3.1011 \(\int \frac{\sqrt [4]{a+b x^4}}{x^4} \, dx\)

Optimal. Leaf size=82 \[ -\frac{b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} \left (a+b x^4\right )^{3/4}}-\frac{\sqrt [4]{a+b x^4}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0993653, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} \left (a+b x^4\right )^{3/4}}-\frac{\sqrt [4]{a+b x^4}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(1/4)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.4854, size = 71, normalized size = 0.87 \[ - \frac{\sqrt [4]{a + b x^{4}}}{3 x^{3}} - \frac{b^{\frac{3}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{3 \sqrt{a} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0396075, size = 66, normalized size = 0.8 \[ \frac{b x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )-a-b x^4}{3 x^3 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-a - b*x^4 + b*x^4*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*
x^4)/a)])/(3*x^3*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}}\sqrt [4]{b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(1/4)/x^4,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(1/4)/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a)^(1/4)/x^4, x)

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Sympy [A]  time = 3.01266, size = 31, normalized size = 0.38 \[ - \frac{\sqrt [4]{b}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(1/4)/x^4, x)