3.10.84 \(\int \frac {x^2}{\sqrt {3-b x^4}} \, dx\) [984]

Optimal. Leaf size=54 \[ \frac {\sqrt [4]{3} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}}-\frac {\sqrt [4]{3} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}} \]

[Out]

3^(1/4)*EllipticE(1/3*b^(1/4)*x*3^(3/4),I)/b^(3/4)-3^(1/4)*EllipticF(1/3*b^(1/4)*x*3^(3/4),I)/b^(3/4)

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Rubi [A]
time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {313, 227, 1213, 435} \begin {gather*} \frac {\sqrt [4]{3} E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}}-\frac {\sqrt [4]{3} F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[3 - b*x^4],x]

[Out]

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

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 313

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Dist[-q^(-1), Int[1/Sqrt[a + b*x^4]
, x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 1213

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + e*(x^2/d)]/Sqrt
[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {x^2}{\sqrt {3-b x^4}} \, dx &=-\frac {\sqrt {3} \int \frac {1}{\sqrt {3-b x^4}} \, dx}{\sqrt {b}}+\frac {\sqrt {3} \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {3}}}{\sqrt {3-b x^4}} \, dx}{\sqrt {b}}\\ &=-\frac {\sqrt [4]{3} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}}+\frac {\int \frac {\sqrt {1+\frac {\sqrt {b} x^2}{\sqrt {3}}}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {3}}}} \, dx}{\sqrt {b}}\\ &=\frac {\sqrt [4]{3} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}}-\frac {\sqrt [4]{3} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.02, size = 30, normalized size = 0.56 \begin {gather*} \frac {x^3 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};\frac {b x^4}{3}\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[3 - b*x^4],x]

[Out]

(x^3*Hypergeometric2F1[1/2, 3/4, 7/4, (b*x^4)/3])/(3*Sqrt[3])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (40 ) = 80\).
time = 0.15, size = 94, normalized size = 1.74

method	result	size
meijerg	\(\frac {\sqrt {3}\, x^{3} \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], \frac {b \,x^{4}}{3}\right )}{9}\)	\(21\)
default	\(-\frac {\sqrt {9-3 \sqrt {3}\, \sqrt {b}\, x^{2}}\, \sqrt {9+3 \sqrt {3}\, \sqrt {b}\, x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {3}\, \sqrt {\sqrt {3}\, \sqrt {b}}}{3}, i\right )-\EllipticE \left (\frac {x \sqrt {3}\, \sqrt {\sqrt {3}\, \sqrt {b}}}{3}, i\right )\right )}{3 \sqrt {\sqrt {3}\, \sqrt {b}}\, \sqrt {-b \,x^{4}+3}\, \sqrt {b}}\)	\(94\)
elliptic	\(-\frac {\sqrt {9-3 \sqrt {3}\, \sqrt {b}\, x^{2}}\, \sqrt {9+3 \sqrt {3}\, \sqrt {b}\, x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {3}\, \sqrt {\sqrt {3}\, \sqrt {b}}}{3}, i\right )-\EllipticE \left (\frac {x \sqrt {3}\, \sqrt {\sqrt {3}\, \sqrt {b}}}{3}, i\right )\right )}{3 \sqrt {\sqrt {3}\, \sqrt {b}}\, \sqrt {-b \,x^{4}+3}\, \sqrt {b}}\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(-b*x^4+3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3/(3^(1/2)*b^(1/2))^(1/2)*(9-3*3^(1/2)*b^(1/2)*x^2)^(1/2)*(9+3*3^(1/2)*b^(1/2)*x^2)^(1/2)/(-b*x^4+3)^(1/2)/
b^(1/2)*(EllipticF(1/3*x*3^(1/2)*(3^(1/2)*b^(1/2))^(1/2),I)-EllipticE(1/3*x*3^(1/2)*(3^(1/2)*b^(1/2))^(1/2),I)
)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-b*x^4+3)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^2/sqrt(-b*x^4 + 3), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (38) = 76\).
time = 0.08, size = 96, normalized size = 1.78 \begin {gather*} -\frac {\frac {\sqrt {3} \sqrt {-b} x \sqrt {\frac {\sqrt {3}}{\sqrt {b}}} E(\arcsin \left (\frac {\sqrt {\frac {\sqrt {3}}{\sqrt {b}}}}{x}\right )\,|\,-1)}{\sqrt {b}} - \frac {\sqrt {3} \sqrt {-b} x \sqrt {\frac {\sqrt {3}}{\sqrt {b}}} F(\arcsin \left (\frac {\sqrt {\frac {\sqrt {3}}{\sqrt {b}}}}{x}\right )\,|\,-1)}{\sqrt {b}} + \sqrt {-b x^{4} + 3}}{b x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-b*x^4+3)^(1/2),x, algorithm="fricas")

[Out]

-(sqrt(3)*sqrt(-b)*x*sqrt(sqrt(3)/sqrt(b))*elliptic_e(arcsin(sqrt(sqrt(3)/sqrt(b))/x), -1)/sqrt(b) - sqrt(3)*s
qrt(-b)*x*sqrt(sqrt(3)/sqrt(b))*elliptic_f(arcsin(sqrt(sqrt(3)/sqrt(b))/x), -1)/sqrt(b) + sqrt(-b*x^4 + 3))/(b
*x)

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Sympy [A]
time = 0.37, size = 39, normalized size = 0.72 \begin {gather*} \frac {\sqrt {3} 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 }}{3}} \right )}}{12 \Gamma \left (\frac {7}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(-b*x**4+3)**(1/2),x)

[Out]

sqrt(3)*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), b*x**4*exp_polar(2*I*pi)/3)/(12*gamma(7/4))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-b*x^4+3)^(1/2),x, algorithm="giac")

[Out]

integrate(x^2/sqrt(-b*x^4 + 3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^2}{\sqrt {3-b\,x^4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(3 - b*x^4)^(1/2),x)

[Out]

int(x^2/(3 - b*x^4)^(1/2), x)

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