3.841 \(\int \frac{1}{\sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=53 \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt{a-b x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0325707, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[a - b*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.3708, size = 48, normalized size = 0.91 \[ \frac{\sqrt [4]{a} \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{\sqrt [4]{b} \sqrt{a - b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.053904, size = 72, normalized size = 1.36 \[ -\frac{i \sqrt{1-\frac{b x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} \sqrt{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[a - b*x^4],x]

[Out]

((-I)*Sqrt[1 - (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[a])]*x], -1])/
(Sqrt[-(Sqrt[b]/Sqrt[a])]*Sqrt[a - b*x^4])

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Maple [A]  time = 0.008, size = 64, normalized size = 1.2 \[{1\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x^4+a)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(-b*x^4 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/sqrt(-b*x^4 + a), x)

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Sympy [A]  time = 2.08008, size = 37, normalized size = 0.7 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*ga
mma(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(-b*x^4 + a), x)