3.1182 \(\int x^5 \sqrt [4]{a-b x^4} \, dx\)

Optimal. Leaf size=105 \[ \frac{2 a^{5/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 b^{3/2} \left (a-b x^4\right )^{3/4}}+\frac{1}{7} x^6 \sqrt [4]{a-b x^4}-\frac{a x^2 \sqrt [4]{a-b x^4}}{21 b} \]

[Out]

-(a*x^2*(a - b*x^4)^(1/4))/(21*b) + (x^6*(a - b*x^4)^(1/4))/7 + (2*a^(5/2)*(1 -
(b*x^4)/a)^(3/4)*EllipticF[ArcSin[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(21*b^(3/2)*(a -
b*x^4)^(3/4))

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Rubi [A]  time = 0.156576, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ \frac{2 a^{5/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 b^{3/2} \left (a-b x^4\right )^{3/4}}+\frac{1}{7} x^6 \sqrt [4]{a-b x^4}-\frac{a x^2 \sqrt [4]{a-b x^4}}{21 b} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a - b*x^4)^(1/4),x]

[Out]

-(a*x^2*(a - b*x^4)^(1/4))/(21*b) + (x^6*(a - b*x^4)^(1/4))/7 + (2*a^(5/2)*(1 -
(b*x^4)/a)^(3/4)*EllipticF[ArcSin[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(21*b^(3/2)*(a -
b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 17.5685, size = 87, normalized size = 0.83 \[ \frac{2 a^{\frac{5}{2}} \left (1 - \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{21 b^{\frac{3}{2}} \left (a - b x^{4}\right )^{\frac{3}{4}}} - \frac{a x^{2} \sqrt [4]{a - b x^{4}}}{21 b} + \frac{x^{6} \sqrt [4]{a - b x^{4}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(-b*x**4+a)**(1/4),x)

[Out]

2*a**(5/2)*(1 - b*x**4/a)**(3/4)*elliptic_f(asin(sqrt(b)*x**2/sqrt(a))/2, 2)/(21
*b**(3/2)*(a - b*x**4)**(3/4)) - a*x**2*(a - b*x**4)**(1/4)/(21*b) + x**6*(a - b
*x**4)**(1/4)/7

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Mathematica [C]  time = 0.0742565, size = 80, normalized size = 0.76 \[ \frac{x^2 \left (a^2 \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^4}{a}\right )-a^2+4 a b x^4-3 b^2 x^8\right )}{21 b \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a - b*x^4)^(1/4),x]

[Out]

(x^2*(-a^2 + 4*a*b*x^4 - 3*b^2*x^8 + a^2*(1 - (b*x^4)/a)^(3/4)*Hypergeometric2F1
[1/2, 3/4, 3/2, (b*x^4)/a]))/(21*b*(a - b*x^4)^(3/4))

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \[ \int{x}^{5}\sqrt [4]{-b{x}^{4}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(-b*x^4+a)^(1/4),x)

[Out]

int(x^5*(-b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^5,x, algorithm="maxima")

[Out]

integrate((-b*x^4 + a)^(1/4)*x^5, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{5}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^5,x, algorithm="fricas")

[Out]

integral((-b*x^4 + a)^(1/4)*x^5, x)

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Sympy [A]  time = 3.30475, size = 31, normalized size = 0.3 \[ \frac{\sqrt [4]{a} x^{6}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(-b*x**4+a)**(1/4),x)

[Out]

a**(1/4)*x**6*hyper((-1/4, 3/2), (5/2,), b*x**4*exp_polar(2*I*pi)/a)/6

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^5,x, algorithm="giac")

[Out]

integrate((-b*x^4 + a)^(1/4)*x^5, x)