3.1016 \(\int \frac{\sqrt [6]{a+b x^2}}{x^6} \, dx\)

Optimal. Leaf size=323 \[ \frac{8 b^2 \sqrt [6]{a+b x^2}}{135 a^2 x}+\frac{16 \sqrt{2-\sqrt{3}} b^2 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{135 \sqrt [4]{3} a^2 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{\sqrt [6]{a+b x^2}}{5 x^5}-\frac{b \sqrt [6]{a+b x^2}}{45 a x^3} \]

[Out]

-(a + b*x^2)^(1/6)/(5*x^5) - (b*(a + b*x^2)^(1/6))/(45*a*x^3) + (8*b^2*(a + b*x^
2)^(1/6))/(135*a^2*x) + (16*Sqrt[2 - Sqrt[3]]*b^2*(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 - Sq
rt[3] - (a/(a + b*x^2))^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (a/(a + b*x^2)
)^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(135*3^(1/4)*a
^2*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 = 0.645614, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{8 b^2 \sqrt [6]{a+b x^2}}{135 a^2 x}+\frac{16 \sqrt{2-\sqrt{3}} b^2 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{135 \sqrt [4]{3} a^2 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{\sqrt [6]{a+b x^2}}{5 x^5}-\frac{b \sqrt [6]{a+b x^2}}{45 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(1/6)/x^6,x]

[Out]

-(a + b*x^2)^(1/6)/(5*x^5) - (b*(a + b*x^2)^(1/6))/(45*a*x^3) + (8*b^2*(a + b*x^
2)^(1/6))/(135*a^2*x) + (16*Sqrt[2 - Sqrt[3]]*b^2*(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 - Sq
rt[3] - (a/(a + b*x^2))^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (a/(a + b*x^2)
)^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(135*3^(1/4)*a
^2*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 = 22.8995, size = 292, normalized size = 0.9 \[ - \frac{\sqrt [6]{a + b x^{2}}}{5 x^{5}} - \frac{b \sqrt [6]{a + b x^{2}}}{45 a x^{3}} + \frac{8 b^{2} \sqrt [6]{a + b x^{2}}}{135 a^{2} x} + \frac{16 \cdot 3^{\frac{3}{4}} b^{2} \sqrt{\frac{\left (- \frac{b x^{2}}{a + b x^{2}} + 1\right )^{\frac{2}{3}} + \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1}{\left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \sqrt [6]{a + b x^{2}} \left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{405 a^{2} x \sqrt [3]{\frac{a}{a + b x^{2}}} \sqrt{\frac{\sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - 1}{\left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1\right )^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x**2)**(1/6)/(5*x**5) - b*(a + b*x**2)**(1/6)/(45*a*x**3) + 8*b**2*(a +
b*x**2)**(1/6)/(135*a**2*x) + 16*3**(3/4)*b**2*sqrt(((-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) - sqrt(3) + 1)), -7 + 4*sqrt(3))/(
405*a**2*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 = 0.0596304, size = 94, normalized size = 0.29 \[ \frac{-81 a^3-90 a^2 b x^2+16 b^3 x^6 \left (\frac{b x^2}{a}+1\right )^{5/6} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};-\frac{b x^2}{a}\right )+15 a b^2 x^4+24 b^3 x^6}{405 a^2 x^5 \left (a+b x^2\right )^{5/6}} \]

Antiderivative was successfully verified.

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

[Out]

(-81*a^3 - 90*a^2*b*x^2 + 15*a*b^2*x^4 + 24*b^3*x^6 + 16*b^3*x^6*(1 + (b*x^2)/a)
^(5/6)*Hypergeometric2F1[1/2, 5/6, 3/2, -((b*x^2)/a)])/(405*a^2*x^5*(a + b*x^2)^
(5/6))

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{6}}\sqrt [6]{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/6)/x^6,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(1/6)/x^6, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^2 + a)^(1/6)/x^6, x)

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Sympy [A]  time = 5.26074, size = 34, normalized size = 0.11 \[ - \frac{\sqrt [6]{a}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, - \frac{1}{6} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(1/6)*hyper((-5/2, -1/6), (-3/2,), b*x**2*exp_polar(I*pi)/a)/(5*x**5)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(1/6)/x^6, x)