3.1229 \(\int \frac{1}{x^6 \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=86 \[ -\frac{2 b^{3/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a-b x^4}}-\frac{\left (a-b x^4\right )^{3/4}}{5 a x^5} \]

[Out]

-(a - b*x^4)^(3/4)/(5*a*x^5) - (2*b^(3/2)*(1 - a/(b*x^4))^(1/4)*x*EllipticE[ArcC
sc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(5*a^(3/2)*(a - b*x^4)^(1/4))

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Rubi [A]  time = 0.121565, 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 \[ -\frac{2 b^{3/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a-b x^4}}-\frac{\left (a-b x^4\right )^{3/4}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a - b*x^4)^(1/4)),x]

[Out]

-(a - b*x^4)^(3/4)/(5*a*x^5) - (2*b^(3/2)*(1 - a/(b*x^4))^(1/4)*x*EllipticE[ArcC
sc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(5*a^(3/2)*(a - b*x^4)^(1/4))

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Rubi in Sympy [A]  time = 16.1433, size = 73, normalized size = 0.85 \[ - \frac{\left (a - b x^{4}\right )^{\frac{3}{4}}}{5 a x^{5}} - \frac{2 b^{\frac{3}{2}} x \sqrt [4]{- \frac{a}{b x^{4}} + 1} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{5 a^{\frac{3}{2}} \sqrt [4]{a - b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(-b*x**4+a)**(1/4),x)

[Out]

-(a - b*x**4)**(3/4)/(5*a*x**5) - 2*b**(3/2)*x*(-a/(b*x**4) + 1)**(1/4)*elliptic
_e(asin(sqrt(a)/(sqrt(b)*x**2))/2, 2)/(5*a**(3/2)*(a - b*x**4)**(1/4))

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Mathematica [C]  time = 0.0629119, size = 84, normalized size = 0.98 \[ \frac{-3 \left (a^2+a b x^4-2 b^2 x^8\right )-4 b^2 x^8 \sqrt [4]{1-\frac{b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )}{15 a^2 x^5 \sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(a - b*x^4)^(1/4)),x]

[Out]

(-3*(a^2 + a*b*x^4 - 2*b^2*x^8) - 4*b^2*x^8*(1 - (b*x^4)/a)^(1/4)*Hypergeometric
2F1[1/4, 3/4, 7/4, (b*x^4)/a])/(15*a^2*x^5*(a - b*x^4)^(1/4))

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{6}}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(-b*x^4+a)^(1/4),x)

[Out]

int(1/x^6/(-b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^6),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^6),x, algorithm="fricas")

[Out]

integral(1/((-b*x^4 + a)^(1/4)*x^6), x)

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Sympy [A]  time = 4.04249, size = 34, normalized size = 0.4 \[ - \frac{i e^{\frac{5 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{6 \sqrt [4]{b} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(-b*x**4+a)**(1/4),x)

[Out]

-I*exp(5*I*pi/4)*hyper((1/4, 3/2), (5/2,), a/(b*x**4))/(6*b**(1/4)*x**6)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^6),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^6), x)