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

Optimal. Leaf size=126 \[ \frac{8 a^{7/2} \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 b^{5/2} \left (a-b x^2\right )^{3/4}}-\frac{4 a^2 x \sqrt [4]{a-b x^2}}{77 b^2}+\frac{2}{11} x^5 \sqrt [4]{a-b x^2}-\frac{2 a x^3 \sqrt [4]{a-b x^2}}{77 b} \]

[Out]

(-4*a^2*x*(a - b*x^2)^(1/4))/(77*b^2) - (2*a*x^3*(a - b*x^2)^(1/4))/(77*b) + (2*
x^5*(a - b*x^2)^(1/4))/11 + (8*a^(7/2)*(1 - (b*x^2)/a)^(3/4)*EllipticF[ArcSin[(S
qrt[b]*x)/Sqrt[a]]/2, 2])/(77*b^(5/2)*(a - b*x^2)^(3/4))

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Rubi [A]  time = 0.146479, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{8 a^{7/2} \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 b^{5/2} \left (a-b x^2\right )^{3/4}}-\frac{4 a^2 x \sqrt [4]{a-b x^2}}{77 b^2}+\frac{2}{11} x^5 \sqrt [4]{a-b x^2}-\frac{2 a x^3 \sqrt [4]{a-b x^2}}{77 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a - b*x^2)^(1/4),x]

[Out]

(-4*a^2*x*(a - b*x^2)^(1/4))/(77*b^2) - (2*a*x^3*(a - b*x^2)^(1/4))/(77*b) + (2*
x^5*(a - b*x^2)^(1/4))/11 + (8*a^(7/2)*(1 - (b*x^2)/a)^(3/4)*EllipticF[ArcSin[(S
qrt[b]*x)/Sqrt[a]]/2, 2])/(77*b^(5/2)*(a - b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 18.9377, size = 110, normalized size = 0.87 \[ \frac{8 a^{\frac{7}{2}} \left (1 - \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{77 b^{\frac{5}{2}} \left (a - b x^{2}\right )^{\frac{3}{4}}} - \frac{4 a^{2} x \sqrt [4]{a - b x^{2}}}{77 b^{2}} - \frac{2 a x^{3} \sqrt [4]{a - b x^{2}}}{77 b} + \frac{2 x^{5} \sqrt [4]{a - b x^{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*a**(7/2)*(1 - b*x**2/a)**(3/4)*elliptic_f(asin(sqrt(b)*x/sqrt(a))/2, 2)/(77*b*
*(5/2)*(a - b*x**2)**(3/4)) - 4*a**2*x*(a - b*x**2)**(1/4)/(77*b**2) - 2*a*x**3*
(a - b*x**2)**(1/4)/(77*b) + 2*x**5*(a - b*x**2)**(1/4)/11

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Mathematica [C]  time = 0.0881803, size = 89, normalized size = 0.71 \[ \frac{2 x \left (2 a^3 \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^2}{a}\right )-2 a^3+a^2 b x^2+8 a b^2 x^4-7 b^3 x^6\right )}{77 b^2 \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(-2*a^3 + a^2*b*x^2 + 8*a*b^2*x^4 - 7*b^3*x^6 + 2*a^3*(1 - (b*x^2)/a)^(3/4)
*Hypergeometric2F1[1/2, 3/4, 3/2, (b*x^2)/a]))/(77*b^2*(a - b*x^2)^(3/4))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{x}^{4}\sqrt [4]{-b{x}^{2}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(-b*x^2+a)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-b*x^2 + a)^(1/4)*x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-b*x^2 + a)^(1/4)*x^4, x)

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Sympy [A]  time = 3.0579, size = 31, normalized size = 0.25 \[ \frac{\sqrt [4]{a} x^{5}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/4)*x**5*hyper((-1/4, 5/2), (7/2,), b*x**2*exp_polar(2*I*pi)/a)/5

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-b*x^2 + a)^(1/4)*x^4, x)