3.2062 \(\int \frac{\sqrt{a+\frac{b}{x^4}}}{x^2} \, dx\)

Optimal. Leaf size=107 \[ -\frac{a^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{b} \sqrt{a+\frac{b}{x^4}}}-\frac{\sqrt{a+\frac{b}{x^4}}}{3 x} \]

[Out]

-Sqrt[a + b/x^4]/(3*x) - (a^(3/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(S
qrt[a] + Sqrt[b]/x^2)*EllipticF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(3*b^(1/4)*
Sqrt[a + b/x^4])

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Rubi [A]  time = 0.12151, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{a^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{b} \sqrt{a+\frac{b}{x^4}}}-\frac{\sqrt{a+\frac{b}{x^4}}}{3 x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^4]/x^2,x]

[Out]

-Sqrt[a + b/x^4]/(3*x) - (a^(3/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(S
qrt[a] + Sqrt[b]/x^2)*EllipticF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(3*b^(1/4)*
Sqrt[a + b/x^4])

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Rubi in Sympy [A]  time = 7.21882, size = 94, normalized size = 0.88 \[ - \frac{a^{\frac{3}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt [4]{b} \sqrt{a + \frac{b}{x^{4}}}} - \frac{\sqrt{a + \frac{b}{x^{4}}}}{3 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.291115, size = 96, normalized size = 0.9 \[ \frac{\sqrt{a+\frac{b}{x^4}} \left (-1-\frac{2 i a x^3 \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a x^4+b\right )}\right )}{3 x} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^4]/x^2,x]

[Out]

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

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Maple [C]  time = 0.023, size = 132, normalized size = 1.2 \[ -{\frac{1}{3\,x \left ( a{x}^{4}+b \right ) }\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ( -2\,a\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ){x}^{3}+\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{x}^{4}a+\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}b \right ){\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(1/2)/x^2,x)

[Out]

-1/3*((a*x^4+b)/x^4)^(1/2)*(-2*a*(-(I*a^(1/2)*x^2-b^(1/2))/b^(1/2))^(1/2)*((I*a^
(1/2)*x^2+b^(1/2))/b^(1/2))^(1/2)*EllipticF(x*(I*a^(1/2)/b^(1/2))^(1/2),I)*x^3+(
I*a^(1/2)/b^(1/2))^(1/2)*x^4*a+(I*a^(1/2)/b^(1/2))^(1/2)*b)/x/(a*x^4+b)/(I*a^(1/
2)/b^(1/2))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a + b/x^4)/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt((a*x^4 + b)/x^4)/x^2, x)

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Sympy [A]  time = 3.42677, size = 39, normalized size = 0.36 \[ - \frac{\sqrt{a} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 x \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a + b/x^4)/x^2, x)