3.878 \(\int \frac{x^4}{(2+3 x^2)^{3/4}} \, dx\)

Optimal. Leaf size=65 \[ \frac{16\ 2^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )}{63 \sqrt{3}}+\frac{2}{21} \sqrt [4]{3 x^2+2} x^3-\frac{8}{63} \sqrt [4]{3 x^2+2} x \]

[Out]

(-8*x*(2 + 3*x^2)^(1/4))/63 + (2*x^3*(2 + 3*x^2)^(1/4))/21 + (16*2^(3/4)*EllipticF[ArcTan[Sqrt[3/2]*x]/2, 2])/
(63*Sqrt[3])

________________________________________________________________________________________

Rubi [A]  time = 0.0148995, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {321, 231} \[ \frac{2}{21} \sqrt [4]{3 x^2+2} x^3-\frac{8}{63} \sqrt [4]{3 x^2+2} x+\frac{16\ 2^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{63 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(2 + 3*x^2)^(3/4),x]

[Out]

(-8*x*(2 + 3*x^2)^(1/4))/63 + (2*x^3*(2 + 3*x^2)^(1/4))/21 + (16*2^(3/4)*EllipticF[ArcTan[Sqrt[3/2]*x]/2, 2])/
(63*Sqrt[3])

Rule 321

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]

Rule 231

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

Rubi steps

\begin{align*} \int \frac{x^4}{\left (2+3 x^2\right )^{3/4}} \, dx &=\frac{2}{21} x^3 \sqrt [4]{2+3 x^2}-\frac{4}{7} \int \frac{x^2}{\left (2+3 x^2\right )^{3/4}} \, dx\\ &=-\frac{8}{63} x \sqrt [4]{2+3 x^2}+\frac{2}{21} x^3 \sqrt [4]{2+3 x^2}+\frac{16}{63} \int \frac{1}{\left (2+3 x^2\right )^{3/4}} \, dx\\ &=-\frac{8}{63} x \sqrt [4]{2+3 x^2}+\frac{2}{21} x^3 \sqrt [4]{2+3 x^2}+\frac{16\ 2^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{63 \sqrt{3}}\\ \end{align*}

Mathematica [C]  time = 0.0170208, size = 49, normalized size = 0.75 \[ \frac{2}{63} x \left (4 \sqrt [4]{2} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right )+\sqrt [4]{3 x^2+2} \left (3 x^2-4\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(2 + 3*x^2)^(3/4),x]

[Out]

(2*x*((-4 + 3*x^2)*(2 + 3*x^2)^(1/4) + 4*2^(1/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (-3*x^2)/2]))/63

________________________________________________________________________________________

Maple [C]  time = 0.02, size = 38, normalized size = 0.6 \begin{align*}{\frac{2\,x \left ( 3\,{x}^{2}-4 \right ) }{63}\sqrt [4]{3\,{x}^{2}+2}}+{\frac{8\,\sqrt [4]{2}x}{63}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\frac{3}{2}};\,-{\frac{3\,{x}^{2}}{2}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/63*x*(3*x^2-4)*(3*x^2+2)^(1/4)+8/63*2^(1/4)*x*hypergeom([1/2,3/4],[3/2],-3/2*x^2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/(3*x^2 + 2)^(3/4), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{4}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^4/(3*x^2 + 2)^(3/4), x)

________________________________________________________________________________________

Sympy [C]  time = 0.651126, size = 27, normalized size = 0.42 \begin{align*} \frac{\sqrt [4]{2} x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2**(1/4)*x**5*hyper((3/4, 5/2), (7/2,), 3*x**2*exp_polar(I*pi)/2)/10

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/(3*x^2 + 2)^(3/4), x)