3.1196 \(\int x^4 \sqrt [4]{a-b x^4} \, dx\)

Optimal. Leaf size=106 \[ -\frac{a^{3/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 \sqrt{b} \left (a-b x^4\right )^{3/4}}-\frac{a x \sqrt [4]{a-b x^4}}{12 b}+\frac{1}{6} x^5 \sqrt [4]{a-b x^4} \]

[Out]

-(a*x*(a - b*x^4)^(1/4))/(12*b) + (x^5*(a - b*x^4)^(1/4))/6 - (a^(3/2)*(1 - a/(b
*x^4))^(3/4)*x^3*EllipticF[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(12*Sqrt[b]*(a -
b*x^4)^(3/4))

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Rubi [A]  time = 0.139298, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{a^{3/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 \sqrt{b} \left (a-b x^4\right )^{3/4}}-\frac{a x \sqrt [4]{a-b x^4}}{12 b}+\frac{1}{6} x^5 \sqrt [4]{a-b x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a - b*x^4)^(1/4),x]

[Out]

-(a*x*(a - b*x^4)^(1/4))/(12*b) + (x^5*(a - b*x^4)^(1/4))/6 - (a^(3/2)*(1 - a/(b
*x^4))^(3/4)*x^3*EllipticF[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(12*Sqrt[b]*(a -
b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 18.4683, size = 87, normalized size = 0.82 \[ - \frac{a^{\frac{3}{2}} x^{3} \left (- \frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{12 \sqrt{b} \left (a - b x^{4}\right )^{\frac{3}{4}}} - \frac{a x \sqrt [4]{a - b x^{4}}}{12 b} + \frac{x^{5} \sqrt [4]{a - b x^{4}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(3/2)*x**3*(-a/(b*x**4) + 1)**(3/4)*elliptic_f(asin(sqrt(a)/(sqrt(b)*x**2))/
2, 2)/(12*sqrt(b)*(a - b*x**4)**(3/4)) - a*x*(a - b*x**4)**(1/4)/(12*b) + x**5*(
a - b*x**4)**(1/4)/6

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Mathematica [C]  time = 0.0471236, size = 79, normalized size = 0.75 \[ \frac{a^2 x \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^4}{a}\right )-a^2 x+3 a b x^5-2 b^2 x^9}{12 b \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-(a^2*x) + 3*a*b*x^5 - 2*b^2*x^9 + a^2*x*(1 - (b*x^4)/a)^(3/4)*Hypergeometric2F
1[1/4, 3/4, 5/4, (b*x^4)/a])/(12*b*(a - b*x^4)^(3/4))

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{x}^{4}\sqrt [4]{-b{x}^{4}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(-b*x^4+a)^(1/4),x)

[Out]

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

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

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. \[{\rm integral}\left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{4}, x\right ) \]

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 = 3.04522, size = 41, normalized size = 0.39 \[ \frac{\sqrt [4]{a} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/4)*x**5*gamma(5/4)*hyper((-1/4, 5/4), (9/4,), b*x**4*exp_polar(2*I*pi)/a)/
(4*gamma(9/4))

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

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)