3.1214 \(\int \frac{1}{x^3 \sqrt [4]{a-b x^4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.112398, antiderivative size = 85, 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 \[ -\frac{\left (a-b x^4\right )^{3/4}}{2 a x^2}-\frac{\sqrt{b} \sqrt [4]{1-\frac{b x^4}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.4015, size = 70, normalized size = 0.82 \[ - \frac{\left (a - b x^{4}\right )^{\frac{3}{4}}}{2 a x^{2}} - \frac{\sqrt{b} \sqrt [4]{1 - \frac{b x^{4}}{a}} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{2 \sqrt{a} \sqrt [4]{a - b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0520952, size = 71, normalized size = 0.84 \[ \frac{-b x^4 \sqrt [4]{1-\frac{b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^4}{a}\right )-2 a+2 b x^4}{4 a x^2 \sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(-b*x^4+a)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.8398, size = 32, normalized size = 0.38 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{2 \sqrt [4]{a} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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