3.1015 \(\int \frac{x^2}{\sqrt [3]{1-x^2} (3+x^2)} \, dx\)

Optimal. Leaf size=515 \[ \frac{\sqrt{2} 3^{3/4} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{\sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{3 x}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}-\frac{3 \tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2 \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3}}+\frac{\tanh ^{-1}(x)}{2\ 2^{2/3}} \]

[Out]

(-3*x)/(1 - Sqrt[3] - (1 - x^2)^(1/3)) - (Sqrt[3]*ArcTan[Sqrt[3]/x])/(2*2^(2/3)) - (Sqrt[3]*ArcTan[(Sqrt[3]*(1
- 2^(1/3)*(1 - x^2)^(1/3)))/x])/(2*2^(2/3)) + ArcTanh[x]/(2*2^(2/3)) - (3*ArcTanh[x/(1 + 2^(1/3)*(1 - x^2)^(1
/3))])/(2*2^(2/3)) - (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1 - x^2)^(1/3) + (1 - x^2)^
(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 -
x^2)^(1/3))], -7 + 4*Sqrt[3]])/(2*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)]) + (Sqrt
[2]*3^(3/4)*(1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))
^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))], -7 + 4*Sqrt[3]])/(x*Sqr
t[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)])

________________________________________________________________________________________

Rubi [A]  time = 0.167402, antiderivative size = 515, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {483, 235, 304, 219, 1879, 393} \[ -\frac{3 x}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}-\frac{3 \tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}+\frac{\sqrt{2} 3^{3/4} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2 \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3}}+\frac{\tanh ^{-1}(x)}{2\ 2^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((1 - x^2)^(1/3)*(3 + x^2)),x]

[Out]

(-3*x)/(1 - Sqrt[3] - (1 - x^2)^(1/3)) - (Sqrt[3]*ArcTan[Sqrt[3]/x])/(2*2^(2/3)) - (Sqrt[3]*ArcTan[(Sqrt[3]*(1
- 2^(1/3)*(1 - x^2)^(1/3)))/x])/(2*2^(2/3)) + ArcTanh[x]/(2*2^(2/3)) - (3*ArcTanh[x/(1 + 2^(1/3)*(1 - x^2)^(1
/3))])/(2*2^(2/3)) - (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1 - x^2)^(1/3) + (1 - x^2)^
(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 -
x^2)^(1/3))], -7 + 4*Sqrt[3]])/(2*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)]) + (Sqrt
[2]*3^(3/4)*(1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))
^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))], -7 + 4*Sqrt[3]])/(x*Sqr
t[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)])

Rule 483

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Dist[e^n/b, Int[
(e*x)^(m - n)*(c + d*x^n)^q, x], x] - Dist[(a*e^n)/b, Int[((e*x)^(m - n)*(c + d*x^n)^q)/(a + b*x^n), x], x] /;
FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b
, c, d, e, m, n, -1, q, x]

Rule 235

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Dist[(3*Sqrt[b*x^2])/(2*b*x), Subst[Int[x/Sqrt[-a + x^3], x], x
, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b}, x]

Rule 304

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, -Dist[(S
qrt[2]*s)/(Sqrt[2 - Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a
+ b*x^3], x], x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 219

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3
])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - S
qrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 1879

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 + Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 + Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 - Sqrt[3])*s + r*x)), x
] + Simp[(3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[-((s
*(s + r*x))/((1 - Sqrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rule 393

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-(b/a), 2]}, Simp[(q*ArcT
an[Sqrt[3]/(q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x] + (Simp[(q*ArcTanh[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a +
b*x^2)^(1/3))])/(2*2^(2/3)*a^(1/3)*d), x] - Simp[(q*ArcTanh[q*x])/(6*2^(2/3)*a^(1/3)*d), x] + Simp[(q*ArcTan[(
Sqrt[3]*(a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3)))/(a^(1/3)*q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x])] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && NegQ[b/a]

Rubi steps

\begin{align*} \int \frac{x^2}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx &=-\left (3 \int \frac{1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\right )+\int \frac{1}{\sqrt [3]{1-x^2}} \, dx\\ &=-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}+\frac{\tanh ^{-1}(x)}{2\ 2^{2/3}}-\frac{3 \tanh ^{-1}\left (\frac{x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{2\ 2^{2/3}}-\frac{\left (3 \sqrt{-x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{2 x}\\ &=-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}+\frac{\tanh ^{-1}(x)}{2\ 2^{2/3}}-\frac{3 \tanh ^{-1}\left (\frac{x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{2\ 2^{2/3}}+\frac{\left (3 \sqrt{-x^2}\right ) \operatorname{Subst}\left (\int \frac{1+\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{2 x}-\frac{\left (3 \sqrt{\frac{1}{2} \left (2+\sqrt{3}\right )} \sqrt{-x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{x}\\ &=-\frac{3 x}{1-\sqrt{3}-\sqrt [3]{1-x^2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}+\frac{\tanh ^{-1}(x)}{2\ 2^{2/3}}-\frac{3 \tanh ^{-1}\left (\frac{x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{2\ 2^{2/3}}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{1-x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{1-x^2}}{1-\sqrt{3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt{3}\right )}{2 x \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (1-\sqrt{3}-\sqrt [3]{1-x^2}\right )^2}}}+\frac{\sqrt{2} 3^{3/4} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{1-x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{1-x^2}}{1-\sqrt{3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt{3}\right )}{x \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (1-\sqrt{3}-\sqrt [3]{1-x^2}\right )^2}}}\\ \end{align*}

Mathematica [C]  time = 0.0184728, size = 28, normalized size = 0.05 \[ \frac{1}{9} x^3 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2/((1 - x^2)^(1/3)*(3 + x^2)),x]

[Out]

(x^3*AppellF1[3/2, 1/3, 1, 5/2, x^2, -x^2/3])/9

________________________________________________________________________________________

Maple [F]  time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{{x}^{2}+3}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(-x^2+1)^(1/3)/(x^2+3),x)

[Out]

int(x^2/(-x^2+1)^(1/3)/(x^2+3),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (x^{2} + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (-x^{2} + 1\right )}^{\frac{2}{3}} x^{2}}{x^{4} + 2 \, x^{2} - 3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(-x^2 + 1)^(2/3)*x^2/(x^4 + 2*x^2 - 3), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/((-(x - 1)*(x + 1))**(1/3)*(x**2 + 3)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (x^{2} + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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