3.2063 \(\int \frac{\sqrt{a+\frac{b}{x^4}}}{x^4} \, dx\)

Optimal. Leaf size=236 \[ -\frac{a^{5/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{2 a^{5/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{\sqrt{a+\frac{b}{x^4}}}{5 x^3}-\frac{2 a \sqrt{a+\frac{b}{x^4}}}{5 \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )} \]

[Out]

-Sqrt[a + b/x^4]/(5*x^3) - (2*a*Sqrt[a + b/x^4])/(5*Sqrt[b]*(Sqrt[a] + Sqrt[b]/x
^2)*x) + (2*a^(5/4)*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])/(5*b^(3/4)*Sqrt[a + b/x^4
]) - (a^(5/4)*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])/(5*b^(3/4)*Sqrt[a + b/x^4])

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Rubi [A]  time = 0.30885, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{a^{5/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{2 a^{5/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{\sqrt{a+\frac{b}{x^4}}}{5 x^3}-\frac{2 a \sqrt{a+\frac{b}{x^4}}}{5 \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^4]/x^4,x]

[Out]

-Sqrt[a + b/x^4]/(5*x^3) - (2*a*Sqrt[a + b/x^4])/(5*Sqrt[b]*(Sqrt[a] + Sqrt[b]/x
^2)*x) + (2*a^(5/4)*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])/(5*b^(3/4)*Sqrt[a + b/x^4
]) - (a^(5/4)*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])/(5*b^(3/4)*Sqrt[a + b/x^4])

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Rubi in Sympy [A]  time = 24.3141, size = 211, normalized size = 0.89 \[ \frac{2 a^{\frac{5}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} \sqrt{a + \frac{b}{x^{4}}}} - \frac{a^{\frac{5}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} \sqrt{a + \frac{b}{x^{4}}}} - \frac{2 a \sqrt{a + \frac{b}{x^{4}}}}{5 \sqrt{b} x \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )} - \frac{\sqrt{a + \frac{b}{x^{4}}}}{5 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**(5/4)*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)/(5*b**(3/4)*sqrt(a + b/x**4))
- a**(5/4)*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)/(5*b**(3/4)*sqrt(a + b/x**4))
- 2*a*sqrt(a + b/x**4)/(5*sqrt(b)*x*(sqrt(a) + sqrt(b)/x**2)) - sqrt(a + b/x**4)
/(5*x**3)

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Mathematica [C]  time = 0.829784, size = 138, normalized size = 0.58 \[ \frac{1}{5} x^2 \sqrt{a+\frac{b}{x^4}} \left (-\frac{2 a x^4+b}{b x^5}-\frac{2 i a \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{\frac{a x^4}{b}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )\right )}{a x^4+b}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x^4]*x^2*(-((b + 2*a*x^4)/(b*x^5)) - ((2*I)*a*Sqrt[(I*Sqrt[a])/Sqrt[
b]]*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]))/(b + a*x^4)))/5

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Maple [C]  time = 0.026, size = 234, normalized size = 1. \[ -{\frac{1}{5\,{x}^{3} \left ( a{x}^{4}+b \right ) }\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ( -2\,i{a}^{{\frac{3}{2}}}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{5}b{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +2\,i{a}^{{\frac{3}{2}}}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{5}b{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +2\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}\sqrt{b}{x}^{8}{a}^{2}+3\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{3/2}{x}^{4}a+\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{b}^{{\frac{5}{2}}} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(1/2)/x^4,x)

[Out]

-1/5*((a*x^4+b)/x^4)^(1/2)*(-2*I*a^(3/2)*(-(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^5*b*EllipticF(x*(I*a^(1/2)/b^(1/2))^
(1/2),I)+2*I*a^(3/2)*(-(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^5*b*EllipticE(x*(I*a^(1/2)/b^(1/2))^(1/2),I)+2*(I*a^(1/2
)/b^(1/2))^(1/2)*b^(1/2)*x^8*a^2+3*(I*a^(1/2)/b^(1/2))^(1/2)*b^(3/2)*x^4*a+(I*a^
(1/2)/b^(1/2))^(1/2)*b^(5/2))/x^3/(a*x^4+b)/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. \[ \int \frac{\sqrt{a + \frac{b}{x^{4}}}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a + b/x^4)/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.69721, size = 41, normalized size = 0.17 \[ - \frac{\sqrt{a} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a + b/x^4)/x^4, x)