptic_e(asin(sqrt(a)/(sqrt(b)*x**2))/2, 2)
/(2*sqrt(b)*(a - b*x**4)**(1/4)) - (a - b*x**4)**(3/4)/(2*b*x)

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Mathematica [C]  time = 0.0300266, size = 53, normalized size = 0.62 \[ \frac{x^3 \sqrt [4]{\frac{a-b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )}{3 \sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \[ \int{{x}^{2}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-b*x^4+a)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(-b*x^4 + a)^(1/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^2/(-b*x^4 + a)^(1/4), x)

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Sympy [A]  time = 2.36131, size = 39, normalized size = 0.45 \[ \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(-b*x^4 + a)^(1/4), x)