3.2061 \(\int \sqrt{a+\frac{b}{x^4}} \, dx\)

Optimal. Leaf size=224 \[ x \sqrt{a+\frac{b}{x^4}}-\frac{2 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{\sqrt [4]{a} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{\sqrt{a+\frac{b}{x^4}}}+\frac{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{\sqrt{a+\frac{b}{x^4}}} \]

[Out]

(-2*Sqrt[b]*Sqrt[a + b/x^4])/((Sqrt[a] + Sqrt[b]/x^2)*x) + Sqrt[a + b/x^4]*x + (
2*a^(1/4)*b^(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])/Sqrt[a + b/x^4] - (a^(1/4)*
b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*Elli
pticF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/Sqrt[a + b/x^4]

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Rubi [A]  time = 0.28856, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ x \sqrt{a+\frac{b}{x^4}}-\frac{2 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{\sqrt [4]{a} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{\sqrt{a+\frac{b}{x^4}}}+\frac{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{\sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^4],x]

[Out]

(-2*Sqrt[b]*Sqrt[a + b/x^4])/((Sqrt[a] + Sqrt[b]/x^2)*x) + Sqrt[a + b/x^4]*x + (
2*a^(1/4)*b^(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])/Sqrt[a + b/x^4] - (a^(1/4)*
b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*Elli
pticF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/Sqrt[a + b/x^4]

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Rubi in Sympy [A]  time = 23.1021, size = 201, normalized size = 0.9 \[ \frac{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{\sqrt{a + \frac{b}{x^{4}}}} - \frac{\sqrt [4]{a} \sqrt [4]{b} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{\sqrt{a + \frac{b}{x^{4}}}} - \frac{2 \sqrt{b} \sqrt{a + \frac{b}{x^{4}}}}{x \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )} + x \sqrt{a + \frac{b}{x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.730353, size = 119, normalized size = 0.53 \[ x \sqrt{a+\frac{b}{x^4}} \left (-1+\frac{2 i a x \sqrt{\frac{a x^4}{b}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{a}}{\sqrt{b}}\right )^{3/2} \left (a x^4+b\right )}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^4],x]

[Out]

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

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Maple [C]  time = 0.021, size = 201, normalized size = 0.9 \[ -{\frac{x}{a{x}^{4}+b}\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ( -2\,i\sqrt{a}\sqrt{b}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}x{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +2\,i\sqrt{a}\sqrt{b}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}x{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{x}^{4}a+\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}b \right ){\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(1/2),x)

[Out]

-((a*x^4+b)/x^4)^(1/2)*x*(-2*I*a^(1/2)*b^(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*EllipticF(x*(I*a^(1/2)/b^(1/2)
)^(1/2),I)+2*I*a^(1/2)*b^(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*EllipticE(x*(I*a^(1/2)/b^(1/2))^(1/2),I)+(I*a^
(1/2)/b^(1/2))^(1/2)*x^4*a+(I*a^(1/2)/b^(1/2))^(1/2)*b)/(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. \[ \int \sqrt{a + \frac{b}{x^{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a + b/x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt((a*x^4 + b)/x^4), x)

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Sympy [A]  time = 3.06519, size = 42, normalized size = 0.19 \[ - \frac{\sqrt{a} x \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**4)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a + b/x^4), x)