3.9.79 \(\int \frac {x^2}{(2+3 x^2)^{3/4}} \, dx\) [879]

Optimal. Leaf size=47 \[ \frac {2}{9} x \sqrt [4]{2+3 x^2}-\frac {4\ 2^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{9 \sqrt {3}} \]

[Out]

2/9*x*(3*x^2+2)^(1/4)-4/27*2^(3/4)*(cos(1/2*arctan(1/2*x*6^(1/2)))^2)^(1/2)/cos(1/2*arctan(1/2*x*6^(1/2)))*Ell
ipticF(sin(1/2*arctan(1/2*x*6^(1/2))),2^(1/2))*3^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {327, 237} \begin {gather*} \frac {2}{9} x \sqrt [4]{3 x^2+2}-\frac {4\ 2^{3/4} F\left (\left .\frac {1}{2} \text {ArcTan}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{9 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(2 + 3*x^2)^(3/4),x]

[Out]

(2*x*(2 + 3*x^2)^(1/4))/9 - (4*2^(3/4)*EllipticF[ArcTan[Sqrt[3/2]*x]/2, 2])/(9*Sqrt[3])

Rule 237

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]))*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]
*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {x^2}{\left (2+3 x^2\right )^{3/4}} \, dx &=\frac {2}{9} x \sqrt [4]{2+3 x^2}-\frac {4}{9} \int \frac {1}{\left (2+3 x^2\right )^{3/4}} \, dx\\ &=\frac {2}{9} x \sqrt [4]{2+3 x^2}-\frac {4\ 2^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{9 \sqrt {3}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 5.56, size = 41, normalized size = 0.87 \begin {gather*} \frac {2}{9} x \left (\sqrt [4]{2+3 x^2}-\sqrt [4]{2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};-\frac {3 x^2}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(2 + 3*x^2)^(3/4),x]

[Out]

(2*x*((2 + 3*x^2)^(1/4) - 2^(1/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (-3*x^2)/2]))/9

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.06, size = 20, normalized size = 0.43

method	result	size
meijerg	\(\frac {2^{\frac {1}{4}} x^{3} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{2}\right ], \left [\frac {5}{2}\right ], -\frac {3 x^{2}}{2}\right )}{6}\)	\(20\)
risch	\(\frac {2 x \left (3 x^{2}+2\right )^{\frac {1}{4}}}{9}-\frac {2 \,2^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}\right ], -\frac {3 x^{2}}{2}\right )}{9}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*2^(1/4)*x^3*hypergeom([3/4,3/2],[5/2],-3/2*x^2)

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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/(3*x^2+2)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^2/(3*x^2 + 2)^(3/4), x)

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Fricas [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/(3*x^2+2)^(3/4),x, algorithm="fricas")

[Out]

integral(x^2/(3*x^2 + 2)^(3/4), x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.39, size = 27, normalized size = 0.57 \begin {gather*} \frac {\sqrt [4]{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(3*x**2+2)**(3/4),x)

[Out]

2**(1/4)*x**3*hyper((3/4, 3/2), (5/2,), 3*x**2*exp_polar(I*pi)/2)/6

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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/(3*x^2+2)^(3/4),x, algorithm="giac")

[Out]

integrate(x^2/(3*x^2 + 2)^(3/4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^2}{{\left (3\,x^2+2\right )}^{3/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(3*x^2 + 2)^(3/4),x)

[Out]

int(x^2/(3*x^2 + 2)^(3/4), x)

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