3.2094 \(\int \frac{1}{\left (a+\frac{b}{x^4}\right )^{3/2} x^2} \, dx\)

Optimal. Leaf size=110 \[ -\frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{5/4} \sqrt [4]{b} \sqrt{a+\frac{b}{x^4}}}-\frac{1}{2 a x \sqrt{a+\frac{b}{x^4}}} \]

[Out]

-1/(2*a*Sqrt[a + b/x^4]*x) - (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])/(4*a^(5/4)*b^(1
/4)*Sqrt[a + b/x^4])

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Rubi [A]  time = 0.127289, antiderivative size = 110, 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 \[ -\frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{5/4} \sqrt [4]{b} \sqrt{a+\frac{b}{x^4}}}-\frac{1}{2 a x \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^4)^(3/2)*x^2),x]

[Out]

-1/(2*a*Sqrt[a + b/x^4]*x) - (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])/(4*a^(5/4)*b^(1
/4)*Sqrt[a + b/x^4])

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Rubi in Sympy [A]  time = 7.4791, size = 97, normalized size = 0.88 \[ - \frac{1}{2 a x \sqrt{a + \frac{b}{x^{4}}}} - \frac{\sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{5}{4}} \sqrt [4]{b} \sqrt{a + \frac{b}{x^{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**4)**(3/2)/x**2,x)

[Out]

-1/(2*a*x*sqrt(a + b/x**4)) - sqrt((a + b/x**4)/(sqrt(a) + sqrt(b)/x**2)**2)*(sq
rt(a) + sqrt(b)/x**2)*elliptic_f(2*atan(b**(1/4)/(a**(1/4)*x)), 1/2)/(4*a**(5/4)
*b**(1/4)*sqrt(a + b/x**4))

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Mathematica [C]  time = 0.138541, size = 105, normalized size = 0.95 \[ -\frac{i \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )+x \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{2 a x^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^4)^(3/2)*x^2),x]

[Out]

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

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Maple [C]  time = 0.02, size = 113, normalized size = 1. \[ -{\frac{a{x}^{4}+b}{2\,{x}^{6}a} \left ( -\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}} \right ) \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^4)^(3/2)/x^2,x)

[Out]

-1/2*(a*x^4+b)*(-(-(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*(I*a^(1/2)/b^(1/2))
^(1/2))/((a*x^4+b)/x^4)^(3/2)/x^6/a/(I*a^(1/2)/b^(1/2))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(3/2)*x^2),x, algorithm="maxima")

[Out]

integrate(1/((a + b/x^4)^(3/2)*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(3/2)*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.00984, size = 37, normalized size = 0.34 \[ - \frac{\Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac{3}{2}} x \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**4)**(3/2)/x**2,x)

[Out]

-gamma(1/4)*hyper((1/4, 3/2), (5/4,), b*exp_polar(I*pi)/(a*x**4))/(4*a**(3/2)*x*
gamma(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(3/2)*x^2),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^4)^(3/2)*x^2), x)