3.1012 \(\int \frac{\sqrt [4]{a+b x^4}}{x^8} \, dx\)

Optimal. Leaf size=104 \[ \frac{2 b^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 a^{3/2} \left (a+b x^4\right )^{3/4}}-\frac{\sqrt [4]{a+b x^4}}{7 x^7}-\frac{b \sqrt [4]{a+b x^4}}{21 a x^3} \]

[Out]

-(a + b*x^4)^(1/4)/(7*x^7) - (b*(a + b*x^4)^(1/4))/(21*a*x^3) + (2*b^(5/2)*(1 +
a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(21*a^(3/2)*
(a + b*x^4)^(3/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.128622, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{2 b^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 a^{3/2} \left (a+b x^4\right )^{3/4}}-\frac{\sqrt [4]{a+b x^4}}{7 x^7}-\frac{b \sqrt [4]{a+b x^4}}{21 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(1/4)/x^8,x]

[Out]

-(a + b*x^4)^(1/4)/(7*x^7) - (b*(a + b*x^4)^(1/4))/(21*a*x^3) + (2*b^(5/2)*(1 +
a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(21*a^(3/2)*
(a + b*x^4)^(3/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.9674, size = 90, normalized size = 0.87 \[ - \frac{\sqrt [4]{a + b x^{4}}}{7 x^{7}} - \frac{b \sqrt [4]{a + b x^{4}}}{21 a x^{3}} + \frac{2 b^{\frac{5}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{21 a^{\frac{3}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(1/4)/x**8,x)

[Out]

-(a + b*x**4)**(1/4)/(7*x**7) - b*(a + b*x**4)**(1/4)/(21*a*x**3) + 2*b**(5/2)*x
**3*(a/(b*x**4) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x**2))/2, 2)/(21*a*
*(3/2)*(a + b*x**4)**(3/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0446194, size = 83, normalized size = 0.8 \[ \frac{-3 a^2-2 b^2 x^8 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )-4 a b x^4-b^2 x^8}{21 a x^7 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(1/4)/x^8,x]

[Out]

(-3*a^2 - 4*a*b*x^4 - b^2*x^8 - 2*b^2*x^8*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F
1[1/4, 3/4, 5/4, -((b*x^4)/a)])/(21*a*x^7*(a + b*x^4)^(3/4))

_______________________________________________________________________________________

Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{8}}\sqrt [4]{b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(1/4)/x^8,x)

[Out]

int((b*x^4+a)^(1/4)/x^8,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^8,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(1/4)/x^8, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^8,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(1/4)/x^8, x)

_______________________________________________________________________________________

Sympy [A]  time = 5.30616, size = 31, normalized size = 0.3 \[ - \frac{\sqrt [4]{b}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(1/4)/x**8,x)

[Out]

-b**(1/4)*hyper((-1/4, 3/2), (5/2,), a*exp_polar(I*pi)/(b*x**4))/(6*x**6)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^8,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(1/4)/x^8, x)