3.822 \(\int \frac{x^6}{\sqrt [4]{a-b x^2}} \, dx\)

Optimal. Leaf size=129 \[ \frac{16 a^{7/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{39 b^{7/2} \sqrt [4]{a-b x^2}}-\frac{8 a^2 x \left (a-b x^2\right )^{3/4}}{39 b^3}-\frac{20 a x^3 \left (a-b x^2\right )^{3/4}}{117 b^2}-\frac{2 x^5 \left (a-b x^2\right )^{3/4}}{13 b} \]

[Out]

(-8*a^2*x*(a - b*x^2)^(3/4))/(39*b^3) - (20*a*x^3*(a - b*x^2)^(3/4))/(117*b^2) -
(2*x^5*(a - b*x^2)^(3/4))/(13*b) + (16*a^(7/2)*(1 - (b*x^2)/a)^(1/4)*EllipticE[
ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(39*b^(7/2)*(a - b*x^2)^(1/4))

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Rubi [A]  time = 0.146651, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{16 a^{7/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{39 b^{7/2} \sqrt [4]{a-b x^2}}-\frac{8 a^2 x \left (a-b x^2\right )^{3/4}}{39 b^3}-\frac{20 a x^3 \left (a-b x^2\right )^{3/4}}{117 b^2}-\frac{2 x^5 \left (a-b x^2\right )^{3/4}}{13 b} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a - b*x^2)^(1/4),x]

[Out]

(-8*a^2*x*(a - b*x^2)^(3/4))/(39*b^3) - (20*a*x^3*(a - b*x^2)^(3/4))/(117*b^2) -
(2*x^5*(a - b*x^2)^(3/4))/(13*b) + (16*a^(7/2)*(1 - (b*x^2)/a)^(1/4)*EllipticE[
ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(39*b^(7/2)*(a - b*x^2)^(1/4))

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Rubi in Sympy [A]  time = 19.2399, size = 114, normalized size = 0.88 \[ \frac{16 a^{\frac{7}{2}} \sqrt [4]{1 - \frac{b x^{2}}{a}} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{39 b^{\frac{7}{2}} \sqrt [4]{a - b x^{2}}} - \frac{8 a^{2} x \left (a - b x^{2}\right )^{\frac{3}{4}}}{39 b^{3}} - \frac{20 a x^{3} \left (a - b x^{2}\right )^{\frac{3}{4}}}{117 b^{2}} - \frac{2 x^{5} \left (a - b x^{2}\right )^{\frac{3}{4}}}{13 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(-b*x**2+a)**(1/4),x)

[Out]

16*a**(7/2)*(1 - b*x**2/a)**(1/4)*elliptic_e(asin(sqrt(b)*x/sqrt(a))/2, 2)/(39*b
**(7/2)*(a - b*x**2)**(1/4)) - 8*a**2*x*(a - b*x**2)**(3/4)/(39*b**3) - 20*a*x**
3*(a - b*x**2)**(3/4)/(117*b**2) - 2*x**5*(a - b*x**2)**(3/4)/(13*b)

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Mathematica [C]  time = 0.0930933, size = 89, normalized size = 0.69 \[ \frac{2 x \left (12 a^3 \sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )-12 a^3+2 a^2 b x^2+a b^2 x^4+9 b^3 x^6\right )}{117 b^3 \sqrt [4]{a-b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a - b*x^2)^(1/4),x]

[Out]

(2*x*(-12*a^3 + 2*a^2*b*x^2 + a*b^2*x^4 + 9*b^3*x^6 + 12*a^3*(1 - (b*x^2)/a)^(1/
4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b*x^2)/a]))/(117*b^3*(a - b*x^2)^(1/4))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{{x}^{6}{\frac{1}{\sqrt [4]{-b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(-b*x^2+a)^(1/4),x)

[Out]

int(x^6/(-b*x^2+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(-b*x^2 + a)^(1/4),x, algorithm="maxima")

[Out]

integrate(x^6/(-b*x^2 + a)^(1/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6}}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(-b*x^2 + a)^(1/4),x, algorithm="fricas")

[Out]

integral(x^6/(-b*x^2 + a)^(1/4), x)

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Sympy [A]  time = 3.28356, size = 29, normalized size = 0.22 \[ \frac{x^{7}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{7 \sqrt [4]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(-b*x**2+a)**(1/4),x)

[Out]

x**7*hyper((1/4, 7/2), (9/2,), b*x**2*exp_polar(2*I*pi)/a)/(7*a**(1/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(-b*x^2 + a)^(1/4),x, algorithm="giac")

[Out]

integrate(x^6/(-b*x^2 + a)^(1/4), x)