3.846 \(\int \frac{x^2}{\sqrt{a-b x^4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.196045, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{a^{3/4} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt{a-b x^4}}-\frac{a^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[a - b*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 36.291, size = 97, normalized size = 0.9 \[ \frac{a^{\frac{3}{4}} \sqrt{1 - \frac{b x^{4}}{a}} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{b^{\frac{3}{4}} \sqrt{a - b x^{4}}} - \frac{a^{\frac{3}{4}} \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{b^{\frac{3}{4}} \sqrt{a - b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.104973, size = 100, normalized size = 0.93 \[ \frac{i \sqrt{1-\frac{b x^4}{a}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{\left (-\frac{\sqrt{b}}{\sqrt{a}}\right )^{3/2} \sqrt{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[a - b*x^4],x]

[Out]

(I*Sqrt[1 - (b*x^4)/a]*(EllipticE[I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[a])]*x], -1] - E
llipticF[I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[a])]*x], -1]))/((-(Sqrt[b]/Sqrt[a]))^(3/2
)*Sqrt[a - b*x^4])

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Maple [A]  time = 0.01, size = 88, normalized size = 0.8 \[ -{1\sqrt{a}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-b*x^4+a)^(1/2),x)

[Out]

-a^(1/2)/(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)/b^(1/2)*(EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I
)-EllipticE(x*(1/a^(1/2)*b^(1/2))^(1/2),I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-b x^{4} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/sqrt(-b*x^4 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^2/sqrt(-b*x^4 + a), x)

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Sympy [A]  time = 2.21856, size = 39, normalized size = 0.36 \[ \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)
*gamma(7/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-b x^{4} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/sqrt(-b*x^4 + a), x)