3.2084 \(\int \frac{x^2}{\sqrt{a+\frac{b}{x^4}}} \, dx\)

Optimal. Leaf size=110 \[ \frac{b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{6 a^{5/4} \sqrt{a+\frac{b}{x^4}}}+\frac{x^3 \sqrt{a+\frac{b}{x^4}}}{3 a} \]

[Out]

(Sqrt[a + b/x^4]*x^3)/(3*a) + (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])/(6*a^(
5/4)*Sqrt[a + b/x^4])

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Rubi [A]  time = 0.140947, 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 \[ \frac{b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{6 a^{5/4} \sqrt{a+\frac{b}{x^4}}}+\frac{x^3 \sqrt{a+\frac{b}{x^4}}}{3 a} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[a + b/x^4],x]

[Out]

(Sqrt[a + b/x^4]*x^3)/(3*a) + (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])/(6*a^(
5/4)*Sqrt[a + b/x^4])

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Rubi in Sympy [A]  time = 9.18552, size = 95, normalized size = 0.86 \[ \frac{x^{3} \sqrt{a + \frac{b}{x^{4}}}}{3 a} + \frac{b^{\frac{3}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{6 a^{\frac{5}{4}} \sqrt{a + \frac{b}{x^{4}}}} \]

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*a) + 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)/(6
*a**(5/4)*sqrt(a + b/x**4))

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Mathematica [C]  time = 0.129054, size = 113, normalized size = 1.03 \[ \frac{x \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a x^4+b\right )+i b \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )}{3 a x^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.016, size = 124, normalized size = 1.1 \[{\frac{1}{3\,a{x}^{2}} \left ( \sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{x}^{5}a-b\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}xb \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b/x^4)^(1/2),x)

[Out]

1/3*((I*a^(1/2)/b^(1/2))^(1/2)*x^5*a-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)/x^4)^(1/2)/x^2/a/(I*a^(1/2)/b^(1/2))^
(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/sqrt(a + b/x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.4267, size = 42, normalized size = 0.38 \[ - \frac{x^{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/sqrt(a + b/x^4), x)