3.1023 \(\int \frac{1}{x^4 \sqrt [6]{a+b x^2}} \, dx\)

Optimal. Leaf size=633 \[ -\frac{4 b^2 x}{9 a^2 \sqrt [6]{a+b x^2}}+\frac{4 b \left (a+b x^2\right )^{5/6}}{9 a^2 x}-\frac{4 b^2 x}{9 a \left (\frac{a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )}+\frac{4 \sqrt{2} b \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{9 \sqrt [4]{3} a x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{2 \sqrt{2+\sqrt{3}} b \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3\ 3^{3/4} a x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{\left (a+b x^2\right )^{5/6}}{3 a x^3} \]

[Out]

(-4*b^2*x)/(9*a^2*(a + b*x^2)^(1/6)) - (a + b*x^2)^(5/6)/(3*a*x^3) + (4*b*(a + b
*x^2)^(5/6))/(9*a^2*x) - (4*b^2*x)/(9*a*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*
(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))) - (2*Sqrt[2 + Sqrt[3]]*b*(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]*EllipticE[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]])/(3*3^(3/4)*a*x*(a/
(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*Sqrt[-((1 - (a/(a + b*x^2))^(1/3))/(1 - Sqr
t[3] - (a/(a + b*x^2))^(1/3))^2)]) + (4*Sqrt[2]*b*(1 - (a/(a + b*x^2))^(1/3))*Sq
rt[(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]])/(9*3^(1/4)*a*x*(a/(a + b*x^2))^(
2/3)*(a + b*x^2)^(1/6)*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 = 1.34764, antiderivative size = 633, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ -\frac{4 b^2 x}{9 a^2 \sqrt [6]{a+b x^2}}+\frac{4 b \left (a+b x^2\right )^{5/6}}{9 a^2 x}-\frac{4 b^2 x}{9 a \left (\frac{a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )}+\frac{4 \sqrt{2} b \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{9 \sqrt [4]{3} a x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{2 \sqrt{2+\sqrt{3}} b \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3\ 3^{3/4} a x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{\left (a+b x^2\right )^{5/6}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^2)^(1/6)),x]

[Out]

(-4*b^2*x)/(9*a^2*(a + b*x^2)^(1/6)) - (a + b*x^2)^(5/6)/(3*a*x^3) + (4*b*(a + b
*x^2)^(5/6))/(9*a^2*x) - (4*b^2*x)/(9*a*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*
(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))) - (2*Sqrt[2 + Sqrt[3]]*b*(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]*EllipticE[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]])/(3*3^(3/4)*a*x*(a/
(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*Sqrt[-((1 - (a/(a + b*x^2))^(1/3))/(1 - Sqr
t[3] - (a/(a + b*x^2))^(1/3))^2)]) + (4*Sqrt[2]*b*(1 - (a/(a + b*x^2))^(1/3))*Sq
rt[(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]])/(9*3^(1/4)*a*x*(a/(a + b*x^2))^(
2/3)*(a + b*x^2)^(1/6)*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 [F]  time = 0., size = 0, normalized size = 0. \[ \frac{4 b^{2} \int \frac{1}{\left (a + b x^{2}\right )^{\frac{7}{6}}}\, dx}{27 a} - \frac{\left (a + b x^{2}\right )^{\frac{5}{6}}}{3 a x^{3}} - \frac{4 b^{2} x}{9 a^{2} \sqrt [6]{a + b x^{2}}} + \frac{4 b \left (a + b x^{2}\right )^{\frac{5}{6}}}{9 a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**2+a)**(1/6),x)

[Out]

4*b**2*Integral((a + b*x**2)**(-7/6), x)/(27*a) - (a + b*x**2)**(5/6)/(3*a*x**3)
- 4*b**2*x/(9*a**2*(a + b*x**2)**(1/6)) + 4*b*(a + b*x**2)**(5/6)/(9*a**2*x)

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Mathematica [C]  time = 0.0548304, size = 83, normalized size = 0.13 \[ \frac{-9 a^2-8 b^2 x^4 \sqrt [6]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )+3 a b x^2+12 b^2 x^4}{27 a^2 x^3 \sqrt [6]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x^2)^(1/6)),x]

[Out]

(-9*a^2 + 3*a*b*x^2 + 12*b^2*x^4 - 8*b^2*x^4*(1 + (b*x^2)/a)^(1/6)*Hypergeometri
c2F1[1/6, 1/2, 3/2, -((b*x^2)/a)])/(27*a^2*x^3*(a + b*x^2)^(1/6))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}}{\frac{1}{\sqrt [6]{b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^2+a)^(1/6),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{1}{6}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(1/6)*x^4),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(1/6)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.06412, size = 32, normalized size = 0.05 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{6} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3 \sqrt [6]{a} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x**2+a)**(1/6),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{1}{6}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(1/6)*x^4),x, algorithm="giac")

[Out]

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