3.2087 \(\int \frac{1}{\sqrt{a+\frac{b}{x^4}} x^4} \, dx\)

Optimal. Leaf size=212 \[ -\frac{\sqrt [4]{a} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt [4]{a} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )} \]

[Out]

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

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Rubi [A]  time = 0.254998, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\sqrt [4]{a} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt [4]{a} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x^4]*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.4923, size = 187, normalized size = 0.88 \[ \frac{\sqrt [4]{a} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{3}{4}} \sqrt{a + \frac{b}{x^{4}}}} - \frac{\sqrt [4]{a} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{2 b^{\frac{3}{4}} \sqrt{a + \frac{b}{x^{4}}}} - \frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{b} x \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/4)*sqrt((a + b/x**4)/(sqrt(a) + sqrt(b)/x**2)**2)*(sqrt(a) + sqrt(b)/x**2)
*elliptic_e(2*atan(b**(1/4)/(a**(1/4)*x)), 1/2)/(b**(3/4)*sqrt(a + b/x**4)) - a*
*(1/4)*sqrt((a + b/x**4)/(sqrt(a) + sqrt(b)/x**2)**2)*(sqrt(a) + sqrt(b)/x**2)*e
lliptic_f(2*atan(b**(1/4)/(a**(1/4)*x)), 1/2)/(2*b**(3/4)*sqrt(a + b/x**4)) - sq
rt(a + b/x**4)/(sqrt(b)*x*(sqrt(a) + sqrt(b)/x**2))

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Mathematica [C]  time = 0.222969, size = 173, normalized size = 0.82 \[ -\frac{a x^4+b}{b x^3 \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt{a} \sqrt{1-\frac{i \sqrt{a} x^2}{\sqrt{b}}} \sqrt{1+\frac{i \sqrt{a} x^2}{\sqrt{b}}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )\right )}{\sqrt{b} x^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.02, size = 198, normalized size = 0.9 \[ -{\frac{1}{{x}^{3}} \left ( \sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}\sqrt{b}{x}^{4}a-i\sqrt{a}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}xb{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +i\sqrt{a}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}xb{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{b}^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(a+b/x^4)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(a + b/x^4)*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.47512, size = 39, normalized size = 0.18 \[ - \frac{\Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 \sqrt{a} x^{3} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-gamma(3/4)*hyper((1/2, 3/4), (7/4,), b*exp_polar(I*pi)/(a*x**4))/(4*sqrt(a)*x**
3*gamma(7/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(a + b/x^4)*x^4), x)