3.1241 \(\int \frac{x^9}{\left (a-b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=108 \[ \frac{4 a^{5/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{7 b^{5/2} \left (a-b x^4\right )^{3/4}}-\frac{2 a x^2 \sqrt [4]{a-b x^4}}{7 b^2}-\frac{x^6 \sqrt [4]{a-b x^4}}{7 b} \]

[Out]

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

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Rubi [A]  time = 0.159211, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{4 a^{5/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{7 b^{5/2} \left (a-b x^4\right )^{3/4}}-\frac{2 a x^2 \sqrt [4]{a-b x^4}}{7 b^2}-\frac{x^6 \sqrt [4]{a-b x^4}}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a - b*x^4)^(3/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.3386, size = 92, normalized size = 0.85 \[ \frac{4 a^{\frac{5}{2}} \left (1 - \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{7 b^{\frac{5}{2}} \left (a - b x^{4}\right )^{\frac{3}{4}}} - \frac{2 a x^{2} \sqrt [4]{a - b x^{4}}}{7 b^{2}} - \frac{x^{6} \sqrt [4]{a - b x^{4}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(-b*x**4+a)**(3/4),x)

[Out]

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

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Mathematica [C]  time = 0.06398, size = 79, normalized size = 0.73 \[ \frac{x^2 \left (2 a^2 \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^4}{a}\right )-2 a^2+a b x^4+b^2 x^8\right )}{7 b^2 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a - b*x^4)^(3/4),x]

[Out]

(x^2*(-2*a^2 + a*b*x^4 + b^2*x^8 + 2*a^2*(1 - (b*x^4)/a)^(3/4)*Hypergeometric2F1
[1/2, 3/4, 3/2, (b*x^4)/a]))/(7*b^2*(a - b*x^4)^(3/4))

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{{x}^{9} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(-b*x^4+a)^(3/4),x)

[Out]

int(x^9/(-b*x^4+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(-b*x^4 + a)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^9/(-b*x^4 + a)^(3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{9}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(-b*x^4 + a)^(3/4),x, algorithm="fricas")

[Out]

integral(x^9/(-b*x^4 + a)^(3/4), x)

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Sympy [A]  time = 4.44315, size = 29, normalized size = 0.27 \[ \frac{x^{10}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{10 a^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(-b*x**4+a)**(3/4),x)

[Out]

x**10*hyper((3/4, 5/2), (7/2,), b*x**4*exp_polar(2*I*pi)/a)/(10*a**(3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(-b*x^4 + a)^(3/4),x, algorithm="giac")

[Out]

integrate(x^9/(-b*x^4 + a)^(3/4), x)