3.2068 \(\int \left (a+\frac{b}{x^4}\right )^{3/2} x^2 \, dx\)

Optimal. Leaf size=126 \[ -\frac{2 a^{3/4} b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt{a+\frac{b}{x^4}}}-\frac{2 b \sqrt{a+\frac{b}{x^4}}}{3 x}+\frac{1}{3} x^3 \left (a+\frac{b}{x^4}\right )^{3/2} \]

[Out]

(-2*b*Sqrt[a + b/x^4])/(3*x) + ((a + b/x^4)^(3/2)*x^3)/3 - (2*a^(3/4)*b^(3/4)*Sq
rt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticF[2*Ar
cCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(3*Sqrt[a + b/x^4])

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Rubi [A]  time = 0.164566, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{2 a^{3/4} b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt{a+\frac{b}{x^4}}}-\frac{2 b \sqrt{a+\frac{b}{x^4}}}{3 x}+\frac{1}{3} x^3 \left (a+\frac{b}{x^4}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(3/2)*x^2,x]

[Out]

(-2*b*Sqrt[a + b/x^4])/(3*x) + ((a + b/x^4)^(3/2)*x^3)/3 - (2*a^(3/4)*b^(3/4)*Sq
rt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticF[2*Ar
cCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(3*Sqrt[a + b/x^4])

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Rubi in Sympy [A]  time = 10.6396, size = 112, normalized size = 0.89 \[ - \frac{2 a^{\frac{3}{4}} b^{\frac{3}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt{a + \frac{b}{x^{4}}}} - \frac{2 b \sqrt{a + \frac{b}{x^{4}}}}{3 x} + \frac{x^{3} \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(3/4)*b**(3/4)*sqrt((a + b/x**4)/(sqrt(a) + sqrt(b)/x**2)**2)*(sqrt(a) + s
qrt(b)/x**2)*elliptic_f(2*atan(b**(1/4)/(a**(1/4)*x)), 1/2)/(3*sqrt(a + b/x**4))
- 2*b*sqrt(a + b/x**4)/(3*x) + x**3*(a + b/x**4)**(3/2)/3

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Mathematica [C]  time = 0.172501, size = 128, normalized size = 1.02 \[ \frac{\sqrt{a+\frac{b}{x^4}} \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a^2 x^8-b^2\right )-4 i a b x^3 \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )\right )}{3 x \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a x^4+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x^4]*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(-b^2 + a^2*x^8) - (4*I)*a*b*x^3*Sqr
t[1 + (a*x^4)/b]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]*x], -1]))/(3*Sqrt
[(I*Sqrt[a])/Sqrt[b]]*x*(b + a*x^4))

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Maple [C]  time = 0.024, size = 138, normalized size = 1.1 \[{\frac{{x}^{3}}{3\, \left ( a{x}^{4}+b \right ) ^{2}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{3}{2}}} \left ( \sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{x}^{8}{a}^{2}+4\,ab\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ){x}^{3}-\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{b}^{2} \right ){\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(3/2)*x^2,x)

[Out]

1/3*((a*x^4+b)/x^4)^(3/2)*x^3*((I*a^(1/2)/b^(1/2))^(1/2)*x^8*a^2+4*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)*Elliptic
F(x*(I*a^(1/2)/b^(1/2))^(1/2),I)*x^3-(I*a^(1/2)/b^(1/2))^(1/2)*b^2)/(a*x^4+b)^2/
(I*a^(1/2)/b^(1/2))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(3/2)*x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.78336, size = 44, normalized size = 0.35 \[ - \frac{a^{\frac{3}{2}} x^{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{3}{4} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(3/2)*x**3*gamma(-3/4)*hyper((-3/2, -3/4), (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. \[ \int{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(3/2)*x^2, x)