3.840 \(\int \frac{x^4}{\sqrt{a-b x^4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0631304, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{a^{5/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 b^{5/4} \sqrt{a-b x^4}}-\frac{x \sqrt{a-b x^4}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[a - b*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 9.27301, size = 65, normalized size = 0.84 \[ \frac{a^{\frac{5}{4}} \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{3 b^{\frac{5}{4}} \sqrt{a - b x^{4}}} - \frac{x \sqrt{a - b x^{4}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.133509, size = 108, normalized size = 1.4 \[ \frac{x \sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} \left (b x^4-a\right )-i a \sqrt{1-\frac{b x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{3 b \sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} \sqrt{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/Sqrt[a - b*x^4],x]

[Out]

(Sqrt[-(Sqrt[b]/Sqrt[a])]*x*(-a + b*x^4) - I*a*Sqrt[1 - (b*x^4)/a]*EllipticF[I*A
rcSinh[Sqrt[-(Sqrt[b]/Sqrt[a])]*x], -1])/(3*Sqrt[-(Sqrt[b]/Sqrt[a])]*b*Sqrt[a -
b*x^4])

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Maple [A]  time = 0.013, size = 86, normalized size = 1.1 \[ -{\frac{x}{3\,b}\sqrt{-b{x}^{4}+a}}+{\frac{a}{3\,b}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(-b*x^4+a)^(1/2),x)

[Out]

-1/3*x*(-b*x^4+a)^(1/2)/b+1/3*a/b/(1/a^(1/2)*b^(1/2))^(1/2)*(1-b^(1/2)*x^2/a^(1/
2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2))^(1/2)/(-b*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*
b^(1/2))^(1/2),I)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/sqrt(-b*x^4 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^4/sqrt(-b*x^4 + a), x)

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Sympy [A]  time = 2.3729, size = 39, normalized size = 0.51 \[ \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*gamma(5/4)*hyper((1/2, 5/4), (9/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)
*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/sqrt(-b*x^4 + a), x)