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

Optimal. Leaf size=289 \[ \frac{\sqrt{2-\sqrt{3}} \left (\frac{b x^2}{a}+1\right )^{2/3} \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} x \sqrt [3]{a^2+2 a b x^2+b^2 x^4} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}}-\frac{a+b x^2}{a x \sqrt [3]{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

-((a + b*x^2)/(a*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/3))) + (Sqrt[2 - Sqrt[3]]*(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 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))], -
7 + 4*Sqrt[3]])/(3^(1/4)*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 = 0.383851, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{2-\sqrt{3}} \left (\frac{b x^2}{a}+1\right )^{2/3} \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} x \sqrt [3]{a^2+2 a b x^2+b^2 x^4} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}}-\frac{a+b x^2}{a x \sqrt [3]{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

-((a + b*x^2)/(a*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/3))) + (Sqrt[2 - Sqrt[3]]*(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 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))], -
7 + 4*Sqrt[3]])/(3^(1/4)*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 = 33.1114, size = 340, normalized size = 1.18 \[ \frac{3^{\frac{3}{4}} b \sqrt{\frac{a^{\frac{2}{3}} b^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{a b + b^{2} x^{2}} + \left (a b + b^{2} x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \sqrt [3]{b} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a b + b^{2} x^{2}}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} \sqrt [3]{b} - \sqrt [3]{a b + b^{2} x^{2}}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{2}{3}} F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \sqrt [3]{b} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a b + b^{2} x^{2}}}{- \sqrt [3]{a} \sqrt [3]{b} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a b + b^{2} x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 a x \sqrt{- \frac{\sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a} \sqrt [3]{b} - \sqrt [3]{a b + b^{2} x^{2}}\right )}{\left (\sqrt [3]{a} \sqrt [3]{b} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a b + b^{2} x^{2}}\right )^{2}}} \left (a b + b^{2} x^{2}\right )^{\frac{4}{3}}} - \frac{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{2}{3}}}{a x \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3**(3/4)*b*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 + s
qrt(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))/(3*a*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)) - (a**2 + 2*a*b*x**2 + b**2*x**4)
**(2/3)/(a*x*(a + b*x**2))

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Mathematica [C]  time = 0.0470247, size = 72, normalized size = 0.25 \[ \frac{-b x^2 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{3}{2};-\frac{b x^2}{a}\right )-3 \left (a+b x^2\right )}{3 a x \sqrt [3]{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt [3]{{b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/x^2/(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. \[ \int \frac{1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{3}} x^{2}}\,{d x} \]

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^2),x, algorithm="maxima")

[Out]

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

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

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^2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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^2),x, algorithm="giac")

[Out]

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