3.847 \(\int \frac{1}{x^2 \sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=128 \[ -\frac{\sqrt{a-b x^4}}{a x}+\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}-\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}} \]

[Out]

-(Sqrt[a - b*x^4]/(a*x)) - (b^(1/4)*Sqrt[1 - (b*x^4)/a]*EllipticE[ArcSin[(b^(1/4
)*x)/a^(1/4)], -1])/(a^(1/4)*Sqrt[a - b*x^4]) + (b^(1/4)*Sqrt[1 - (b*x^4)/a]*Ell
ipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(a^(1/4)*Sqrt[a - b*x^4])

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Rubi [A]  time = 0.233451, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ -\frac{\sqrt{a-b x^4}}{a x}+\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}-\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[a - b*x^4]),x]

[Out]

-(Sqrt[a - b*x^4]/(a*x)) - (b^(1/4)*Sqrt[1 - (b*x^4)/a]*EllipticE[ArcSin[(b^(1/4
)*x)/a^(1/4)], -1])/(a^(1/4)*Sqrt[a - b*x^4]) + (b^(1/4)*Sqrt[1 - (b*x^4)/a]*Ell
ipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(a^(1/4)*Sqrt[a - b*x^4])

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Rubi in Sympy [A]  time = 39.9791, size = 110, normalized size = 0.86 \[ - \frac{\sqrt{a - b x^{4}}}{a x} - \frac{\sqrt [4]{b} \sqrt{1 - \frac{b x^{4}}{a}} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{\sqrt [4]{a} \sqrt{a - b x^{4}}} + \frac{\sqrt [4]{b} \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{\sqrt [4]{a} \sqrt{a - b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a - b*x**4)/(a*x) - b**(1/4)*sqrt(1 - b*x**4/a)*elliptic_e(asin(b**(1/4)*x
/a**(1/4)), -1)/(a**(1/4)*sqrt(a - b*x**4)) + b**(1/4)*sqrt(1 - b*x**4/a)*ellipt
ic_f(asin(b**(1/4)*x/a**(1/4)), -1)/(a**(1/4)*sqrt(a - b*x**4))

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Mathematica [C]  time = 0.490198, size = 115, normalized size = 0.9 \[ \frac{-i \sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} \sqrt{1-\frac{b x^4}{a}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )\right )+\frac{b x^3}{a}-\frac{1}{x}}{\sqrt{a-b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-x^(-1) + (b*x^3)/a - I*Sqrt[-(Sqrt[b]/Sqrt[a])]*Sqrt[1 - (b*x^4)/a]*(EllipticE
[I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[a])]*x], -1] - EllipticF[I*ArcSinh[Sqrt[-(Sqrt[b]
/Sqrt[a])]*x], -1]))/Sqrt[a - b*x^4]

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Maple [A]  time = 0.015, size = 106, normalized size = 0.8 \[ -{\frac{1}{ax}\sqrt{-b{x}^{4}+a}}+{1\sqrt{b}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(-b*x^4+a)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(-b*x^4 + a)*x^2), x)

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Sympy [A]  time = 2.46498, size = 41, normalized size = 0.32 \[ \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} x \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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