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

Optimal. Leaf size=599 \[ \frac{55 b \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{9 \sqrt{2} \sqrt [4]{3} a^{8/3} x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac{55 \sqrt{2+\sqrt{3}} b \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{12\ 3^{3/4} a^{8/3} x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac{55 b^2 x}{18 a^3 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}+\frac{55 b \left (a+b x^2\right )^{2/3}}{18 a^3 x}-\frac{11 \left (a+b x^2\right )^{2/3}}{6 a^2 x^3}+\frac{3}{2 a x^3 \sqrt [3]{a+b x^2}} \]

[Out]

3/(2*a*x^3*(a + b*x^2)^(1/3)) - (11*(a + b*x^2)^(2/3))/(6*a^2*x^3) + (55*b*(a +
b*x^2)^(2/3))/(18*a^3*x) + (55*b^2*x)/(18*a^3*((1 - Sqrt[3])*a^(1/3) - (a + b*x^
2)^(1/3))) - (55*Sqrt[2 + Sqrt[3]]*b*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[(a^(2/3)
+ a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a +
b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))/((
1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(12*3^(3/4)*a^(8/3)
*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a +
b*x^2)^(1/3))^2)]) + (55*b*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)
*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/
3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))/((1 - Sqrt[3
])*a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(9*Sqrt[2]*3^(1/4)*a^(8/3)*x*
Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a + b*x
^2)^(1/3))^2)])

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Rubi [A]  time = 1.0218, antiderivative size = 599, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{55 b \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{9 \sqrt{2} \sqrt [4]{3} a^{8/3} x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac{55 \sqrt{2+\sqrt{3}} b \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{12\ 3^{3/4} a^{8/3} x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac{55 b^2 x}{18 a^3 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}+\frac{55 b \left (a+b x^2\right )^{2/3}}{18 a^3 x}-\frac{11 \left (a+b x^2\right )^{2/3}}{6 a^2 x^3}+\frac{3}{2 a x^3 \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

3/(2*a*x^3*(a + b*x^2)^(1/3)) - (11*(a + b*x^2)^(2/3))/(6*a^2*x^3) + (55*b*(a +
b*x^2)^(2/3))/(18*a^3*x) + (55*b^2*x)/(18*a^3*((1 - Sqrt[3])*a^(1/3) - (a + b*x^
2)^(1/3))) - (55*Sqrt[2 + Sqrt[3]]*b*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[(a^(2/3)
+ a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a +
b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))/((
1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(12*3^(3/4)*a^(8/3)
*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a +
b*x^2)^(1/3))^2)]) + (55*b*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)
*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/
3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))/((1 - Sqrt[3
])*a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(9*Sqrt[2]*3^(1/4)*a^(8/3)*x*
Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a + b*x
^2)^(1/3))^2)])

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Rubi in Sympy [A]  time = 45.6966, size = 498, normalized size = 0.83 \[ \frac{3}{2 a x^{3} \sqrt [3]{a + b x^{2}}} - \frac{11 \left (a + b x^{2}\right )^{\frac{2}{3}}}{6 a^{2} x^{3}} - \frac{55 b^{2} x}{18 a^{3} \left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )} + \frac{55 b \left (a + b x^{2}\right )^{\frac{2}{3}}}{18 a^{3} x} - \frac{55 \sqrt [4]{3} b \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a + b x^{2}} + \left (a + b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{36 a^{\frac{8}{3}} x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}}} + \frac{55 \sqrt{2} \cdot 3^{\frac{3}{4}} b \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a + b x^{2}} + \left (a + b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{54 a^{\frac{8}{3}} x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/(2*a*x**3*(a + b*x**2)**(1/3)) - 11*(a + b*x**2)**(2/3)/(6*a**2*x**3) - 55*b**
2*x/(18*a**3*(a**(1/3)*(-1 + sqrt(3)) + (a + b*x**2)**(1/3))) + 55*b*(a + b*x**2
)**(2/3)/(18*a**3*x) - 55*3**(1/4)*b*sqrt((a**(2/3) + a**(1/3)*(a + b*x**2)**(1/
3) + (a + b*x**2)**(2/3))/(a**(1/3)*(-1 + sqrt(3)) + (a + b*x**2)**(1/3))**2)*sq
rt(sqrt(3) + 2)*(a**(1/3) - (a + b*x**2)**(1/3))*elliptic_e(asin((a**(1/3)*(1 +
sqrt(3)) - (a + b*x**2)**(1/3))/(-a**(1/3)*(-1 + sqrt(3)) - (a + b*x**2)**(1/3))
), -7 + 4*sqrt(3))/(36*a**(8/3)*x*sqrt(-a**(1/3)*(a**(1/3) - (a + b*x**2)**(1/3)
)/(a**(1/3)*(-1 + sqrt(3)) + (a + b*x**2)**(1/3))**2)) + 55*sqrt(2)*3**(3/4)*b*s
qrt((a**(2/3) + a**(1/3)*(a + b*x**2)**(1/3) + (a + b*x**2)**(2/3))/(a**(1/3)*(-
1 + sqrt(3)) + (a + b*x**2)**(1/3))**2)*(a**(1/3) - (a + b*x**2)**(1/3))*ellipti
c_f(asin((a**(1/3)*(1 + sqrt(3)) - (a + b*x**2)**(1/3))/(-a**(1/3)*(-1 + sqrt(3)
) - (a + b*x**2)**(1/3))), -7 + 4*sqrt(3))/(54*a**(8/3)*x*sqrt(-a**(1/3)*(a**(1/
3) - (a + b*x**2)**(1/3))/(a**(1/3)*(-1 + sqrt(3)) + (a + b*x**2)**(1/3))**2))

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Mathematica [C]  time = 0.0564722, size = 83, normalized size = 0.14 \[ \frac{-18 a^2-55 b^2 x^4 \sqrt [3]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )+66 a b x^2+165 b^2 x^4}{54 a^3 x^3 \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-18*a^2 + 66*a*b*x^2 + 165*b^2*x^4 - 55*b^2*x^4*(1 + (b*x^2)/a)^(1/3)*Hypergeom
etric2F1[1/3, 1/2, 3/2, -((b*x^2)/a)])/(54*a^3*x^3*(a + b*x^2)^(1/3))

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^2+a)^(4/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{6} + a x^{4}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((b*x^6 + a*x^4)*(b*x^2 + a)^(1/3)), x)

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Sympy [A]  time = 4.12367, size = 32, normalized size = 0.05 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{4}{3} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac{4}{3}} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-hyper((-3/2, 4/3), (-1/2,), b*x**2*exp_polar(I*pi)/a)/(3*a**(4/3)*x**3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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