3.1256 \(\int \frac{x^4}{\left (a-b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=86 \[ -\frac{x \sqrt [4]{a-b x^4}}{2 b}-\frac{\sqrt{a} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{b} \left (a-b x^4\right )^{3/4}} \]

[Out]

-(x*(a - b*x^4)^(1/4))/(2*b) - (Sqrt[a]*(1 - a/(b*x^4))^(3/4)*x^3*EllipticF[ArcC
sc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(2*Sqrt[b]*(a - b*x^4)^(3/4))

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Rubi [A]  time = 0.108521, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{x \sqrt [4]{a-b x^4}}{2 b}-\frac{\sqrt{a} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{b} \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a - b*x^4)^(3/4),x]

[Out]

-(x*(a - b*x^4)^(1/4))/(2*b) - (Sqrt[a]*(1 - a/(b*x^4))^(3/4)*x^3*EllipticF[ArcC
sc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(2*Sqrt[b]*(a - b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 15.725, size = 71, normalized size = 0.83 \[ - \frac{\sqrt{a} x^{3} \left (- \frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{2 \sqrt{b} \left (a - b x^{4}\right )^{\frac{3}{4}}} - \frac{x \sqrt [4]{a - b x^{4}}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0510693, size = 64, normalized size = 0.74 \[ \frac{x \left (a \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^4}{a}\right )-a+b x^4\right )}{2 b \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(-b*x^4+a)^(3/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(-b*x^4 + a)^(3/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^4/(-b*x^4 + a)^(3/4), x)

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Sympy [A]  time = 2.65272, size = 39, normalized size = 0.45 \[ \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*gamma(5/4)*hyper((3/4, 5/4), (9/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a**(3/4
)*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(-b*x^4 + a)^(3/4), x)