\(\int \frac {x^4}{\sqrt [3]{1-x^2} (3+x^2)} \, dx\) [1014]

Optimal result

Integrand size = 22, antiderivative size = 536 \[ \int \frac {x^4}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {3}{7} x \left (1-x^2\right )^{2/3}+\frac {54 x}{7 \left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )}+\frac {3 \sqrt {3} \arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3}}+\frac {3 \sqrt {3} \arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}-\frac {3 \text {arctanh}(x)}{2\ 2^{2/3}}+\frac {9 \text {arctanh}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{2\ 2^{2/3}}+\frac {27 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt {3}\right )}{7 x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}-\frac {18 \sqrt {2} 3^{3/4} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right ),-7+4 \sqrt {3}\right )}{7 x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}} \]

[Out]

-3/7*x*(-x^2+1)^(2/3)-3/4*arctanh(x)*2^(1/3)+9/4*arctanh(x/(1+2^(1/3)*(-x^2+1)^(1/3)))*2^(1/3)+54/7*x/(1-(-x^2
+1)^(1/3)-3^(1/2))+3/4*arctan(3^(1/2)/x)*2^(1/3)*3^(1/2)+3/4*arctan((1-2^(1/3)*(-x^2+1)^(1/3))*3^(1/2)/x)*2^(1
/3)*3^(1/2)-18/7*3^(3/4)*(1-(-x^2+1)^(1/3))*EllipticF((1-(-x^2+1)^(1/3)+3^(1/2))/(1-(-x^2+1)^(1/3)-3^(1/2)),2*
I-I*3^(1/2))*2^(1/2)*((1+(-x^2+1)^(1/3)+(-x^2+1)^(2/3))/(1-(-x^2+1)^(1/3)-3^(1/2))^2)^(1/2)/x/((-1+(-x^2+1)^(1
/3))/(1-(-x^2+1)^(1/3)-3^(1/2))^2)^(1/2)+27/7*3^(1/4)*(1-(-x^2+1)^(1/3))*EllipticE((1-(-x^2+1)^(1/3)+3^(1/2))/
(1-(-x^2+1)^(1/3)-3^(1/2)),2*I-I*3^(1/2))*((1+(-x^2+1)^(1/3)+(-x^2+1)^(2/3))/(1-(-x^2+1)^(1/3)-3^(1/2))^2)^(1/
2)*(1/2*6^(1/2)+1/2*2^(1/2))/x/((-1+(-x^2+1)^(1/3))/(1-(-x^2+1)^(1/3)-3^(1/2))^2)^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {490, 544, 241, 310, 225, 1893, 402} \[ \int \frac {x^4}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {18 \sqrt {2} 3^{3/4} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{7 \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}+\frac {27 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{7 \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}+\frac {3 \sqrt {3} \arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}+\frac {3 \sqrt {3} \arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3}}+\frac {9 \text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac {3 \text {arctanh}(x)}{2\ 2^{2/3}}-\frac {3}{7} \left (1-x^2\right )^{2/3} x+\frac {54 x}{7 \left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )} \]

[In]

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

[Out]

(-3*x*(1 - x^2)^(2/3))/7 + (54*x)/(7*(1 - Sqrt[3] - (1 - x^2)^(1/3))) + (3*Sqrt[3]*ArcTan[Sqrt[3]/x])/(2*2^(2/
3)) + (3*Sqrt[3]*ArcTan[(Sqrt[3]*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x])/(2*2^(2/3)) - (3*ArcTanh[x])/(2*2^(2/3)) +
(9*ArcTanh[x/(1 + 2^(1/3)*(1 - x^2)^(1/3))])/(2*2^(2/3)) + (27*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]])/(7*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 -
Sqrt[3] - (1 - x^2)^(1/3))^2)]) - (18*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]])/(7*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)])

Rule 225

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

Rule 241

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 310

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 + Sqrt[3]))*(s/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 402

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[q*(ArcTan
[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[Sqr
t[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]

Rule 490

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

Rule 544

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[f/d,
Int[(a + b*x^n)^p, x], x] + Dist[(d*e - c*f)/d, Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, p, n}, x]

Rule 1893

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]/(r^2
*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/(
(1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {3}{7} x \left (1-x^2\right )^{2/3}+\frac {3}{7} \int \frac {3-6 x^2}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx \\ & = -\frac {3}{7} x \left (1-x^2\right )^{2/3}-\frac {18}{7} \int \frac {1}{\sqrt [3]{1-x^2}} \, dx+9 \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx \\ & = -\frac {3}{7} x \left (1-x^2\right )^{2/3}+\frac {3 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3}}+\frac {3 \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}(x)}{2\ 2^{2/3}}+\frac {9 \tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{2\ 2^{2/3}}+\frac {\left (27 \sqrt {-x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{7 x} \\ & = -\frac {3}{7} x \left (1-x^2\right )^{2/3}+\frac {3 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3}}+\frac {3 \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}(x)}{2\ 2^{2/3}}+\frac {9 \tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{2\ 2^{2/3}}-\frac {\left (27 \sqrt {-x^2}\right ) \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{7 x}+\frac {\left (27 \left (1+\sqrt {3}\right ) \sqrt {-x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{7 x} \\ & = -\frac {3}{7} x \left (1-x^2\right )^{2/3}+\frac {54 x}{7 \left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )}+\frac {3 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3}}+\frac {3 \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}(x)}{2\ 2^{2/3}}+\frac {9 \tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{2\ 2^{2/3}}+\frac {27 \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 )}{7 x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}-\frac {18 \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 )}{7 x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 5.18 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.29 \[ \int \frac {x^4}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {1}{7} x \left (-2 x^2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )+\frac {3 \left (-1+x^2-\frac {27 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )}{\left (3+x^2\right ) \left (-9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},x^2,-\frac {x^2}{3}\right )-\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )\right )\right )}\right )}{\sqrt [3]{1-x^2}}\right ) \]

[In]

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

[Out]

(x*(-2*x^2*AppellF1[3/2, 1/3, 1, 5/2, x^2, -1/3*x^2] + (3*(-1 + x^2 - (27*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3
*x^2])/((3 + x^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3*
x^2] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2])))))/(1 - x^2)^(1/3)))/7

Maple [F]

\[\int \frac {x^{4}}{\left (-x^{2}+1\right )^{\frac {1}{3}} \left (x^{2}+3\right )}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {x^4}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int { \frac {x^{4}}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^4}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int \frac {x^{4}}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^4}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int { \frac {x^{4}}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {x^4}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int { \frac {x^{4}}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int \frac {x^4}{{\left (1-x^2\right )}^{1/3}\,\left (x^2+3\right )} \,d x \]

[In]

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

[Out]

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