3.1183 \(\int x \sqrt [4]{a-b x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.100571, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{a^{3/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{b} \left (a-b x^4\right )^{3/4}}+\frac{1}{3} x^2 \sqrt [4]{a-b x^4} \]

Antiderivative was successfully verified.

[In]  Int[x*(a - b*x^4)^(1/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.2816, size = 66, normalized size = 0.8 \[ \frac{a^{\frac{3}{2}} \left (1 - \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{3 \sqrt{b} \left (a - b x^{4}\right )^{\frac{3}{4}}} + \frac{x^{2} \sqrt [4]{a - b x^{4}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0532167, size = 64, normalized size = 0.78 \[ \frac{x^2 \left (a \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^4}{a}\right )+2 a-2 b x^4\right )}{6 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(2*a - 2*b*x^4 + a*(1 - (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (
b*x^4)/a]))/(6*(a - b*x^4)^(3/4))

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Maple [F]  time = 0.025, size = 0, normalized size = 0. \[ \int x\sqrt [4]{-b{x}^{4}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(-b*x^4+a)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-b*x^4 + a)^(1/4)*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-b*x^4 + a)^(1/4)*x, x)

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Sympy [A]  time = 2.40872, size = 31, normalized size = 0.38 \[ \frac{\sqrt [4]{a} x^{2}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-b*x^4 + a)^(1/4)*x, x)