3.790 \(\int \frac{\sqrt [4]{a+b x^2}}{x^2} \, dx\)

Optimal. Leaf size=72 \[ \frac{\sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\left (a+b x^2\right )^{3/4}}-\frac{\sqrt [4]{a+b x^2}}{x} \]

[Out]

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

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Rubi [A]  time = 0.0633, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\left (a+b x^2\right )^{3/4}}-\frac{\sqrt [4]{a+b x^2}}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(1/4)/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.64162, size = 60, normalized size = 0.83 \[ \frac{\sqrt{a} \sqrt{b} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{\left (a + b x^{2}\right )^{\frac{3}{4}}} - \frac{\sqrt [4]{a + b x^{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/4)/x**2,x)

[Out]

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

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Mathematica [C]  time = 0.0399432, size = 68, normalized size = 0.94 \[ \frac{b x \left (\frac{a+b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right )}{2 \left (a+b x^2\right )^{3/4}}-\frac{\sqrt [4]{a+b x^2}}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(1/4)/x^2,x]

[Out]

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

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}}\sqrt [4]{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/4)/x^2,x)

[Out]

int((b*x^2+a)^(1/4)/x^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/4)/x^2,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(1/4)/x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/4)/x^2,x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(1/4)/x^2, x)

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Sympy [A]  time = 2.56579, size = 29, normalized size = 0.4 \[ - \frac{\sqrt [4]{a}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(1/4)/x**2,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/4)/x^2,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/4)/x^2, x)