\(\int \frac {1}{x^2 \sqrt [3]{1-x^2} (3+x^2)^2} \, dx\) [1421]

Optimal result

Integrand size = 22, antiderivative size = 563 \[ \int \frac {1}{x^2 \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=-\frac {\left (1-x^2\right )^{2/3}}{8 x}+\frac {\left (1-x^2\right )^{2/3}}{24 x \left (3+x^2\right )}+\frac {x}{8 \left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )}-\frac {7 \arctan \left (\frac {\sqrt {3}}{x}\right )}{72\ 2^{2/3} \sqrt {3}}-\frac {7 \arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{72\ 2^{2/3} \sqrt {3}}+\frac {7 \text {arctanh}(x)}{216\ 2^{2/3}}-\frac {7 \text {arctanh}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{72\ 2^{2/3}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt {3}\right )}{16 x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}-\frac {\left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right ),-7+4 \sqrt {3}\right )}{4 \sqrt {2} \sqrt [4]{3} x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}} \] Output:

-1/8*(-x^2+1)^(2/3)/x+1/24*(-x^2+1)^(2/3)/x/(x^2+3)+x/(8-8*3^(1/2)-8*(-x^2 
+1)^(1/3))-7/432*arctan(3^(1/2)/x)*2^(1/3)*3^(1/2)-7/432*arctan((1-2^(1/3) 
*(-x^2+1)^(1/3))*3^(1/2)/x)*2^(1/3)*3^(1/2)+7/432*arctanh(x)*2^(1/3)-7/144 
*arctanh(x/(1+2^(1/3)*(-x^2+1)^(1/3)))*2^(1/3)+1/16*3^(1/4)*(1/2*6^(1/2)+1 
/2*2^(1/2))*(1-(-x^2+1)^(1/3))*((1+(-x^2+1)^(1/3)+(-x^2+1)^(2/3))/(1-3^(1/ 
2)-(-x^2+1)^(1/3))^2)^(1/2)*EllipticE((1+3^(1/2)-(-x^2+1)^(1/3))/(1-3^(1/2 
)-(-x^2+1)^(1/3)),2*I-I*3^(1/2))/x/(-(1-(-x^2+1)^(1/3))/(1-3^(1/2)-(-x^2+1 
)^(1/3))^2)^(1/2)-1/24*2^(1/2)*3^(3/4)*(1-(-x^2+1)^(1/3))*((1+(-x^2+1)^(1/ 
3)+(-x^2+1)^(2/3))/(1-3^(1/2)-(-x^2+1)^(1/3))^2)^(1/2)*EllipticF((1+3^(1/2 
)-(-x^2+1)^(1/3))/(1-3^(1/2)-(-x^2+1)^(1/3)),2*I-I*3^(1/2))/x/(-(1-(-x^2+1 
)^(1/3))/(1-3^(1/2)-(-x^2+1)^(1/3))^2)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

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

Time = 10.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.30 \[ \int \frac {1}{x^2 \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\frac {-x^4 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )+\frac {9 \left (-8+5 x^2+3 x^4+\frac {69 x^2 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )}{-9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},x^2,-\frac {x^2}{3}\right )-\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )\right )}\right )}{\sqrt [3]{1-x^2} \left (3+x^2\right )}}{216 x} \] Input:

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

Output:

(-(x^4*AppellF1[3/2, 1/3, 1, 5/2, x^2, -1/3*x^2]) + (9*(-8 + 5*x^2 + 3*x^4 
 + (69*x^2*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*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)*(3 + x 
^2)))/(216*x)
 

Rubi [A] (warning: unable to verify)

Time = 0.56 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.09, 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, 27, 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.

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

\(\Big \downarrow \) 374

\(\displaystyle \frac {\left (1-x^2\right )^{2/3}}{24 x \left (x^2+3\right )}-\frac {1}{24} \int -\frac {27-5 x^2}{3 x^2 \sqrt [3]{1-x^2} \left (x^2+3\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{72} \int \frac {27-5 x^2}{x^2 \sqrt [3]{1-x^2} \left (x^2+3\right )}dx+\frac {\left (1-x^2\right )^{2/3}}{24 x \left (x^2+3\right )}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {1}{72} \left (-\frac {1}{3} \int \frac {3 \left (3 x^2+23\right )}{\sqrt [3]{1-x^2} \left (x^2+3\right )}dx-\frac {9 \left (1-x^2\right )^{2/3}}{x}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x \left (x^2+3\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{72} \left (-\int \frac {3 x^2+23}{\sqrt [3]{1-x^2} \left (x^2+3\right )}dx-\frac {9 \left (1-x^2\right )^{2/3}}{x}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x \left (x^2+3\right )}\)

\(\Big \downarrow \) 405

\(\displaystyle \frac {1}{72} \left (-3 \int \frac {1}{\sqrt [3]{1-x^2}}dx-14 \int \frac {1}{\sqrt [3]{1-x^2} \left (x^2+3\right )}dx-\frac {9 \left (1-x^2\right )^{2/3}}{x}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x \left (x^2+3\right )}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {1}{72} \left (\frac {9 \sqrt {-x^2} \int \frac {\sqrt [3]{1-x^2}}{\sqrt {-x^2}}d\sqrt [3]{1-x^2}}{2 x}-14 \int \frac {1}{\sqrt [3]{1-x^2} \left (x^2+3\right )}dx-\frac {9 \left (1-x^2\right )^{2/3}}{x}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x \left (x^2+3\right )}\)

\(\Big \downarrow \) 305

\(\displaystyle \frac {1}{72} \left (\frac {9 \sqrt {-x^2} \int \frac {\sqrt [3]{1-x^2}}{\sqrt {-x^2}}d\sqrt [3]{1-x^2}}{2 x}-14 \left (\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac {\text {arctanh}(x)}{6\ 2^{2/3}}\right )-\frac {9 \left (1-x^2\right )^{2/3}}{x}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x \left (x^2+3\right )}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {1}{72} \left (\frac {9 \sqrt {-x^2} \left (\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {-x^2}}d\sqrt [3]{1-x^2}-\int \frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{\sqrt {-x^2}}d\sqrt [3]{1-x^2}\right )}{2 x}-14 \left (\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac {\text {arctanh}(x)}{6\ 2^{2/3}}\right )-\frac {9 \left (1-x^2\right )^{2/3}}{x}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x \left (x^2+3\right )}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {1}{72} \left (\frac {9 \sqrt {-x^2} \left (-\int \frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{\sqrt {-x^2}}d\sqrt [3]{1-x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-x^2} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}}}\right )}{2 x}-14 \left (\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac {\text {arctanh}(x)}{6\ 2^{2/3}}\right )-\frac {9 \left (1-x^2\right )^{2/3}}{x}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x \left (x^2+3\right )}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {1}{72} \left (\frac {9 \sqrt {-x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-x^2} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-x^2} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {-x^2}}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )}{2 x}-14 \left (\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac {\text {arctanh}(x)}{6\ 2^{2/3}}\right )-\frac {9 \left (1-x^2\right )^{2/3}}{x}\right )+\frac {\left (1-x^2\right )^{2/3}}{24 x \left (x^2+3\right )}\)

Input:

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

Output:

(1 - x^2)^(2/3)/(24*x*(3 + x^2)) + ((-9*(1 - x^2)^(2/3))/x - 14*(ArcTan[Sq 
rt[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))) + (9*Sqrt[-x^2]*((-2*Sqrt[-x^2])/(1 - Sq 
rt[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*Sqrt[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 - S 
qrt[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]])/(3^(1/4)*Sqrt[-x^2] 
*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)])))/(2*x) 
)/72
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 233

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]
 

rule 305

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]
 

rule 374

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]
 

rule 405

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]
 

rule 445

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]
 

rule 760

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]
 

rule 833

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]
 

rule 2418

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(-(-x^2 + 1)^(2/3)/(x^8 + 5*x^6 + 3*x^4 - 9*x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^2 \sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\int \frac {1}{\left (-x^{2}+1\right )^{\frac {1}{3}} x^{6}+6 \left (-x^{2}+1\right )^{\frac {1}{3}} x^{4}+9 \left (-x^{2}+1\right )^{\frac {1}{3}} x^{2}}d x \] Input:

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

Output:

int(1/(( - x**2 + 1)**(1/3)*x**6 + 6*( - x**2 + 1)**(1/3)*x**4 + 9*( - x** 
2 + 1)**(1/3)*x**2),x)