3.1015 \(\int \frac{\sqrt [6]{a+b x^2}}{x^4} \, dx\)

Optimal. Leaf size=297 \[ -\frac{b \sqrt [6]{a+b x^2}}{9 a x}-\frac{2 \sqrt{2-\sqrt{3}} b \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{9 \sqrt [4]{3} a x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{\sqrt [6]{a+b x^2}}{3 x^3} \]

[Out]

-(a + b*x^2)^(1/6)/(3*x^3) - (b*(a + b*x^2)^(1/6))/(9*a*x) - (2*Sqrt[2 - Sqrt[3]
]*b*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*Sqrt[(1 + (a/(a + b*x^2))^(1/3
) + (a/(a + b*x^2))^(2/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))^2]*EllipticF[Ar
cSin[(1 + Sqrt[3] - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))
], -7 + 4*Sqrt[3]])/(9*3^(1/4)*a*x*(a/(a + b*x^2))^(1/3)*Sqrt[-((1 - (a/(a + b*x
^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))^2)])

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Rubi [A]  time = 0.558486, antiderivative size = 297, 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 \[ -\frac{b \sqrt [6]{a+b x^2}}{9 a x}-\frac{2 \sqrt{2-\sqrt{3}} b \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{9 \sqrt [4]{3} a x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{\sqrt [6]{a+b x^2}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(1/6)/x^4,x]

[Out]

-(a + b*x^2)^(1/6)/(3*x^3) - (b*(a + b*x^2)^(1/6))/(9*a*x) - (2*Sqrt[2 - Sqrt[3]
]*b*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*Sqrt[(1 + (a/(a + b*x^2))^(1/3
) + (a/(a + b*x^2))^(2/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))^2]*EllipticF[Ar
cSin[(1 + Sqrt[3] - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))
], -7 + 4*Sqrt[3]])/(9*3^(1/4)*a*x*(a/(a + b*x^2))^(1/3)*Sqrt[-((1 - (a/(a + b*x
^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))^2)])

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Rubi in Sympy [A]  time = 17.2399, size = 267, normalized size = 0.9 \[ - \frac{\sqrt [6]{a + b x^{2}}}{3 x^{3}} - \frac{b \sqrt [6]{a + b x^{2}}}{9 a x} - \frac{2 \cdot 3^{\frac{3}{4}} b \sqrt{\frac{\left (- \frac{b x^{2}}{a + b x^{2}} + 1\right )^{\frac{2}{3}} + \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1}{\left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \sqrt [6]{a + b x^{2}} \left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{27 a x \sqrt [3]{\frac{a}{a + b x^{2}}} \sqrt{\frac{\sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - 1}{\left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1\right )^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/6)/x**4,x)

[Out]

-(a + b*x**2)**(1/6)/(3*x**3) - b*(a + b*x**2)**(1/6)/(9*a*x) - 2*3**(3/4)*b*sqr
t(((-b*x**2/(a + b*x**2) + 1)**(2/3) + (-b*x**2/(a + b*x**2) + 1)**(1/3) + 1)/(-
(-b*x**2/(a + b*x**2) + 1)**(1/3) - sqrt(3) + 1)**2)*sqrt(-sqrt(3) + 2)*(a + b*x
**2)**(1/6)*(-(-b*x**2/(a + b*x**2) + 1)**(1/3) + 1)*elliptic_f(asin((-(-b*x**2/
(a + b*x**2) + 1)**(1/3) + 1 + sqrt(3))/(-(-b*x**2/(a + b*x**2) + 1)**(1/3) - sq
rt(3) + 1)), -7 + 4*sqrt(3))/(27*a*x*(a/(a + b*x**2))**(1/3)*sqrt(((-b*x**2/(a +
b*x**2) + 1)**(1/3) - 1)/(-(-b*x**2/(a + b*x**2) + 1)**(1/3) - sqrt(3) + 1)**2)
)

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Mathematica [C]  time = 0.0504287, size = 85, normalized size = 0.29 \[ \frac{-3 \left (3 a^2+4 a b x^2+b^2 x^4\right )-2 b^2 x^4 \left (\frac{b x^2}{a}+1\right )^{5/6} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};-\frac{b x^2}{a}\right )}{27 a x^3 \left (a+b x^2\right )^{5/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(1/6)/x^4,x]

[Out]

(-3*(3*a^2 + 4*a*b*x^2 + b^2*x^4) - 2*b^2*x^4*(1 + (b*x^2)/a)^(5/6)*Hypergeometr
ic2F1[1/2, 5/6, 3/2, -((b*x^2)/a)])/(27*a*x^3*(a + b*x^2)^(5/6))

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}}\sqrt [6]{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/6)/x^4,x)

[Out]

int((b*x^2+a)^(1/6)/x^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{6}}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/6)/x^4,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(1/6)/x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{1}{6}}}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/6)/x^4,x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(1/6)/x^4, x)

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Sympy [A]  time = 4.05234, size = 34, normalized size = 0.11 \[ - \frac{\sqrt [6]{a}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{1}{6} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(1/6)/x**4,x)

[Out]

-a**(1/6)*hyper((-3/2, -1/6), (-1/2,), b*x**2*exp_polar(I*pi)/a)/(3*x**3)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{6}}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/6)/x^4,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/6)/x^4, x)