3.997 \(\int \frac{\sqrt [4]{a+b x^4}}{x^3} \, dx\)

Optimal. Leaf size=79 \[ \frac{\sqrt{a} \sqrt{b} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \left (a+b x^4\right )^{3/4}}-\frac{\sqrt [4]{a+b x^4}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.100554, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\sqrt{a} \sqrt{b} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \left (a+b x^4\right )^{3/4}}-\frac{\sqrt [4]{a+b x^4}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(1/4)/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.2915, size = 66, normalized size = 0.84 \[ \frac{\sqrt{a} \sqrt{b} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{2 \left (a + b x^{4}\right )^{\frac{3}{4}}} - \frac{\sqrt [4]{a + b x^{4}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(1/4)/x**3,x)

[Out]

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

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Mathematica [C]  time = 0.0416064, size = 66, normalized size = 0.84 \[ \frac{b x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )-2 \left (a+b x^4\right )}{4 x^2 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(1/4)/x^3,x]

[Out]

(-2*(a + b*x^4) + b*x^4*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -
((b*x^4)/a)])/(4*x^2*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}\sqrt [4]{b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(1/4)/x^3,x)

[Out]

int((b*x^4+a)^(1/4)/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^3,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(1/4)/x^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^3,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(1/4)/x^3, x)

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Sympy [A]  time = 2.66805, size = 32, normalized size = 0.41 \[ - \frac{\sqrt [4]{a}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(1/4)/x**3,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^3,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(1/4)/x^3, x)