3.1226 \(\int \frac{x^6}{\sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=109 \[ \frac{a^{3/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 b^{3/2} \sqrt [4]{a-b x^4}}-\frac{a \left (a-b x^4\right )^{3/4}}{4 b^2 x}-\frac{x^3 \left (a-b x^4\right )^{3/4}}{6 b} \]

[Out]

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

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Rubi [A]  time = 0.153926, antiderivative size = 109, 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 \[ \frac{a^{3/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 b^{3/2} \sqrt [4]{a-b x^4}}-\frac{a \left (a-b x^4\right )^{3/4}}{4 b^2 x}-\frac{x^3 \left (a-b x^4\right )^{3/4}}{6 b} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a - b*x^4)^(1/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.6817, size = 88, normalized size = 0.81 \[ \frac{a^{\frac{3}{2}} x \sqrt [4]{- \frac{a}{b x^{4}} + 1} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{4 b^{\frac{3}{2}} \sqrt [4]{a - b x^{4}}} - \frac{a \left (a - b x^{4}\right )^{\frac{3}{4}}}{4 b^{2} x} - \frac{x^{3} \left (a - b x^{4}\right )^{\frac{3}{4}}}{6 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(-b*x**4+a)**(1/4),x)

[Out]

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

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Mathematica [C]  time = 0.0527921, size = 66, normalized size = 0.61 \[ \frac{x^3 \left (a \sqrt [4]{1-\frac{b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )-a+b x^4\right )}{6 b \sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a - b*x^4)^(1/4),x]

[Out]

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

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{{x}^{6}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(-b*x^4+a)^(1/4),x)

[Out]

int(x^6/(-b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(-b*x^4 + a)^(1/4),x, algorithm="maxima")

[Out]

integrate(x^6/(-b*x^4 + a)^(1/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6}}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(-b*x^4 + a)^(1/4),x, algorithm="fricas")

[Out]

integral(x^6/(-b*x^4 + a)^(1/4), x)

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Sympy [A]  time = 3.05551, size = 39, normalized size = 0.36 \[ \frac{x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(-b*x**4+a)**(1/4),x)

[Out]

x**7*gamma(7/4)*hyper((1/4, 7/4), (11/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a**(1/
4)*gamma(11/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(-b*x^4 + a)^(1/4),x, algorithm="giac")

[Out]

integrate(x^6/(-b*x^4 + a)^(1/4), x)