3.1198 \(\int \frac{\sqrt [4]{a-b x^4}}{x^4} \, dx\)

Optimal. Leaf size=85 \[ \frac{b^{3/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} \left (a-b x^4\right )^{3/4}}-\frac{\sqrt [4]{a-b x^4}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.105709, antiderivative size = 85, 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 \[ \frac{b^{3/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} \left (a-b x^4\right )^{3/4}}-\frac{\sqrt [4]{a-b x^4}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^4)^(1/4)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 14.2382, size = 70, normalized size = 0.82 \[ - \frac{\sqrt [4]{a - b x^{4}}}{3 x^{3}} + \frac{b^{\frac{3}{2}} x^{3} \left (- \frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{3 \sqrt{a} \left (a - b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0399864, size = 67, normalized size = 0.79 \[ \frac{-b x^4 \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^4}{a}\right )-a+b x^4}{3 x^3 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-a + b*x^4 - b*x^4*(1 - (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, (b*x^
4)/a])/(3*x^3*(a - b*x^4)^(3/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^4+a)^(1/4)/x^4,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-b*x^4 + a)^(1/4)/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-b*x^4 + a)^(1/4)/x^4, x)

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Sympy [A]  time = 3.14495, size = 36, normalized size = 0.42 \[ - \frac{i \sqrt [4]{b} e^{\frac{7 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-b*x^4 + a)^(1/4)/x^4, x)