3.2060 \(\int \sqrt{a+\frac{b}{x^4}} x^2 \, dx\)

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

[Out]

(Sqrt[a + b/x^4]*x^3)/3 - (b^(3/4)*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])/(3*a^(1/4)
*Sqrt[a + b/x^4])

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Rubi [A]  time = 0.137301, 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 \[ \frac{1}{3} x^3 \sqrt{a+\frac{b}{x^4}}-\frac{b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{a} \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^4]*x^2,x]

[Out]

(Sqrt[a + b/x^4]*x^3)/3 - (b^(3/4)*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])/(3*a^(1/4)
*Sqrt[a + b/x^4])

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Rubi in Sympy [A]  time = 8.71533, size = 94, normalized size = 0.88 \[ \frac{x^{3} \sqrt{a + \frac{b}{x^{4}}}}{3} - \frac{b^{\frac{3}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt [4]{a} \sqrt{a + \frac{b}{x^{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*sqrt(a + b/x**4)/3 - b**(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*a**
(1/4)*sqrt(a + b/x**4))

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Mathematica [C]  time = 0.303075, size = 93, normalized size = 0.87 \[ \frac{1}{3} x^2 \sqrt{a+\frac{b}{x^4}} \left (x-\frac{2 i b \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a x^4+b\right )}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x^4]*x^2*(x - ((2*I)*b*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

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Maple [C]  time = 0.052, size = 130, normalized size = 1.2 \[{\frac{{x}^{2}}{3\,a{x}^{4}+3\,b}\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ( \sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{x}^{5}a+2\,b\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ) +\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}xb \right ){\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a+b/x^4)^(1/2),x)

[Out]

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

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. \[{\rm integral}\left (x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.79714, size = 44, normalized size = 0.41 \[ - \frac{\sqrt{a} x^{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)