3.999 \(\int \frac{\sqrt [4]{a+b x^4}}{x^{11}} \, dx\)

Optimal. Leaf size=125 \[ \frac{b^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{24 a^{3/2} \left (a+b x^4\right )^{3/4}}+\frac{b^2 \sqrt [4]{a+b x^4}}{24 a^2 x^2}-\frac{\sqrt [4]{a+b x^4}}{10 x^{10}}-\frac{b \sqrt [4]{a+b x^4}}{60 a x^6} \]

[Out]

-(a + b*x^4)^(1/4)/(10*x^10) - (b*(a + b*x^4)^(1/4))/(60*a*x^6) + (b^2*(a + b*x^
4)^(1/4))/(24*a^2*x^2) + (b^(5/2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b
]*x^2)/Sqrt[a]]/2, 2])/(24*a^(3/2)*(a + b*x^4)^(3/4))

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Rubi [A]  time = 0.18376, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{b^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{24 a^{3/2} \left (a+b x^4\right )^{3/4}}+\frac{b^2 \sqrt [4]{a+b x^4}}{24 a^2 x^2}-\frac{\sqrt [4]{a+b x^4}}{10 x^{10}}-\frac{b \sqrt [4]{a+b x^4}}{60 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(1/4)/x^11,x]

[Out]

-(a + b*x^4)^(1/4)/(10*x^10) - (b*(a + b*x^4)^(1/4))/(60*a*x^6) + (b^2*(a + b*x^
4)^(1/4))/(24*a^2*x^2) + (b^(5/2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b
]*x^2)/Sqrt[a]]/2, 2])/(24*a^(3/2)*(a + b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 18.991, size = 107, normalized size = 0.86 \[ - \frac{\sqrt [4]{a + b x^{4}}}{10 x^{10}} - \frac{b \sqrt [4]{a + b x^{4}}}{60 a x^{6}} + \frac{b^{2} \sqrt [4]{a + b x^{4}}}{24 a^{2} x^{2}} + \frac{b^{\frac{5}{2}} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{24 a^{\frac{3}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(1/4)/x**11,x)

[Out]

-(a + b*x**4)**(1/4)/(10*x**10) - b*(a + b*x**4)**(1/4)/(60*a*x**6) + b**2*(a +
b*x**4)**(1/4)/(24*a**2*x**2) + b**(5/2)*(1 + b*x**4/a)**(3/4)*elliptic_f(atan(s
qrt(b)*x**2/sqrt(a))/2, 2)/(24*a**(3/2)*(a + b*x**4)**(3/4))

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Mathematica [C]  time = 0.0536903, size = 94, normalized size = 0.75 \[ \frac{-24 a^3-28 a^2 b x^4+5 b^3 x^{12} \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )+6 a b^2 x^8+10 b^3 x^{12}}{240 a^2 x^{10} \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(1/4)/x^11,x]

[Out]

(-24*a^3 - 28*a^2*b*x^4 + 6*a*b^2*x^8 + 10*b^3*x^12 + 5*b^3*x^12*(1 + (b*x^4)/a)
^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -((b*x^4)/a)])/(240*a^2*x^10*(a + b*x^4)
^(3/4))

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{11}}\sqrt [4]{b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(1/4)/x^11,x)

[Out]

int((b*x^4+a)^(1/4)/x^11,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^11,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(1/4)/x^11, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{11}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^11,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(1/4)/x^11, x)

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Sympy [A]  time = 8.72474, size = 34, normalized size = 0.27 \[ - \frac{\sqrt [4]{a}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, - \frac{1}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{10 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(1/4)/x**11,x)

[Out]

-a**(1/4)*hyper((-5/2, -1/4), (-3/2,), b*x**4*exp_polar(I*pi)/a)/(10*x**10)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^11,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(1/4)/x^11, x)