3.795 \(\int \sqrt [4]{a-b x^2} \, dx\)

Optimal. Leaf size=78 \[ \frac{2 a^{3/2} \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{b} \left (a-b x^2\right )^{3/4}}+\frac{2}{3} x \sqrt [4]{a-b x^2} \]

[Out]

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

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Rubi [A]  time = 0.0561887, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 a^{3/2} \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{b} \left (a-b x^2\right )^{3/4}}+\frac{2}{3} x \sqrt [4]{a-b x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^2)^(1/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.52594, size = 66, normalized size = 0.85 \[ \frac{2 a^{\frac{3}{2}} \left (1 - \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{3 \sqrt{b} \left (a - b x^{2}\right )^{\frac{3}{4}}} + \frac{2 x \sqrt [4]{a - b x^{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0397393, size = 63, normalized size = 0.81 \[ \frac{a x \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^2}{a}\right )+2 a x-2 b x^3}{3 \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^2)^(1/4),x]

[Out]

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

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int \sqrt [4]{-b{x}^{2}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+a)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-b*x^2 + a)^(1/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-b*x^2 + a)^(1/4), x)

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Sympy [A]  time = 2.37684, size = 27, normalized size = 0.35 \[ \sqrt [4]{a} x{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**2+a)**(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-b*x^2 + a)^(1/4), x)