3.1013 \(\int \frac{\sqrt [4]{a+b x^4}}{x^{12}} \, dx\)

Optimal. Leaf size=128 \[ -\frac{4 b^{7/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 a^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{2 b^2 \sqrt [4]{a+b x^4}}{77 a^2 x^3}-\frac{\sqrt [4]{a+b x^4}}{11 x^{11}}-\frac{b \sqrt [4]{a+b x^4}}{77 a x^7} \]

[Out]

-(a + b*x^4)^(1/4)/(11*x^11) - (b*(a + b*x^4)^(1/4))/(77*a*x^7) + (2*b^2*(a + b*
x^4)^(1/4))/(77*a^2*x^3) - (4*b^(7/2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot
[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(77*a^(5/2)*(a + b*x^4)^(3/4))

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Rubi [A]  time = 0.159284, antiderivative size = 128, 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 \[ -\frac{4 b^{7/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 a^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{2 b^2 \sqrt [4]{a+b x^4}}{77 a^2 x^3}-\frac{\sqrt [4]{a+b x^4}}{11 x^{11}}-\frac{b \sqrt [4]{a+b x^4}}{77 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(1/4)/x^12,x]

[Out]

-(a + b*x^4)^(1/4)/(11*x^11) - (b*(a + b*x^4)^(1/4))/(77*a*x^7) + (2*b^2*(a + b*
x^4)^(1/4))/(77*a^2*x^3) - (4*b^(7/2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot
[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(77*a^(5/2)*(a + b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 18.7729, size = 114, normalized size = 0.89 \[ - \frac{\sqrt [4]{a + b x^{4}}}{11 x^{11}} - \frac{b \sqrt [4]{a + b x^{4}}}{77 a x^{7}} + \frac{2 b^{2} \sqrt [4]{a + b x^{4}}}{77 a^{2} x^{3}} - \frac{4 b^{\frac{7}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{77 a^{\frac{5}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(1/4)/x**12,x)

[Out]

-(a + b*x**4)**(1/4)/(11*x**11) - b*(a + b*x**4)**(1/4)/(77*a*x**7) + 2*b**2*(a
+ b*x**4)**(1/4)/(77*a**2*x**3) - 4*b**(7/2)*x**3*(a/(b*x**4) + 1)**(3/4)*ellipt
ic_f(atan(sqrt(a)/(sqrt(b)*x**2))/2, 2)/(77*a**(5/2)*(a + b*x**4)**(3/4))

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Mathematica [C]  time = 0.0543245, size = 93, normalized size = 0.73 \[ \frac{-7 a^3-8 a^2 b x^4+4 b^3 x^{12} \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+a b^2 x^8+2 b^3 x^{12}}{77 a^2 x^{11} \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(1/4)/x^12,x]

[Out]

(-7*a^3 - 8*a^2*b*x^4 + a*b^2*x^8 + 2*b^3*x^12 + 4*b^3*x^12*(1 + (b*x^4)/a)^(3/4
)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^4)/a)])/(77*a^2*x^11*(a + b*x^4)^(3/4)
)

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{12}}\sqrt [4]{b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(1/4)/x^12,x)

[Out]

int((b*x^4+a)^(1/4)/x^12,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^12,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(1/4)/x^12, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{12}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^12,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(1/4)/x^12, x)

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Sympy [A]  time = 10.5732, size = 46, normalized size = 0.36 \[ \frac{\sqrt [4]{a} \Gamma \left (- \frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{11}{4}, - \frac{1}{4} \\ - \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{11} \Gamma \left (- \frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(1/4)/x**12,x)

[Out]

a**(1/4)*gamma(-11/4)*hyper((-11/4, -1/4), (-7/4,), b*x**4*exp_polar(I*pi)/a)/(4
*x**11*gamma(-7/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^12,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(1/4)/x^12, x)