3.836 \(\int \frac{x^6}{\left (a-b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=129 \[ \frac{80 a^{7/2} \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 b^{7/2} \left (a-b x^2\right )^{3/4}}-\frac{40 a^2 x \sqrt [4]{a-b x^2}}{77 b^3}-\frac{20 a x^3 \sqrt [4]{a-b x^2}}{77 b^2}-\frac{2 x^5 \sqrt [4]{a-b x^2}}{11 b} \]

[Out]

(-40*a^2*x*(a - b*x^2)^(1/4))/(77*b^3) - (20*a*x^3*(a - b*x^2)^(1/4))/(77*b^2) -
(2*x^5*(a - b*x^2)^(1/4))/(11*b) + (80*a^(7/2)*(1 - (b*x^2)/a)^(3/4)*EllipticF[
ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(77*b^(7/2)*(a - b*x^2)^(3/4))

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Rubi [A]  time = 0.143882, 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 \[ \frac{80 a^{7/2} \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 b^{7/2} \left (a-b x^2\right )^{3/4}}-\frac{40 a^2 x \sqrt [4]{a-b x^2}}{77 b^3}-\frac{20 a x^3 \sqrt [4]{a-b x^2}}{77 b^2}-\frac{2 x^5 \sqrt [4]{a-b x^2}}{11 b} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a - b*x^2)^(3/4),x]

[Out]

(-40*a^2*x*(a - b*x^2)^(1/4))/(77*b^3) - (20*a*x^3*(a - b*x^2)^(1/4))/(77*b^2) -
(2*x^5*(a - b*x^2)^(1/4))/(11*b) + (80*a^(7/2)*(1 - (b*x^2)/a)^(3/4)*EllipticF[
ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(77*b^(7/2)*(a - b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 19.1014, size = 114, normalized size = 0.88 \[ \frac{80 a^{\frac{7}{2}} \left (1 - \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{77 b^{\frac{7}{2}} \left (a - b x^{2}\right )^{\frac{3}{4}}} - \frac{40 a^{2} x \sqrt [4]{a - b x^{2}}}{77 b^{3}} - \frac{20 a x^{3} \sqrt [4]{a - b x^{2}}}{77 b^{2}} - \frac{2 x^{5} \sqrt [4]{a - b x^{2}}}{11 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(-b*x**2+a)**(3/4),x)

[Out]

80*a**(7/2)*(1 - b*x**2/a)**(3/4)*elliptic_f(asin(sqrt(b)*x/sqrt(a))/2, 2)/(77*b
**(7/2)*(a - b*x**2)**(3/4)) - 40*a**2*x*(a - b*x**2)**(1/4)/(77*b**3) - 20*a*x*
*3*(a - b*x**2)**(1/4)/(77*b**2) - 2*x**5*(a - b*x**2)**(1/4)/(11*b)

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Mathematica [C]  time = 0.0822427, size = 91, normalized size = 0.71 \[ \frac{2 \left (20 a^3 x \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^2}{a}\right )-20 a^3 x+10 a^2 b x^3+3 a b^2 x^5+7 b^3 x^7\right )}{77 b^3 \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a - b*x^2)^(3/4),x]

[Out]

(2*(-20*a^3*x + 10*a^2*b*x^3 + 3*a*b^2*x^5 + 7*b^3*x^7 + 20*a^3*x*(1 - (b*x^2)/a
)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (b*x^2)/a]))/(77*b^3*(a - b*x^2)^(3/4))

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{{x}^{6} \left ( -b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(-b*x^2+a)^(3/4),x)

[Out]

int(x^6/(-b*x^2+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(-b*x^2 + a)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^6/(-b*x^2 + a)^(3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6}}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(-b*x^2 + a)^(3/4),x, algorithm="fricas")

[Out]

integral(x^6/(-b*x^2 + a)^(3/4), x)

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Sympy [A]  time = 3.19207, size = 29, normalized size = 0.22 \[ \frac{x^{7}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{7 a^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(-b*x**2+a)**(3/4),x)

[Out]

x**7*hyper((3/4, 7/2), (9/2,), b*x**2*exp_polar(2*I*pi)/a)/(7*a**(3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(-b*x^2 + a)^(3/4),x, algorithm="giac")

[Out]

integrate(x^6/(-b*x^2 + a)^(3/4), x)