3.1021 \(\int \frac{1}{\sqrt [6]{a+b x^2}} \, dx\)

Optimal. Leaf size=577 \[ \frac{3 x}{2 \sqrt [6]{a+b x^2}}+\frac{3 a x}{2 \left (\frac{a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )}-\frac{3^{3/4} a \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{2} b x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}+\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 b x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}} \]

[Out]

(3*x)/(2*(a + b*x^2)^(1/6)) + (3*a*x)/(2*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)
*(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))) + (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*a*(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]*EllipticE[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]])/(4*b*x*(a
/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*Sqrt[-((1 - (a/(a + b*x^2))^(1/3))/(1 - Sq
rt[3] - (a/(a + b*x^2))^(1/3))^2)]) - (3^(3/4)*a*(1 - (a/(a + b*x^2))^(1/3))*Sqr
t[(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[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]])/(Sqrt[2]*b*x*(a/(a + b*x^2))^(2/3
)*(a + b*x^2)^(1/6)*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 = 1.0618, antiderivative size = 577, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546 \[ \frac{3 x}{2 \sqrt [6]{a+b x^2}}+\frac{3 a x}{2 \left (\frac{a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )}-\frac{3^{3/4} a \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{2} b x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}+\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 b x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(-1/6),x]

[Out]

(3*x)/(2*(a + b*x^2)^(1/6)) + (3*a*x)/(2*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)
*(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))) + (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*a*(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]*EllipticE[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]])/(4*b*x*(a
/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*Sqrt[-((1 - (a/(a + b*x^2))^(1/3))/(1 - Sq
rt[3] - (a/(a + b*x^2))^(1/3))^2)]) - (3^(3/4)*a*(1 - (a/(a + b*x^2))^(1/3))*Sqr
t[(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[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]])/(Sqrt[2]*b*x*(a/(a + b*x^2))^(2/3
)*(a + b*x^2)^(1/6)*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 [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a \int \frac{1}{\left (a + b x^{2}\right )^{\frac{7}{6}}}\, dx}{2} + \frac{3 x}{2 \sqrt [6]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*Integral((a + b*x**2)**(-7/6), x)/2 + 3*x/(2*(a + b*x**2)**(1/6))

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Mathematica [C]  time = 0.0251743, size = 47, normalized size = 0.08 \[ \frac{x \sqrt [6]{\frac{a+b x^2}{a}} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )}{\sqrt [6]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(-1/6),x]

[Out]

(x*((a + b*x^2)/a)^(1/6)*Hypergeometric2F1[1/6, 1/2, 3/2, -((b*x^2)/a)])/(a + b*
x^2)^(1/6)

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt [6]{b{x}^{2}+a}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)^(1/6),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(-1/6), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^2 + a)^(-1/6), x)

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Sympy [A]  time = 2.57766, size = 24, normalized size = 0.04 \[ \frac{x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{6}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{\sqrt [6]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(-1/6), x)