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

Optimal. Leaf size=103 \[ -\frac{b^{3/2} \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{6 \sqrt{a} \left (a-b x^2\right )^{3/4}}+\frac{b \sqrt [4]{a-b x^2}}{6 a x}-\frac{\sqrt [4]{a-b x^2}}{3 x^3} \]

[Out]

-(a - b*x^2)^(1/4)/(3*x^3) + (b*(a - b*x^2)^(1/4))/(6*a*x) - (b^(3/2)*(1 - (b*x^
2)/a)^(3/4)*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(6*Sqrt[a]*(a - b*x^2)^
(3/4))

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Rubi [A]  time = 0.110913, antiderivative size = 103, 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 \[ -\frac{b^{3/2} \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{6 \sqrt{a} \left (a-b x^2\right )^{3/4}}+\frac{b \sqrt [4]{a-b x^2}}{6 a x}-\frac{\sqrt [4]{a-b x^2}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^2)^(1/4)/x^4,x]

[Out]

-(a - b*x^2)^(1/4)/(3*x^3) + (b*(a - b*x^2)^(1/4))/(6*a*x) - (b^(3/2)*(1 - (b*x^
2)/a)^(3/4)*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(6*Sqrt[a]*(a - b*x^2)^
(3/4))

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Rubi in Sympy [A]  time = 14.0851, size = 82, normalized size = 0.8 \[ - \frac{\sqrt [4]{a - b x^{2}}}{3 x^{3}} + \frac{b \sqrt [4]{a - b x^{2}}}{6 a x} - \frac{b^{\frac{3}{2}} \left (1 - \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{6 \sqrt{a} \left (a - b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a - b*x**2)**(1/4)/(3*x**3) + b*(a - b*x**2)**(1/4)/(6*a*x) - b**(3/2)*(1 - b*
x**2/a)**(3/4)*elliptic_f(asin(sqrt(b)*x/sqrt(a))/2, 2)/(6*sqrt(a)*(a - b*x**2)*
*(3/4))

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Mathematica [C]  time = 0.0509637, size = 84, normalized size = 0.82 \[ \frac{-4 a^2-b^2 x^4 \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^2}{a}\right )+6 a b x^2-2 b^2 x^4}{12 a x^3 \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*a^2 + 6*a*b*x^2 - 2*b^2*x^4 - b^2*x^4*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F
1[1/2, 3/4, 3/2, (b*x^2)/a])/(12*a*x^3*(a - b*x^2)^(3/4))

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}}\sqrt [4]{-b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+a)^(1/4)/x^4,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-b*x^2 + a)^(1/4)/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-b*x^2 + a)^(1/4)/x^4, x)

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Sympy [A]  time = 3.13489, size = 36, normalized size = 0.35 \[ - \frac{\sqrt [4]{a}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{1}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-b*x^2 + a)^(1/4)/x^4, x)