3.663 \(\int \frac{1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}} \, dx\)

Optimal. Leaf size=609 \[ \frac{3 x \left (\frac{b x^2}{a}+1\right )^{4/3}}{2 \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )}+\frac{3 x \left (a+b x^2\right )}{2 a \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac{3^{3/4} a \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} \left (\frac{b x^2}{a}+1\right )^{4/3} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{2} b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} \left (\frac{b x^2}{a}+1\right )^{4/3} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}} \]

[Out]

(3*x*(a + b*x^2))/(2*a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)) + (3*x*(1 + (b*x^2)/a)
^(4/3))/(2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)*(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3
))) - (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*a*(1 + (b*x^2)/a)^(4/3)*(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]*EllipticE[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]])/(4*b*x*(a^2 + 2*a*b*x^2
+ b^2*x^4)^(2/3)*Sqrt[-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - Sqrt[3] - (1 + (b*x^2)
/a)^(1/3))^2)]) + (3^(3/4)*a*(1 + (b*x^2)/a)^(4/3)*(1 - (1 + (b*x^2)/a)^(1/3))*S
qrt[(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 - Sqr
t[3] - (1 + (b*x^2)/a)^(1/3))], -7 + 4*Sqrt[3]])/(Sqrt[2]*b*x*(a^2 + 2*a*b*x^2 +
b^2*x^4)^(2/3)*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 = 0.894752, antiderivative size = 609, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{3 x \left (\frac{b x^2}{a}+1\right )^{4/3}}{2 \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )}+\frac{3 x \left (a+b x^2\right )}{2 a \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac{3^{3/4} a \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} \left (\frac{b x^2}{a}+1\right )^{4/3} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{2} b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} \left (\frac{b x^2}{a}+1\right )^{4/3} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}} \]

Antiderivative was successfully verified.

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

[Out]

(3*x*(a + b*x^2))/(2*a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)) + (3*x*(1 + (b*x^2)/a)
^(4/3))/(2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)*(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3
))) - (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*a*(1 + (b*x^2)/a)^(4/3)*(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]*EllipticE[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]])/(4*b*x*(a^2 + 2*a*b*x^2
+ b^2*x^4)^(2/3)*Sqrt[-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - Sqrt[3] - (1 + (b*x^2)
/a)^(1/3))^2)]) + (3^(3/4)*a*(1 + (b*x^2)/a)^(4/3)*(1 - (1 + (b*x^2)/a)^(1/3))*S
qrt[(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 - Sqr
t[3] - (1 + (b*x^2)/a)^(1/3))], -7 + 4*Sqrt[3]])/(Sqrt[2]*b*x*(a^2 + 2*a*b*x^2 +
b^2*x^4)^(2/3)*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 = 76.8475, size = 738, normalized size = 1.21 \[ \text{result too large to display} \]

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)**(2/3),x)

[Out]

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

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Mathematica [C]  time = 0.0531306, size = 64, normalized size = 0.11 \[ -\frac{x \left (a+b x^2\right ) \left (\sqrt [3]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )-3\right )}{2 a \left (\left (a+b x^2\right )^2\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.018, size = 0, normalized size = 0. \[ \int \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{-{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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