3.1184 \(\int \frac{\sqrt [4]{a-b x^4}}{x^3} \, dx\)

Optimal. Leaf size=82 \[ -\frac{\sqrt [4]{a-b x^4}}{2 x^2}-\frac{\sqrt{a} \sqrt{b} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \left (a-b x^4\right )^{3/4}} \]

[Out]

-(a - b*x^4)^(1/4)/(2*x^2) - (Sqrt[a]*Sqrt[b]*(1 - (b*x^4)/a)^(3/4)*EllipticF[Ar
cSin[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(2*(a - b*x^4)^(3/4))

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Rubi [A]  time = 0.111953, antiderivative size = 82, 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 \[ -\frac{\sqrt [4]{a-b x^4}}{2 x^2}-\frac{\sqrt{a} \sqrt{b} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^4)^(1/4)/x^3,x]

[Out]

-(a - b*x^4)^(1/4)/(2*x^2) - (Sqrt[a]*Sqrt[b]*(1 - (b*x^4)/a)^(3/4)*EllipticF[Ar
cSin[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(2*(a - b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 12.4827, size = 68, normalized size = 0.83 \[ - \frac{\sqrt{a} \sqrt{b} \left (1 - \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{2 \left (a - b x^{4}\right )^{\frac{3}{4}}} - \frac{\sqrt [4]{a - b x^{4}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.042127, size = 68, normalized size = 0.83 \[ \frac{-b x^4 \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^4}{a}\right )-2 a+2 b x^4}{4 x^2 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^4)^(1/4)/x^3,x]

[Out]

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

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}\sqrt [4]{-b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^4+a)^(1/4)/x^3,x)

[Out]

int((-b*x^4+a)^(1/4)/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)/x^3,x, algorithm="maxima")

[Out]

integrate((-b*x^4 + a)^(1/4)/x^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)/x^3,x, algorithm="fricas")

[Out]

integral((-b*x^4 + a)^(1/4)/x^3, x)

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Sympy [A]  time = 2.76864, size = 34, normalized size = 0.41 \[ - \frac{\sqrt [4]{a}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)/x^3,x, algorithm="giac")

[Out]

integrate((-b*x^4 + a)^(1/4)/x^3, x)