3.1194 \(\int x^{12} \sqrt [4]{a-b x^4} \, dx\)

Optimal. Leaf size=156 \[ -\frac{3 a^{7/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{112 b^{5/2} \left (a-b x^4\right )^{3/4}}-\frac{3 a^3 x \sqrt [4]{a-b x^4}}{112 b^3}-\frac{3 a^2 x^5 \sqrt [4]{a-b x^4}}{280 b^2}+\frac{1}{14} x^{13} \sqrt [4]{a-b x^4}-\frac{a x^9 \sqrt [4]{a-b x^4}}{140 b} \]

[Out]

(-3*a^3*x*(a - b*x^4)^(1/4))/(112*b^3) - (3*a^2*x^5*(a - b*x^4)^(1/4))/(280*b^2)
- (a*x^9*(a - b*x^4)^(1/4))/(140*b) + (x^13*(a - b*x^4)^(1/4))/14 - (3*a^(7/2)*
(1 - a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(112*b^
(5/2)*(a - b*x^4)^(3/4))

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Rubi [A]  time = 0.216627, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{3 a^{7/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{112 b^{5/2} \left (a-b x^4\right )^{3/4}}-\frac{3 a^3 x \sqrt [4]{a-b x^4}}{112 b^3}-\frac{3 a^2 x^5 \sqrt [4]{a-b x^4}}{280 b^2}+\frac{1}{14} x^{13} \sqrt [4]{a-b x^4}-\frac{a x^9 \sqrt [4]{a-b x^4}}{140 b} \]

Antiderivative was successfully verified.

[In]  Int[x^12*(a - b*x^4)^(1/4),x]

[Out]

(-3*a^3*x*(a - b*x^4)^(1/4))/(112*b^3) - (3*a^2*x^5*(a - b*x^4)^(1/4))/(280*b^2)
- (a*x^9*(a - b*x^4)^(1/4))/(140*b) + (x^13*(a - b*x^4)^(1/4))/14 - (3*a^(7/2)*
(1 - a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(112*b^
(5/2)*(a - b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 27.1995, size = 136, normalized size = 0.87 \[ - \frac{3 a^{\frac{7}{2}} x^{3} \left (- \frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{112 b^{\frac{5}{2}} \left (a - b x^{4}\right )^{\frac{3}{4}}} - \frac{3 a^{3} x \sqrt [4]{a - b x^{4}}}{112 b^{3}} - \frac{3 a^{2} x^{5} \sqrt [4]{a - b x^{4}}}{280 b^{2}} - \frac{a x^{9} \sqrt [4]{a - b x^{4}}}{140 b} + \frac{x^{13} \sqrt [4]{a - b x^{4}}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**12*(-b*x**4+a)**(1/4),x)

[Out]

-3*a**(7/2)*x**3*(-a/(b*x**4) + 1)**(3/4)*elliptic_f(asin(sqrt(a)/(sqrt(b)*x**2)
)/2, 2)/(112*b**(5/2)*(a - b*x**4)**(3/4)) - 3*a**3*x*(a - b*x**4)**(1/4)/(112*b
**3) - 3*a**2*x**5*(a - b*x**4)**(1/4)/(280*b**2) - a*x**9*(a - b*x**4)**(1/4)/(
140*b) + x**13*(a - b*x**4)**(1/4)/14

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Mathematica [C]  time = 0.0685212, size = 102, normalized size = 0.65 \[ \frac{15 a^4 x \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^4}{a}\right )-15 a^4 x+9 a^3 b x^5+2 a^2 b^2 x^9+44 a b^3 x^{13}-40 b^4 x^{17}}{560 b^3 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^12*(a - b*x^4)^(1/4),x]

[Out]

(-15*a^4*x + 9*a^3*b*x^5 + 2*a^2*b^2*x^9 + 44*a*b^3*x^13 - 40*b^4*x^17 + 15*a^4*
x*(1 - (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, (b*x^4)/a])/(560*b^3*(a
- b*x^4)^(3/4))

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int{x}^{12}\sqrt [4]{-b{x}^{4}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^12*(-b*x^4+a)^(1/4),x)

[Out]

int(x^12*(-b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{12}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^12,x, algorithm="maxima")

[Out]

integrate((-b*x^4 + a)^(1/4)*x^12, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{12}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^12,x, algorithm="fricas")

[Out]

integral((-b*x^4 + a)^(1/4)*x^12, x)

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Sympy [A]  time = 10.243, size = 41, normalized size = 0.26 \[ \frac{\sqrt [4]{a} x^{13} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{17}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**12*(-b*x**4+a)**(1/4),x)

[Out]

a**(1/4)*x**13*gamma(13/4)*hyper((-1/4, 13/4), (17/4,), b*x**4*exp_polar(2*I*pi)
/a)/(4*gamma(17/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{12}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^12,x, algorithm="giac")

[Out]

integrate((-b*x^4 + a)^(1/4)*x^12, x)