3.1017 \(\int \frac{\sqrt [6]{a+b x^2}}{x^8} \, dx\)

Optimal. Leaf size=347 \[ -\frac{16 b^3 \sqrt [6]{a+b x^2}}{405 a^3 x}-\frac{32 \sqrt{2-\sqrt{3}} b^3 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{405 \sqrt [4]{3} a^3 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}+\frac{2 b^2 \sqrt [6]{a+b x^2}}{135 a^2 x^3}-\frac{\sqrt [6]{a+b x^2}}{7 x^7}-\frac{b \sqrt [6]{a+b x^2}}{105 a x^5} \]

[Out]

-(a + b*x^2)^(1/6)/(7*x^7) - (b*(a + b*x^2)^(1/6))/(105*a*x^5) + (2*b^2*(a + b*x
^2)^(1/6))/(135*a^2*x^3) - (16*b^3*(a + b*x^2)^(1/6))/(405*a^3*x) - (32*Sqrt[2 -
Sqrt[3]]*b^3*(a + b*x^2)^(1/6)*(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]*El
lipticF[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]])/(405*3^(1/4)*a^3*x*(a/(a + b*x^2))^(1/3)*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 = 0.756328, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{16 b^3 \sqrt [6]{a+b x^2}}{405 a^3 x}-\frac{32 \sqrt{2-\sqrt{3}} b^3 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{405 \sqrt [4]{3} a^3 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}+\frac{2 b^2 \sqrt [6]{a+b x^2}}{135 a^2 x^3}-\frac{\sqrt [6]{a+b x^2}}{7 x^7}-\frac{b \sqrt [6]{a+b x^2}}{105 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(1/6)/x^8,x]

[Out]

-(a + b*x^2)^(1/6)/(7*x^7) - (b*(a + b*x^2)^(1/6))/(105*a*x^5) + (2*b^2*(a + b*x
^2)^(1/6))/(135*a^2*x^3) - (16*b^3*(a + b*x^2)^(1/6))/(405*a^3*x) - (32*Sqrt[2 -
Sqrt[3]]*b^3*(a + b*x^2)^(1/6)*(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]*El
lipticF[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]])/(405*3^(1/4)*a^3*x*(a/(a + b*x^2))^(1/3)*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 [A]  time = 29.8831, size = 316, normalized size = 0.91 \[ - \frac{\sqrt [6]{a + b x^{2}}}{7 x^{7}} - \frac{b \sqrt [6]{a + b x^{2}}}{105 a x^{5}} + \frac{2 b^{2} \sqrt [6]{a + b x^{2}}}{135 a^{2} x^{3}} - \frac{16 b^{3} \sqrt [6]{a + b x^{2}}}{405 a^{3} x} - \frac{32 \cdot 3^{\frac{3}{4}} b^{3} \sqrt{\frac{\left (- \frac{b x^{2}}{a + b x^{2}} + 1\right )^{\frac{2}{3}} + \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1}{\left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \sqrt [6]{a + b x^{2}} \left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{1215 a^{3} x \sqrt [3]{\frac{a}{a + b x^{2}}} \sqrt{\frac{\sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - 1}{\left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1\right )^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/6)/x**8,x)

[Out]

-(a + b*x**2)**(1/6)/(7*x**7) - b*(a + b*x**2)**(1/6)/(105*a*x**5) + 2*b**2*(a +
b*x**2)**(1/6)/(135*a**2*x**3) - 16*b**3*(a + b*x**2)**(1/6)/(405*a**3*x) - 32*
3**(3/4)*b**3*sqrt(((-b*x**2/(a + b*x**2) + 1)**(2/3) + (-b*x**2/(a + b*x**2) +
1)**(1/3) + 1)/(-(-b*x**2/(a + b*x**2) + 1)**(1/3) - sqrt(3) + 1)**2)*sqrt(-sqrt
(3) + 2)*(a + b*x**2)**(1/6)*(-(-b*x**2/(a + b*x**2) + 1)**(1/3) + 1)*elliptic_f
(asin((-(-b*x**2/(a + b*x**2) + 1)**(1/3) + 1 + sqrt(3))/(-(-b*x**2/(a + b*x**2)
+ 1)**(1/3) - sqrt(3) + 1)), -7 + 4*sqrt(3))/(1215*a**3*x*(a/(a + b*x**2))**(1/
3)*sqrt(((-b*x**2/(a + b*x**2) + 1)**(1/3) - 1)/(-(-b*x**2/(a + b*x**2) + 1)**(1
/3) - sqrt(3) + 1)**2))

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Mathematica [C]  time = 0.0641691, size = 108, normalized size = 0.31 \[ \frac{-3 \left (405 a^4+432 a^3 b x^2-15 a^2 b^2 x^4+70 a b^3 x^6+112 b^4 x^8\right )-224 b^4 x^8 \left (\frac{b x^2}{a}+1\right )^{5/6} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};-\frac{b x^2}{a}\right )}{8505 a^3 x^7 \left (a+b x^2\right )^{5/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(1/6)/x^8,x]

[Out]

(-3*(405*a^4 + 432*a^3*b*x^2 - 15*a^2*b^2*x^4 + 70*a*b^3*x^6 + 112*b^4*x^8) - 22
4*b^4*x^8*(1 + (b*x^2)/a)^(5/6)*Hypergeometric2F1[1/2, 5/6, 3/2, -((b*x^2)/a)])/
(8505*a^3*x^7*(a + b*x^2)^(5/6))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{8}}\sqrt [6]{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/6)/x^8,x)

[Out]

int((b*x^2+a)^(1/6)/x^8,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{6}}}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/6)/x^8,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(1/6)/x^8, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{1}{6}}}{x^{8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/6)/x^8,x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(1/6)/x^8, x)

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Sympy [A]  time = 7.45747, size = 34, normalized size = 0.1 \[ - \frac{\sqrt [6]{a}{{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{2}, - \frac{1}{6} \\ - \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(1/6)/x**8,x)

[Out]

-a**(1/6)*hyper((-7/2, -1/6), (-5/2,), b*x**2*exp_polar(I*pi)/a)/(7*x**7)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{6}}}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/6)/x^8,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/6)/x^8, x)