3.660 \(\int \frac{1}{\sqrt [3]{a^2+2 a b x^2+b^2 x^4}} \, dx\)

Optimal. Leaf size=256 \[ -\frac{3^{3/4} \sqrt{2-\sqrt{3}} a \left (\frac{b x^2}{a}+1\right )^{2/3} \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{b x \sqrt [3]{a^2+2 a b x^2+b^2 x^4} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}} \]

[Out]

-((3^(3/4)*Sqrt[2 - Sqrt[3]]*a*(1 + (b*x^2)/a)^(2/3)*(1 - (1 + (b*x^2)/a)^(1/3))
*Sqrt[(1 + (1 + (b*x^2)/a)^(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - Sqrt[3] - (1 + (b
*x^2)/a)^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (1 + (b*x^2)/a)^(1/3))/(1 - S
qrt[3] - (1 + (b*x^2)/a)^(1/3))], -7 + 4*Sqrt[3]])/(b*x*(a^2 + 2*a*b*x^2 + b^2*x
^4)^(1/3)*Sqrt[-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3
))^2)]))

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Rubi [A]  time = 0.279655, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{3^{3/4} \sqrt{2-\sqrt{3}} a \left (\frac{b x^2}{a}+1\right )^{2/3} \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{b x \sqrt [3]{a^2+2 a b x^2+b^2 x^4} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-1/3),x]

[Out]

-((3^(3/4)*Sqrt[2 - Sqrt[3]]*a*(1 + (b*x^2)/a)^(2/3)*(1 - (1 + (b*x^2)/a)^(1/3))
*Sqrt[(1 + (1 + (b*x^2)/a)^(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - Sqrt[3] - (1 + (b
*x^2)/a)^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (1 + (b*x^2)/a)^(1/3))/(1 - S
qrt[3] - (1 + (b*x^2)/a)^(1/3))], -7 + 4*Sqrt[3]])/(b*x*(a^2 + 2*a*b*x^2 + b^2*x
^4)^(1/3)*Sqrt[-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3
))^2)]))

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Rubi in Sympy [A]  time = 29.5528, size = 304, normalized size = 1.19 \[ - \frac{3^{\frac{3}{4}} \sqrt{\frac{a^{\frac{2}{3}} b^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{a b + b^{2} x^{2}} + \left (a b + b^{2} x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \sqrt [3]{b} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a b + b^{2} x^{2}}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} \sqrt [3]{b} - \sqrt [3]{a b + b^{2} x^{2}}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{2}{3}} F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \sqrt [3]{b} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a b + b^{2} x^{2}}}{- \sqrt [3]{a} \sqrt [3]{b} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a b + b^{2} x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{x \sqrt{- \frac{\sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a} \sqrt [3]{b} - \sqrt [3]{a b + b^{2} x^{2}}\right )}{\left (\sqrt [3]{a} \sqrt [3]{b} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a b + b^{2} x^{2}}\right )^{2}}} \left (a b + b^{2} x^{2}\right )^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b**2*x**4+2*a*b*x**2+a**2)**(1/3),x)

[Out]

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

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Mathematica [C]  time = 0.0264652, size = 49, normalized size = 0.19 \[ \frac{x \left (\frac{a+b x^2}{a}\right )^{2/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{3}{2};-\frac{b x^2}{a}\right )}{\sqrt [3]{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-1/3),x]

[Out]

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

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Maple [F]  time = 0.013, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt [3]{{b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b^2*x^4+2*a*b*x^2+a^2)^(1/3),x)

[Out]

int(1/(b^2*x^4+2*a*b*x^2+a^2)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(-1/3),x, algorithm="maxima")

[Out]

integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(-1/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(-1/3),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)^(-1/3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [3]{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b**2*x**4+2*a*b*x**2+a**2)**(1/3),x)

[Out]

Integral((a**2 + 2*a*b*x**2 + b**2*x**4)**(-1/3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(-1/3),x, algorithm="giac")

[Out]

integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(-1/3), x)