3.1014 \(\int \frac{\sqrt [4]{a+b x^4}}{x^{16}} \, dx\)

Optimal. Leaf size=152 \[ \frac{8 b^{9/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{231 a^{7/2} \left (a+b x^4\right )^{3/4}}-\frac{4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}+\frac{2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}-\frac{\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac{b \sqrt [4]{a+b x^4}}{165 a x^{11}} \]

[Out]

-(a + b*x^4)^(1/4)/(15*x^15) - (b*(a + b*x^4)^(1/4))/(165*a*x^11) + (2*b^2*(a +
b*x^4)^(1/4))/(231*a^2*x^7) - (4*b^3*(a + b*x^4)^(1/4))/(231*a^3*x^3) + (8*b^(9/
2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(231
*a^(7/2)*(a + b*x^4)^(3/4))

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Rubi [A]  time = 0.201295, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{8 b^{9/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{231 a^{7/2} \left (a+b x^4\right )^{3/4}}-\frac{4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}+\frac{2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}-\frac{\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac{b \sqrt [4]{a+b x^4}}{165 a x^{11}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(1/4)/x^16,x]

[Out]

-(a + b*x^4)^(1/4)/(15*x^15) - (b*(a + b*x^4)^(1/4))/(165*a*x^11) + (2*b^2*(a +
b*x^4)^(1/4))/(231*a^2*x^7) - (4*b^3*(a + b*x^4)^(1/4))/(231*a^3*x^3) + (8*b^(9/
2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(231
*a^(7/2)*(a + b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 23.3178, size = 138, normalized size = 0.91 \[ - \frac{\sqrt [4]{a + b x^{4}}}{15 x^{15}} - \frac{b \sqrt [4]{a + b x^{4}}}{165 a x^{11}} + \frac{2 b^{2} \sqrt [4]{a + b x^{4}}}{231 a^{2} x^{7}} - \frac{4 b^{3} \sqrt [4]{a + b x^{4}}}{231 a^{3} x^{3}} + \frac{8 b^{\frac{9}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{231 a^{\frac{7}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(1/4)/x**16,x)

[Out]

-(a + b*x**4)**(1/4)/(15*x**15) - b*(a + b*x**4)**(1/4)/(165*a*x**11) + 2*b**2*(
a + b*x**4)**(1/4)/(231*a**2*x**7) - 4*b**3*(a + b*x**4)**(1/4)/(231*a**3*x**3)
+ 8*b**(9/2)*x**3*(a/(b*x**4) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x**2)
)/2, 2)/(231*a**(7/2)*(a + b*x**4)**(3/4))

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Mathematica [C]  time = 0.0616777, size = 105, normalized size = 0.69 \[ \frac{-77 a^4-84 a^3 b x^4+3 a^2 b^2 x^8-40 b^4 x^{16} \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )-10 a b^3 x^{12}-20 b^4 x^{16}}{1155 a^3 x^{15} \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(1/4)/x^16,x]

[Out]

(-77*a^4 - 84*a^3*b*x^4 + 3*a^2*b^2*x^8 - 10*a*b^3*x^12 - 20*b^4*x^16 - 40*b^4*x
^16*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^4)/a)])/(1155*
a^3*x^15*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{16}}\sqrt [4]{b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(1/4)/x^16,x)

[Out]

int((b*x^4+a)^(1/4)/x^16,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{16}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^16,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(1/4)/x^16, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{16}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^16,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(1/4)/x^16, x)

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Sympy [A]  time = 21.7158, size = 46, normalized size = 0.3 \[ \frac{\sqrt [4]{a} \Gamma \left (- \frac{15}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{15}{4}, - \frac{1}{4} \\ - \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{15} \Gamma \left (- \frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(1/4)/x**16,x)

[Out]

a**(1/4)*gamma(-15/4)*hyper((-15/4, -1/4), (-11/4,), b*x**4*exp_polar(I*pi)/a)/(
4*x**15*gamma(-11/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{16}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^16,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(1/4)/x^16, x)