Integrand size = 22, antiderivative size = 581 \[ \int \frac {1}{x^4 \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=-\frac {11 \left (1-x^2\right )^{2/3}}{216 x^3}+\frac {11 \left (1-x^2\right )^{2/3}}{648 x}+\frac {\left (1-x^2\right )^{2/3}}{24 x^3 \left (3+x^2\right )}-\frac {11 x}{648 \left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )}+\frac {11 \arctan \left (\frac {\sqrt {3}}{x}\right )}{216\ 2^{2/3} \sqrt {3}}+\frac {11 \arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{216\ 2^{2/3} \sqrt {3}}-\frac {11 \text {arctanh}(x)}{648\ 2^{2/3}}+\frac {11 \text {arctanh}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{216\ 2^{2/3}}-\frac {11 \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt {3}\right )}{432\ 3^{3/4} x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}+\frac {11 \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right ),-7+4 \sqrt {3}\right )}{324 \sqrt {2} \sqrt [4]{3} x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}} \]
-11/216*(-x^2+1)^(2/3)/x^3+11/648*(-x^2+1)^(2/3)/x+1/24*(-x^2+1)^(2/3)/x^3 /(x^2+3)-11/1296*arctanh(x)*2^(1/3)+11/432*arctanh(x/(1+2^(1/3)*(-x^2+1)^( 1/3)))*2^(1/3)-11/648*x/(1-(-x^2+1)^(1/3)-3^(1/2))+11/1296*arctan(3^(1/2)/ x)*2^(1/3)*3^(1/2)+11/1296*arctan((1-2^(1/3)*(-x^2+1)^(1/3))*3^(1/2)/x)*2^ (1/3)*3^(1/2)+11/1944*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)-11/1296*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)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.11 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.30 \[ \int \frac {1}{x^4 \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\frac {11 x^6 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )+\frac {27 \left (-72+72 x^2+11 x^4-11 x^6+\frac {693 x^4 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )}{9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )+2 x^2 \left (-\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},x^2,-\frac {x^2}{3}\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )\right )}\right )}{\sqrt [3]{1-x^2} \left (3+x^2\right )}}{17496 x^3} \]
(11*x^6*AppellF1[3/2, 1/3, 1, 5/2, x^2, -1/3*x^2] + (27*(-72 + 72*x^2 + 11 *x^4 - 11*x^6 + (693*x^4*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/(9*App ellF1[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)*(3 + x^2)))/(17496*x^3)
Time = 0.55 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {374, 27, 445, 445, 405, 233, 305, 833, 760, 2418}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{x^4 \sqrt [3]{1-x^2} \left (x^2+3\right )^2} \, dx\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle \frac {\left (1-x^2\right )^{2/3}}{24 x^3 \left (x^2+3\right )}-\frac {1}{24} \int -\frac {11 \left (3-x^2\right )}{3 x^4 \sqrt [3]{1-x^2} \left (x^2+3\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{72} \int \frac {3-x^2}{x^4 \sqrt [3]{1-x^2} \left (x^2+3\right )}dx+\frac {\left (1-x^2\right )^{2/3}}{24 x^3 \left (x^2+3\right )}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {11}{72} \left (-\frac {1}{9} \int \frac {3-5 x^2}{x^2 \sqrt [3]{1-x^2} \left (x^2+3\right )}dx-\frac {\left (1-x^2\right )^{2/3}}{3 x^3}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x^3 \left (x^2+3\right )}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {11}{72} \left (\frac {1}{9} \left (\frac {1}{3} \int \frac {x^2+21}{\sqrt [3]{1-x^2} \left (x^2+3\right )}dx+\frac {\left (1-x^2\right )^{2/3}}{x}\right )-\frac {\left (1-x^2\right )^{2/3}}{3 x^3}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x^3 \left (x^2+3\right )}\) |
| \(\Big \downarrow \) 405 |
| \(\displaystyle \frac {11}{72} \left (\frac {1}{9} \left (\frac {1}{3} \left (\int \frac {1}{\sqrt [3]{1-x^2}}dx+18 \int \frac {1}{\sqrt [3]{1-x^2} \left (x^2+3\right )}dx\right )+\frac {\left (1-x^2\right )^{2/3}}{x}\right )-\frac {\left (1-x^2\right )^{2/3}}{3 x^3}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x^3 \left (x^2+3\right )}\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {11}{72} \left (\frac {1}{9} \left (\frac {1}{3} \left (18 \int \frac {1}{\sqrt [3]{1-x^2} \left (x^2+3\right )}dx-\frac {3 \sqrt {-x^2} \int \frac {\sqrt [3]{1-x^2}}{\sqrt {-x^2}}d\sqrt [3]{1-x^2}}{2 x}\right )+\frac {\left (1-x^2\right )^{2/3}}{x}\right )-\frac {\left (1-x^2\right )^{2/3}}{3 x^3}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x^3 \left (x^2+3\right )}\) |
| \(\Big \downarrow \) 305 |
| \(\displaystyle \frac {11}{72} \left (\frac {1}{9} \left (\frac {1}{3} \left (18 \left (\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac {\text {arctanh}(x)}{6\ 2^{2/3}}\right )-\frac {3 \sqrt {-x^2} \int \frac {\sqrt [3]{1-x^2}}{\sqrt {-x^2}}d\sqrt [3]{1-x^2}}{2 x}\right )+\frac {\left (1-x^2\right )^{2/3}}{x}\right )-\frac {\left (1-x^2\right )^{2/3}}{3 x^3}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x^3 \left (x^2+3\right )}\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {11}{72} \left (\frac {1}{9} \left (\frac {1}{3} \left (18 \left (\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac {\text {arctanh}(x)}{6\ 2^{2/3}}\right )-\frac {3 \sqrt {-x^2} \left (\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {-x^2}}d\sqrt [3]{1-x^2}-\int \frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{\sqrt {-x^2}}d\sqrt [3]{1-x^2}\right )}{2 x}\right )+\frac {\left (1-x^2\right )^{2/3}}{x}\right )-\frac {\left (1-x^2\right )^{2/3}}{3 x^3}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x^3 \left (x^2+3\right )}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {11}{72} \left (\frac {1}{9} \left (\frac {1}{3} \left (18 \left (\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac {\text {arctanh}(x)}{6\ 2^{2/3}}\right )-\frac {3 \sqrt {-x^2} \left (-\int \frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{\sqrt {-x^2}}d\sqrt [3]{1-x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-x^2} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}}}\right )}{2 x}\right )+\frac {\left (1-x^2\right )^{2/3}}{x}\right )-\frac {\left (1-x^2\right )^{2/3}}{3 x^3}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x^3 \left (x^2+3\right )}\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {11}{72} \left (\frac {1}{9} \left (\frac {1}{3} \left (18 \left (\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac {\text {arctanh}(x)}{6\ 2^{2/3}}\right )-\frac {3 \sqrt {-x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-x^2} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-x^2} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {-x^2}}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )}{2 x}\right )+\frac {\left (1-x^2\right )^{2/3}}{x}\right )-\frac {\left (1-x^2\right )^{2/3}}{3 x^3}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x^3 \left (x^2+3\right )}\) |
(1 - x^2)^(2/3)/(24*x^3*(3 + x^2)) + (11*(-1/3*(1 - x^2)^(2/3)/x^3 + ((1 - x^2)^(2/3)/x + (18*(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)*Sqrt[3]) - ArcTanh[x]/(6*2 ^(2/3)) + ArcTanh[x/(1 + 2^(1/3)*(1 - x^2)^(1/3))]/(2*2^(2/3))) - (3*Sqrt[ -x^2]*((-2*Sqrt[-x^2])/(1 - Sqrt[3] - (1 - x^2)^(1/3)) + (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]])/(Sqrt[-x^2 ]*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)]) - (2*S qrt[2 - Sqrt[3]]*(1 + 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]*EllipticF[ArcSi n[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))], -7 + 4 *Sqrt[3]])/(3^(1/4)*Sqrt[-x^2]*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)])))/(2*x))/3)/9))/72
3.11.30.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[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]
Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit h[{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[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]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[(((a_) + (b_.)*(x_)^2)^(p_)*((e_) + (f_.)*(x_)^2))/((c_) + (d_.)*(x_)^2 ), x_Symbol] :> Simp[f/d Int[(a + b*x^2)^p, x], x] + Simp[(d*e - c*f)/d Int[(a + b*x^2)^p/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, p}, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
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]*Sqrt[(- 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]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && NegQ[a]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[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] + S imp[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 - S qrt[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*Sqrt[3])*a*d^3, 0]
\[\int \frac {1}{x^{4} \left (-x^{2}+1\right )^{\frac {1}{3}} \left (x^{2}+3\right )^{2}}d x\]
\[ \int \frac {1}{x^4 \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\int { \frac {1}{{\left (x^{2} + 3\right )}^{2} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} x^{4}} \,d x } \]
\[ \int \frac {1}{x^4 \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\int \frac {1}{x^{4} \sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )^{2}}\, dx \]
\[ \int \frac {1}{x^4 \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\int { \frac {1}{{\left (x^{2} + 3\right )}^{2} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} x^{4}} \,d x } \]
\[ \int \frac {1}{x^4 \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\int { \frac {1}{{\left (x^{2} + 3\right )}^{2} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} x^{4}} \,d x } \]
Timed out. \[ \int \frac {1}{x^4 \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\int \frac {1}{x^4\,{\left (1-x^2\right )}^{1/3}\,{\left (x^2+3\right )}^2} \,d x \]