3.1013 \(\int \sqrt [6]{a+b x^2} \, dx\)

Optimal. Leaf size=273 \[ \frac{3}{4} x \sqrt [6]{a+b x^2}+\frac{3^{3/4} \sqrt{2-\sqrt{3}} a \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 b x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}} \]

[Out]

(3*x*(a + b*x^2)^(1/6))/4 + (3^(3/4)*Sqrt[2 - Sqrt[3]]*a*(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[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))^(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 = 0.470247, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{3}{4} x \sqrt [6]{a+b x^2}+\frac{3^{3/4} \sqrt{2-\sqrt{3}} a \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 b x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(1/6),x]

[Out]

(3*x*(a + b*x^2)^(1/6))/4 + (3^(3/4)*Sqrt[2 - Sqrt[3]]*a*(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[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))^(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 = 10.0109, size = 246, normalized size = 0.9 \[ \frac{3^{\frac{3}{4}} a \sqrt{\frac{\left (- \frac{b x^{2}}{a + b x^{2}} + 1\right )^{\frac{2}{3}} + \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1}{\left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \sqrt [6]{a + b x^{2}} \left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{4 b x \sqrt [3]{\frac{a}{a + b x^{2}}} \sqrt{\frac{\sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - 1}{\left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1\right )^{2}}}} + \frac{3 x \sqrt [6]{a + b x^{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3**(3/4)*a*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(as
in((-(-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))/(4*b*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) - sqr
t(3) + 1)**2)) + 3*x*(a + b*x**2)**(1/6)/4

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Mathematica [C]  time = 0.0332939, size = 62, normalized size = 0.23 \[ \frac{a x \left (\frac{b x^2}{a}+1\right )^{5/6} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};-\frac{b x^2}{a}\right )+3 x \left (a+b x^2\right )}{4 \left (a+b x^2\right )^{5/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(1/6),x]

[Out]

(3*x*(a + b*x^2) + a*x*(1 + (b*x^2)/a)^(5/6)*Hypergeometric2F1[1/2, 5/6, 3/2, -(
(b*x^2)/a)])/(4*(a + b*x^2)^(5/6))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int \sqrt [6]{b{x}^{2}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/6),x)

[Out]

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

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

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. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac{1}{6}}, x\right ) \]

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 = 2.68838, size = 26, normalized size = 0.1 \[ \sqrt [6]{a} x{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{6}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(1/6),x)

[Out]

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

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

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)